We consider the problem of fair allocation of m indivisible goods to n agents with either subadditive or XOS valuations, in the arbitrary entitlements case. As fairness notions, we consider the anyprice share (APS) ex-post, and the maximum expectation share (MES) ex-ante.

We consider a search problem on trees aiming to find a treasure that an adversary places at one of the nodes. The algorithm can query nodes and extract directional information from them. That is, each node holds a pointer, termed advice , to one of its neighbors. Ideally, this advice points to the neighbor that is closer to the treasure, however, with probability \(q\) this advice points to a uniformly random neighbor. Crucially, the advice is permanent , hence querying the same node again yields the same answer. Let \(\Delta\) denote the maximal degree. Roughly speaking, we show that the expected number of queries incurs a phase transition when \(q\) is about \(1/\sqrt{\Delta}\) . In a recent paper, at TALG’21, we showed that if \(q\) is above the threshold then the expected number of queries is polynomial in \(n\) . Here, we prove that below the threshold, the expected number of queries is \(\mathcal{O}(\sqrt{\Delta}\log\Delta\cdot\log^{2}n)\) , which is tight up to an \(\mathcal{O}(\log n)\) factor when \(\Delta\) is small. We further show that this factor can be reduced to \(\mathcal{O}(\log\log n)\) in the case of regular trees and assuming that \(q for sufficiently small \(c>0\) . In addition, we study the case that the treasure must be found with some given probability. We show that for every fixed \(\varepsilon,\delta>0\) , if \(q<1/\Delta^{\varepsilon}\) then there exists a search strategy that with probability \(1-\delta\) finds the treasure using \((\delta^{-1}\log n)^{O(\frac{1}{\varepsilon})}\) queries, whereas \((\delta^{-1}\log n)^{\Omega(\frac{1}{\varepsilon})}\) queries are necessary.

Abstract In the well known planted clique problem, a clique (or alternatively, an independent set) of size k is planted at random in an Erdos-Renyi random G(n, p) graph, and the goal is to design an algorithm that finds the maximum clique (or independent set) in the resulting graph. We introduce a variation on this problem, where instead of planting the clique at random, the clique is planted by an adversary who attempts to make it difficult to find the maximum clique in the resulting graph. We show that for the standard setting of the parameters of the problem, namely, a clique of size $$k = \sqrt{n}$$ k = n planted in a random $$G(n, \frac{1}{2})$$ G ( n , 1 2 ) graph, the known polynomial time algorithms can be extended (in a non-trivial way) to work also in the adversarial setting. In contrast, we show that for other natural settings of the parameters, such as planting an independent set of size $$k=\frac{n}{2}$$ k = n 2 in a G(n, p) graph with $$p = n^{-\frac{1}{2}}$$ p = n - 1 2 , there is no polynomial time algorithm that finds an independent set of size k, unless NP has randomized polynomial time algorithms.

We consider the problem of allocating m indivisible chores to n agents with additive disvaluation (cost) functions. It is easy to show that there are picking sequences that give every agent (that uses the greedy picking strategy) a bundle of chores of disvalue at most twice her share value (maximin share, MMS, for agents of equal entitlement, and anyprice share, APS, for agents of arbitrary entitlement). Aziz, Li and Wu (2022) designed picking sequences that improve this ratio to 5/3 for the case of equal entitlement. We design picking sequences that improve the ratio to 1.733 for the case of arbitrary entitlement, and to 8/5 for the case of equal entitlement. (In fact, computer assisted analysis suggests that the ratio is smaller than 1.543 in the equal entitlement case.) We also prove a lower bound of 3/2 on the obtainable ratio when n is sufficiently large.

We consider fair allocation of a set M of indivisible goods to n equally-entitled agents, with no monetary transfers. Every agent i has a valuation function vi from some given class of valuation functions. A share s is a function that maps a pair (vi,n) to a non-negative value, with the interpretation that if an allocation of M to n agents fails to give agent i a bundle of value at least equal to s(vi,n), this serves as evidence that the allocation is not fair towards i. For such an interpretation to make sense, we would like the share to be feasible, meaning that for any valuations in the class, there is an allocation that gives every agent at least her share. The maximin share (MMS) was a natural candidate for a feasible share for additive valuations. However, Kurokawa, Procaccia and Wang [2018] show that it is not feasible.

When allocating indivisible items across agents, it is desirable for the allocation to be envy-free, which means that each agent prefers their own bundle over every other bundle. Even though envy-free allocations are not guaranteed to exist for worst-case utilities, they frequently exist in practice. To explain this phenomenon, prior work has shown that, if utilities are drawn from certain probability distributions, then envy-free allocations exist with high probability (as long as the number of items is sufficiently large relative to the number of agents).

We consider the problem of allocating a set on indivisible items to players with private preferences in an efficient and fair way. We focus on valuations that have dichotomous marginals, in which the added value of any item to a set is either 0 or 1, and aim to design truthful allocation mechanisms (without money) that maximize welfare and are fair. For the case that players have submodular valuations with dichotomous marginals, we design such a deterministic truthful allocation mechanism. The allocation output by our mechanism is Lorenz dominating, and consequently satisfies many desired fairness properties, such as being envy-free up to any item (EFX), and maximizing the Nash Social Welfare (NSW). We then show that our mechanism with random priorities is envy-free ex-ante, while having all the above properties ex-post. Furthermore, we present several impossibility results precluding similar results for the larger class of XOS valuations.

We consider the problem of fair allocation of indivisible goods to n agents, with no transfers. When agents have equal entitlements, the well established notion of the maximin share (MMS) serves as an attractive fairness criterion, where to qualify as fair, an allocation needs to give every agent at least a substantial fraction of her MMS. In this paper we consider the case of arbitrary (unequal) entitlements. We explain shortcomings in previous attempts that extend the MMS to unequal entitlements. Our conceptual contribution is the introduction of a new notion of a share, the AnyPrice share (APS), that is appropriate for settings with arbitrary entitlements. The AnyPrice share of an agent is the value she can guarantee to herself if she is given a budget equal to her entitlement, and she buys her highest value affordable set when items are adversarially priced with a total price equal to the total entitlements. Even for the equal entitlements case, this notion is new, and satisfies APS ≥ MMS, where the inequality is sometimes strict. We also present an alternative definition for the APS as a maximization problem (a fractional version of the MMS), and provide comparisons between the APS and previous notions of fairness. Our main result concerns additive valuations and arbitrary entitlements, for which we provide a polynomial-time algorithm that gives every agent at least a 3/5-fraction of her APS. This algorithm can also be viewed as providing a strategy in a certain natural bidding game, and this strategy secures each agent that uses it at least a 3/5-fraction of her APS, regardless of the strategies used by other agents.

The class of gross substitutes (GS) set functions plays a central role in Economics and Computer Science. GS belongs to the hierarchy of complement free valuations introduced by Lehmann, Lehmann and Nisan, along with other prominent classes: GS ⊊ Submodular ⊊ XOS ⊊ Subadditive$. The GS class has always been more enigmatic than its counterpart classes, both in its definition and in its relation to the other classes. For example, while it is well understood how closely the Submodular, XOS and Subadditive classes (point-wise) approximate one another, approximability of these classes by GS remained wide open. In particular, the largest gap known between Submodular and GS valuations was some constant ratio smaller than 2. Our main result is the existence of a submodular valuation (one that is also budget additive) that cannot be approximated by GS within a ratio better than $Ømega(łog m/łogłog m), where m is the number of items. En route, we uncover a new symmetrization operation that preserves GS, which may be of independent interest. We show that our main result is tight with respect to budget additive valuations. However, whether GS approximates general submodular valuations within a poly-logarithmic factor remains open, even in the special case of concave of GS valuations (a subclass of Submodular containing budget additive). For concave of Rado valuations (Rado is a significant subclass of GS, containing, e.g., weighted matroid rank functions and OXS), we show approximability by GS within an O(łog2m) factor.

We consider a search problem on trees in which an agent starts at the root of a tree and aims to locate an adversarially placed treasure, by moving along the edges, while relying on local, partial information. Specifically, each node in the tree holds a pointer to one of its neighbors, termedadvice. A node is faulty with probabilityq. The advice at a non-faulty node points to the neighbor that is closer to the treasure, and the advice at a faulty node points to a uniformly random neighbor. Crucially, the advice ispermanent, in the sense that querying the same node again would yield the same answer. Let Δ denote the maximum degree. For the expected number of moves (edge traversals) until finding the treasure, we show that a phase transition occurs when thenoise parameterqis roughly 1 √Δ. Below the threshold, there exists an algorithm with expected number of movesO(D√Δ), whereDis the depth of the treasure, whereas above the threshold, every search algorithm has an expected number of moves, which is both exponential inDand polynomial in the number of nodes n. In contrast, if we require to find the treasure with probability at least 1 − δ, then for every fixed ɛ > 0, ifq< 1/Δɛ, then there exists a search strategy that with probability 1 − δ finds the treasure using (Δ−1D)O(1/ε)moves. Moreover, we show that (Δ−1D)Ω(1/ε)moves are necessary.

Set functions with convenient properties (such as submodularity) appear in application areas of current interest, such as algorithmic game theory, and allow for improved optimization algorithms. It is natural to ask (e.g., in the context of data driven optimization) how robust such properties are, and whether small deviations from them can be tolerated. We consider two such questions in the important special case of linear set functions. One question that we address is whether any set function that approximately satisfies the modularity equation (linear functions satisfy the modularity equation exactly) is close to a linear function. The answer to this is positive (in a precise formal sense) as shown by Kalton and Roberts [ Trans. Amer. Math. Soc., 278 (1983), pp. 803--816] (and further improved by Bondarenko, Prymak, and Radchenko [ J. Math. Anal. Appl., 402 (2013), pp. 234--241]). We revisit their proof idea that is based on expander graphs and provide significantly stronger upper bounds by combining it with new techniques. Furthermore, we provide improved lower bounds for this problem. Another question that we address is that of how to learn a linear function $h$ that is close to an approximately linear function $f$, while querying the value of $f$ on only a small number of sets. We present a deterministic algorithm that makes only linearly many (in the number of items) nonadaptive queries, and thus improve upon a previous algorithm of Chierichetti, Das, Dasgupta, and Kumar [ Proceedings of the 56th Symposium on Foundations of Computer Science, 2015, pp. 1143--1162] that is randomized and makes more than a quadratic number of queries. Our learning algorithm is based on the Hadamard transform.

Let $G$ be a 2-coloring of a complete graph on $n$ vertices, for sufficiently large $n$. We prove that $G$ contains at least $n^{(\frac{1}{4} - o(1))\log n}$ monochromatic complete subgraphs, thus improving over a lower bound of $n^{0.1576\log n}$ due to Székely [ Combinatorica, 4 (1984), pp. 363--372]. We also present lower bounds concerning the number of monochromatic complete subgraphs within certain ranges of sizes, incomparable in nature to lower bounds previously proved by Conlon [ Combinatorica, 32 (2012), pp. 171--186]. If furthermore one assumes that the largest monochromatic complete subgraph in $G$ is of size $(\frac{1}{2} + o(1))\log n$ (it is a well known open question whether such graphs exist), then for every constant $0 \le c \le \frac{1}{2}$ we determine (up to low order terms) the number of monochromatic complete subgraphs of size $c \log n$. We do so by proving a lower bound that matches (up to low order terms) a previous upper bound of Székely. For example, the number of monochromatic complete subgraphs of size $\frac{1}{2} \log n$ is $n^{\frac{1}{8}(4 - \log e \pm o(1))\log n} \simeq n^{0.32 \log n}$.

We consider the classic scheduling problem of minimizing the total weighted flow-time on a single machine (min-WPFT), when preemption is allowed. In this problem, we are given a set of n jobs, each job having a release time rj, a processing time pj, and a weight wj. The flow-time of a job is defined as the amount of time the job spends in the system before it completes; that is, Fj = Cj – rj, where Cj is the completion time of job. The objective is to minimize the total weighted flow-time of jobs. This NP-hard problem has been studied quite extensively for decades. In a recent breakthrough, Batra, Garg, and Kumar [6] presented a pseudo-polynomial time algorithm that has an O(1) approximation ratio. The design of a truly polynomial time algorithm, however, remained an open problem. In this paper, we show a transformation from pseudo-polynomial time algorithms to polynomial time algorithms in the context of min-WPFT. Our result combined with the result of Batra, Garg, and Kumar [6] settles the long standing conjecture that there is a polynomial time algorithm with O(1)-approximation for min-WPFT.

We consider the problem of robust inference in which inputs may be maliciously corrupted by a powerful adversary, and the learner’s goal is to accurately predict the original, uncorrupted input’s true label given only the adversarially corrupted version of the input. We specifically focus on the multiclass version of this problem in which more than two labels are possible. We substantially extend and generalize previous work which had only considered the binary case, thus uncovering stark differences between the two cases. We show how robust inference can be modeled as a zero-sum game between a learner who maximizes the expected accuracy, and an adversary. The value of this game is the best-attainable accuracy rate of any algorithm. We then show how the optimal policy for both the learner and adversary can be exactly characterized in terms of a particular hypergraph, specifically, as the hypergraph’s maximum fractional independent set and minimum fractional set cover, respectively. This characterization yields efficient algorithms in the size of the domain (number of possible inputs). For the typical setting that the domain is huge, we also design efficient local computation algorithms for approximating maximum fractional independent set in hypergraphs. This leads to a near optimal algorithm for the learner whose complexity is independent of the domain size, instead depending only on the rank and maximum degree of the underlying hypergraph, and on the desired approximation ratio.

In the Local Computation Algorithms (LCA) model, the algorithm is asked to compute a part of the output by reading as little as possible from the input. For example, an LCA for coloring a graph is given a vertex name (as a "query"), and it should output the color assigned to that vertex after inquiring about some part of the graph topology using "probes"; all outputs must be consistent with the same coloring. LCAs are useful when the input is huge, and the output as a whole is not needed simultaneously. Most previous work on LCAs was limited to bounded-degree graphs, which seems inevitable because probes are of the form "what vertex is at the other end of edge i of vertex v?". In this work we study LCAs for unbounded-degree graphs. In particular, such LCAs are expected to probe the graph a number of times that is significantly smaller than the maximum, average, or even minimum degree. We show that there are problems that have very efficient LCAs on any graph - specifically, we show that there is an LCA for the weak coloring problem (where a coloring is legal if every vertex has a neighbor with a different color) that uses log^* n+O(1) probes to reply to any query. As another way of dealing with large degrees, we propose a more powerful type of probe which we call a strong probe: given a vertex name, it returns a list of its neighbors. Lower bounds for strong probes are stronger than ones in the edge probe model (which we call weak probes). Our main result in this model is that roughly Omega(sqrt{n}) strong probes are required to compute a maximal matching. Our findings include interesting separations between closely related problems. For weak probes, we show that while weak 3-coloring can be done with probe complexity log^* n+O(1), weak 2-coloring has probe complexity Omega(log n/log log n). For strong probes, our negative result for maximal matching is complemented by an LCA for (1-epsilon)-approximate maximum matching on regular graphs that uses O(1) strong probes, for any constant epsilon>0.

For a simple (unbiased) random walk on a connected graph with $n$ vertices, the cover time (the expected number of steps it takes to visit all vertices) is at most $O(n^3)$. We consider locally biased random walks, in which the probability of traversing an edge depends on the degrees of its endpoints. We confirm a conjecture of Abdullah, Cooper, and Draief [2015] that the min-degree local bias rule ensures a cover time of $O(n^2)$. For this we formulate and prove the following lemma about spanning trees. Let $R(e)$ denote for edge $e$ the minimum degree among its two endpoints. We say that a weight function $W$ for the edges is feasible if it is nonnegative, dominated by $R$ (for every edge $W(e) \le R(e)$), and the sum over all edges of the ratios $W(e)/R(e)$ equals $n-1$. For example, in trees $W(e) = R(e)$, and in regular graphs the sum of edge weights is $d(n-1)$. We will show in a lemma that for every feasible $W$, the minimum weight spanning tree has total weight $O(n)$. For regular graphs, a similar lemma was proved by Kahn et al. (1989).

For a simple (unbiased) random walk on a connected graph with n vertices, the cover time (the expected number of steps it takes to visit all vertices) is at most O(n3). We consider locally biased random walks, in which the probability of traversing an edge depends on the degrees of its endpoints. We confirm a conjecture of Abdullah, Cooper and Draief [2015] that the min-degree local bias rule ensures a cover time of O(n2). For this we formulate and prove the following lemma about spanning trees. Let R(e) denote for edge e the minimum degree among its two endpoints. We say that a weight function W for the edges is feasible if it is nonnegative, dominated by R (for every edge W (e) < R(e)) and the sum over all edges of the ratios W(e)/R(e) equals n - 1. For example, in trees W (e) = R(e), and in regular graphs the sum of edge weights is d(n - 1). Lemma: for every feasible W, the minimum weight spanning tree has total weight O(n). For regular graphs, a similar lemma was proved by Kahn, Linial, Nisan and Saks [1989].

Set functions with convenient properties (such as submodularity) appear in application areas of current interest, such as algorithmic game theory, and allow for improved optimization algorithms. It is natural to ask (e.g., in the context of data driven optimization) how robust such properties are, and whether small deviations from them can be tolerated. We consider two such questions in the important special case of linear set functions.

We consider sequential decision making in a setting where regret is measured with respect to a set of stateful reference policies, and feedback is limited to observing the rewards of the actions performed (the so-called bandit setting). If either the reference policies are stateless rather than stateful or the feedback includes the rewards of all actions (the so-called experts setting), previous work shows that the optimal regret grows like $\Theta(\sqrt{T})$ in terms of the number of decision rounds $T$. The difficulty in our setting is that the decision maker unavoidably loses track of the internal states of the reference policies and thus cannot reliably attribute rewards observed in a certain round to any of the reference policies. In fact, in this setting it is impossible for the algorithm to estimate which policy gives the highest (or even approximately highest) total reward. Nevertheless, we design an algorithm that achieves expected regret that is sublinear in $T$, of the form $O( T/\log^{1/4}{T} )$. Our algorithm is based on a certain local repetition lemma that may be of independent interest. We also show that no algorithm can guarantee expected regret better than $O( T/\log^{3/2} T )$.

The following paradigm is often used for handling NP-hard combinatorial optimization problems. One first formulates the problem as an integer program, then one relaxes it to a linear program (LP, or more generally, a convex program), then one solves the LP relaxation in polynomial time, and finally one rounds the optimal LP solution, obtaining a feasible solution to the original problem. Many of the commonly used rounding schemes (such as randomized rounding, threshold rounding and others) are "oblivious" in the sense that the rounding is performed based on the LP solution alone, disregarding the objective function. The goal of our work is to better understand in which cases oblivious rounding suffices in order to obtain approximation ratios that match the integrality gap of the underlying LP. Our study is information theoretic - the rounding is restricted to be oblivious but not restricted to run in polynomial time. In this information theoretic setting we characterize the approximation ratio achievable by oblivious rounding. It turns out to equal the integrality gap of the underlying LP on a problem that is the closure of the original combinatorial optimization problem. We apply our findings to the study of the approximation ratios obtainable by oblivious rounding for the maximum welfare problem, showing that when valuation functions are submodular oblivious rounding can match the integrality gap of the configuration LP (though we do not know what this integrality gap is), but when valuation functions are gross substitutes oblivious rounding cannot match the integrality gap (which is 1).

We present the hosted coloring framework for studying al- gorithmic and hardness results for the k-coloring problem. There is a class H of host graphs. One selects a graph H ∈ H and plants in it a balanced k-coloring (by partitioning the vertex set into k roughly equal parts, and removing all edges within each part). The resulting graph G is given as input to a polynomial time algorithm that needs to k-color G (any legal k-coloring would do – the algorithm is not required to recover the planted k-coloring). Earlier planted models correspond to the case that H is the class of all n-vertex d-regular graphs, a member H ∈ H is chosen at random, and then a balanced k-coloring is planted at random. Blum and Spencer [1995] designed algorithms for this model when d = n δ (for 0 < δ ≤ 1), and Alon and Kahale [1997] managed to do so even when d is a sufficiently large constant. The new aspect in our framework is that it need not in- volve randomness. In one model within the framework (with k = 3) H is a d regular spectral expander (meaning that ex- cept for the largest eigenvalue of its adjacency matrix, every other eigenvalue has absolute value much smaller than d) chosen by an adversary, and the planted 3-coloring is ran- dom. We show that the 3-coloring algorithm of Alon and Kahale [1997] can be modified to apply to this case. In an- other model H is a random d-regular graph but the planted balanced 3-coloring is chosen by an adversary, after seeing H. We show that for a certain range of average degrees somewhat below √ n, finding a 3-coloring is NP-hard. To- gether these results (and other results that we have) help clarify which aspects of randomness in the planted coloring model are the key to successful 3-coloring algorithms.

We introduce a new hierarchy over monotone set functions, that we refer to as MPH (Maximum over Positive Hypergraphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, MPH-m (where m is the total number of items) captures all monotone functions. The lowest level, MPH-1, captures all monotone submodular functions, and more generally, the class of functions known as XOS. Every monotone function that has a positive hypergraph representation of rank k (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in MPH-k. Every monotone function that has supermodular degree k (in the sense defined by Feige and Izsak [ITCS 2013]) is in MPH-(k+1). In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of MPH-k.One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the MPH hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of k+1 if all players hold valuation functions in MPH-k. The other is an upper bound of 2k on the price of anarchy of simultaneous first price auctions.

We consider a model where given an uncorrupted input an adversary can corrupt it to one out of m corrupted inputs. We model the classification and inference problems as a zero-sum game between a learner, minimizing the expected error, and an adversary, maximizing the expected error. The value of this game is the optimal error rate achievable. For learning using a limited hypothesis class \mathcalH over corrupted inputs, we give an efficient algorithm that given an uncorrupted sample returns a hypothesis h∈\mathcalH whose error on adversarially corrupted inputs is near optimal. Our algorithm uses as a blackbox an oracle that solves the ERM problem for the hypothesis class \mathcalH. We provide a generalization bound for our setting, showing that for a sufficiently large sample, the performance on the sample and future unseen corrupted inputs will be similar. This gives an efficient learning algorithm for our adversarial setting, based on an ERM oracle. We also consider an inference related setting of the problem, where given a corrupted input, the learner queries the target function on various uncorrupted inputs and generates a prediction regarding the given corrupted input. There is no limitation on the prediction function the learner may generate, so implicitly the hypothesis class includes all possible hypotheses. In this setting we characterize the optimal learner policy as a minimum vertex cover in a given bipartite graph, and the optimal adversary policy as a maximum matching in the same bipartite graph. We design efficient local algorithms for approximating minimum vertex cover in bipartite graphs, which implies an efficient near optimal algorithm for the learner.

It is a well observed phenomenon that natural images are smooth, in the sense that nearby pixels tend to have similar values. We describe a mathematical model of images that makes no assumptions on the nature of the environment that images depict. It only assumes that images can be taken at different scales (zoom levels). We provide quantitative bounds on the smoothness of a typical image in our model, as a function of the number of available scales. These bounds can serve as a baseline against which to compare the observed smoothness of natural images.

We show (under standard assumptions) that there are \textsf{NP} optimization problems for which estimation is easier than approximation. Namely, one can estimate the value of the optimal solution within a ratio of $\rho$, but it is difficult to find a solution whose value is within $\rho$ of optimal. As an important special case, we show that there are linear programming relaxations for which no polynomial time rounding technique matches the integrality gap of the linear program.

We consider the following activation process in undirected graphs: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least r active neighbors, where r > 1 is the activation threshold. A contagious set is a set whose activation results with the entire graph being active. Given a graph G, let m(G, r) be the minimal size of a contagious set. It is known that for every d-regular or nearly d-regular graph on n vertices, . We consider such graphs that additionally have expansion properties, parameterized by the spectral gap and/or the girth of the graphs. The general flavor of our results is that sufficiently strong expansion properties imply that (and more generally, . In addition, we demonstrate that rather weak assumptions on the girth and/or the spectral gap suffice in order to imply that . For example, we show this for graphs of girth at least 7, and for graphs with λ(G) < (1 − ε)d, provided the graph has no 4-cycles. Our results are algorithmic, entailing simple and effcient algorithms for selecting contagious sets.

We consider sequential decision making in a setting where regret is measured with respect to a set of stateful reference policies, and feedback is limited to observing the rewards of the actions performed (the so called “bandit” setting). If either the reference policies are stateless rather than stateful, or the feedback includes the rewards of all actions (the so called “expert” setting), previous work shows that the √ optimal regret grows like Θ(√T) in terms of the number of decision rounds T. The difficulty in our setting is that the decision maker unavoidably loses track of the internal states of the reference policies, and thus cannot reliably attribute rewards observed in a certain round to any of the reference policies. In fact, in this setting it is impossible for the algorithm to estimate which policy gives the highest (or even approximately highest) total reward. Nevertheless, we design an algorithm that achieves expected regret that is sublinear in T, of the form O(T/ log1/4T). Our algorithm is based on a certain local repetition lemma that may be of independent interest. We also show that no algorithm can guarantee expected regret better than O(T/ log3/2T).

We study a multi-round optimization setting in which in each round a player may select one of several actions, and each action produces an outcome vector, not observable to the player until the round ends. The final payoff for the player is computed by applying some known function f to the sum of all outcome vectors (e.g., the minimum of all coordinates of the sum). We show that standard notions of performance measure (such as comparison to the best single action) used in related expert and bandit settings (in which the payoff in each round is scalar) are not useful in our vector setting. Instead, we propose a different performance measure, and design algorithms that have vanishing regret with respect to our new measure.

Given an arbitrary 2-player game G that we refer to as the basic game, we propose a notion of a multiplayer invitation game that proceeds for a fixed number of rounds, where in each round some player (whose identity is determined by a scheduler) gets to invite a player of his choice to play a match of the basic game. The question that we study is how does the price of stability of the invitation game compare to that of the basic game. For a wide range of schedulers we prove a dichotomy result, showing that there are only two types of basic games, those that we call invitation resistant in which the price of stability of the invitation version is equal to that of the basic game, and those that we call asymptotically efficient in which the price of stability tends to 0 as the number of rounds grows. 1 In particular, when the basic game is the prisoners dilemma the game is asymptotically efficient if and only if the payoff when both players defect is nonzero.

We study graph partitioning problems from a min-max perspective, in which an input graph on $n$ vertices should be partitioned into $k$ parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the $k$ parts need to be of equal size, and where they must separate a set of $k$ given terminals. We consider a common generalization of these two problems, and design for it an $O(\sqrt{\log n\log k})$ approximation algorithm. This improves over an $O(\log^2 n)$ approximation for the second version due to Svitkina and Tardos [Min-max multiway cut, in APPROX-RANDOM, 2004, Springer, Berlin, 2004], and roughly $O(k\log n)$ approximation for the first version that follows from other previous work. We also give an $O(1)$ approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the small-set expansion problem. In this problem, we are given a graph $G$ and the goal is to find a nonempty set $S\subseteq V$ of size $|S| \leq \rho n$ with minimum edge expansion. We give an $O(\sqrt{\log{n}\log{(1/\rho)}})$ bicriteria approximation algorithm for small-set expansion in general graphs, and an improved factor of $O(1)$ for graphs that exclude any fixed minor.

In the musical chairs game $MC(n,m)$, a team of $n$ players plays against an adversarial scheduler. The scheduler wins if the game proceeds indefinitely, while termination after a finite number of rounds is declared a win of the team. At each round of the game each player occupies one of the $m$ available chairs. Termination (and a win of the team) is declared as soon as each player occupies a unique chair. Two players that simultaneously occupy the same chair are said to be in conflict. In other words, termination (and a win for the team) is reached as soon as there are no conflicts. The only means of communication throughout the game is this: At every round of the game, the scheduler selects an arbitrary nonempty set of players who are currently in conflict, and notifies each of them separately that it must move. A player who is thus notified changes its chair according to its deterministic program. As we show, for $m\ge 2n-1$ chairs the team has a winning strategy. Moreover, using topological arguments we show that this bound is tight. For $m\leq 2n-2$ the scheduler has a strategy that is guaranteed to make the game continue indefinitely and thus win. We also have some results on additional interesting questions. For example, if $m \ge 2n-1$ (so that the team can win), how quickly can they achieve victory?

We introduce a sealed bid auction of a single item in which the winner is chosen at random among the highest k bidders according to a fixed probability distribution, and the price for the chosen winner is the Vickrey-Clarke-Groves price. We call such an auction a cascade auction. Our analysis suggests that this type of auction may give higher revenues compared to second price auction in cases of collusion.

Given a set of items and a collection of players, each with a nonnegative monotone valuation set function over the items, the welfare maximization problem requires that every item be allocated to exactly one player, and one wishes to maximize the sum of values obtained by the players, as computed by applying the respective valuation function to the bundle of items allocated to the player. This problem in its full generality is NP-hard, and moreover, at least as hard to approximate as set-packing. Better approximation guarantees are known for restricted classes of valuation functions.

Motivated by the market for display advertisement over the Internet, we study competition between firms with a fixed supply whose size cannot be changed, and analyze the resulting revenue. We are most interested in studying the asymmetric case in which one large seller dominates the market and competes against a small new entrant seller. We present a model in which sellers announce selling policies, and given these policies buyers distribute their budget in a strategic fashion among sellers so as to maximize the portion of the supply that they receive. As a function of the policies of the sellers, we analyze revenue of sellers in pure and mixed Nash equilibria for the buyers. Our results show a contrast between the near-symmetric case (sellers with similar supply sizes) and the extremely asymmetric case (a very large seller vs. a very small seller). In particular, in the near-symmetric case, simple policies can ensure each seller a revenue almost proportional to her market share. In contrast, in the asymmetric case the large seller has a selling policy that yields disproportionally low revenue for the small seller. Interestingly, in our abstract model, non-monotone selling policies (namely, sometimes giving more of the supply to a buyer who decreases his bid) can offer advantages to the large seller that are (provably) impossible to achieve via monotone selling policies.

We consider the restricted assignment version of the problem of max-min fair allocation of indivisible goods, also known as the Santa Claus problem . There are m items and n players. Every item has some nonnegative value, and every player is interested in only some of the items. The goal is to distribute the items to the players in a way that maximizes the minimum of the sum of the values of the items given to any player. It was previously shown via a nonconstructive proof that uses the Lovász local lemma that the integrality gap of a certain configuration LP for the problem is no worse than some (unspecified) constant. This gives a polynomial-time algorithm to estimate the optimum value of the problem within a constant factor, but does not provide a polynomial-time algorithm for finding a corresponding allocation. We use a different approach to analyze the integrality gap. Our approach is based upon local search techniques for finding perfect matchings in certain classes of hypergraphs. As a result, we prove that the integrality gap of the configuration LP is no worse than 1/4. Our proof provides a local search algorithm which finds the corresponding allocation, but is nonconstructive in the sense that this algorithm is not known to converge to a local optimum in a polynomial number of steps.

We study graph partitioning problems from a min-max perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are: (i) the k parts need to be of equal size, and (ii) the parts must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O(√log n log k)-approximation algorithm. This improves over an O(log2n) approximation for the second version due to Svitkina and Tardos, and roughly O(k log n) approximation for the first version that follows from other previous work. We also give an improved O(1)-approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the Small Set Expansion problem. In this problem, we are given a graph G and the goal is to find a non-empty subset S of V of size at most pn with minimum edge-expansion. We give an O(√log n log (1/p)) bicriteria approximation algorithm for the general case of Small Set Expansion and O(1) approximation algorithm for graphs that exclude any fixed minor.

We consider a task of scheduling with a common deadline on a single machine. Every player reports to a scheduler the length of his job and the scheduler needs to finish as many jobs as possible by the deadline. For this simple problem, there is a truthful mechanism that achieves maximum welfare in dominant strategies. The new aspect of our work is that in our setting players are uncertain about their own job lengths, and hence are incapable of providing truthful reports (in the strict sense of the word). For a probabilistic model for uncertainty we show that even with relatively little uncertainty, no mechanism can guarantee a constant fraction of the maximum welfare. To remedy this situation, we introduce a new measure of economic efficiency, based on a notion of a fair share of a player, and design mechanisms that are Ω(1)-fair. In addition to its intrinsic appeal, our notion of fairness implies good approximation of maximum welfare in several cases of interest. In our mechanisms the machine is sometimes left idle even though there are jobs that want to use it. We show that this unfavorable aspect is unavoidable, unless one gives up other favorable aspects (e.g., give up Ω(1)-fairness). We also consider a qualitative approach to uncertainty as an alternative to the probabilistic quantitative model. In the qualitative approach we break away from solution concepts such as dominant strategies (they are no longer well defined), and instead suggest an axiomatic approach, which amounts to listing desirable properties for mechanisms. We provide a mechanism that satisfies these properties.

Submodular maximization generalizes many important problems including Max Cut in directed and undirected graphs and hypergraphs, certain constraint satisfaction problems, and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is NP-hard. In this paper, we design the first constant-factor approximation algorithms for maximizing nonnegative (non-monotone) submodular functions. In particular, we give a deterministic local-search $\frac{1}{3}$-approximation and a randomized $\frac{2}{5}$-approximation algorithm for maximizing nonnegative submodular functions. We also show that a uniformly random set gives a $\frac{1}{4}$-approximation. For symmetric submodular functions, we show that a random set gives a $\frac{1}{2}$-approximation, which can also be achieved by deterministic local search. These algorithms work in the value oracle model, where the submodular function is accessible through a black box returning $f(S)$ for a given set S. We show that in this model, a $(\frac{1}{2}+\epsilon)$-approximation for symmetric submodular functions would require an exponential number of queries for any fixed $\epsilon>0$. In the model where f is given explicitly (as a sum of nonnegative submodular functions, each depending only on a constant number of elements), we prove NP-hardness of $(\frac{5}{6}+\epsilon)$-approximation in the symmetric case and NP-hardness of $(\frac{3}{4}+\epsilon)$-approximation in the general case.

It is known that random [Formula: see text]-CNF formulas have a so-called satisfiability threshold at a density (namely, clause-variable ratio) of roughly [Formula: see text]: at densities slightly below this threshold almost all [Formula: see text]-CNF formulas are satisfiable, whereas slightly above this threshold almost no [Formula: see text]-CNF formula is satisfiable. In the current work we consider satisfiable random formulas and inspect another parameter—the diameter of the solution space (that is, the maximal Hamming distance between a pair of satisfying assignments). It was previously shown that for all densities up to a density slightly below the satisfiability threshold the diameter is almost surely at least roughly [Formula: see text] (and [Formula: see text] at much lower densities). At densities very much higher than the satisfiability threshold, the diameter is almost surely zero (a very dense satisfiable formula is expected to have only one satisfying assignment). In this paper we show that for all densities above a density that is slightly above the satisfiability threshold (more precisely, at ratio [Formula: see text], [Formula: see text] tending to 0 as [Formula: see text] grows) the diameter is almost surely [Formula: see text]. This shows that a relatively small change in the density around the satisfiability threshold (a multiplicative [Formula: see text] factor) makes a dramatic change in the diameter. This drop in the diameter cannot be attributed to the fact that a larger fraction of the formulas are not satisfiable (and hence have diameter 0), because the nonsatisfiable formulas are excluded from consideration by our conditioning that the formula be satisfiable.

In the Densest k-Subgraph problem, given a graph G and a parameter k, one needs to find a subgraph of G induced on k vertices that contains the largest number of edges. There is a significant gap between the best known upper and lower bounds for this problem. It is NP-hard, and does not have a PTAS unless NP has subexponential time algorithms. On the other hand, the current best known algorithm of Feige, Kortsarz and Peleg, gives an approximation ratio of n1/3 - c for some fixed c>0 (later estimated at around c= 1/90).

The following problem was introduced by Marcin Krzywkowski as a generalization of a problem of Todd Ebert. After initially coordinating a strategy, n players each occupy a different vertex of a graph. Either blue or red hats are placed randomly and independently on their heads. Each player sees the colors of the hats of players in neighboring vertices and no other hats (and hence, in particular, the player does not see the color of his own hat). Simultaneously, each player either tries to guess the color of his own hat or passes. The players win if at least one player guesses correctly and no player guesses wrong. The value of the game is the winning probability of the strategy that maximizes this probability. Previously, the value of such games was derived for certain families of graphs, including complete graphs of carefully chosen sizes, trees, and the 4-cycle. In this manuscript we conjecture that on every graph there is an optimal strategy in which all players who do not belong to the maximum clique always pass. We provide several results that support this conjecture, and determine among other things the value of the hat game for any bipartite graph and any planar graph that contains a triangle.

An adaptive (n, m, s, t)-scheme is a deterministic scheme for encoding a vector X of m bits with at most n ones by a vector Y of s bits, so that any bit of X can be determined by t adaptive probes to Y. A non-adaptive (n, m, s, t)-scheme is defined analogously. The study of such schemes arises in the investigation of the static membership problem in the bitprobe model. Answering a question of Buhrman, Miltersen, Radhakrishnan and Venkatesh [SICOMP 2002] we present adaptive (n, m, s, 2) schemes with s < m for all n satisfying 4n2 + 4n < m and adaptive (n, m, s, 2) schemes with s = o(m) for all n = o(logm). We further show that there are adaptive (n, m, s, 3)-schemes with s = o(m) for all n = o(m), settling a problem of Radhakrishnan, Raman and Rao [ESA 2001], and prove that there are non-adaptive (n, m, s, 4)-schemes with s = o(m) for all n = o(m). Therefore, three adaptive probes or four non-adaptive probes already suffice to obtain a significant saving in space compared to the total length of the input vector. Lower bounds are discussed as well.

In this paper we study the model of ∊-smoothed k-CNF formulas. Starting from an arbitrary instance F with n variables and m = dn clauses, apply the ∊-smoothing operation of flipping the polarity of every literal in every clause independently at random with probability ∊. Keeping ∊ and k fixed, and letting the density d = m/n grow, it is rather easy to see that for d ≥ ∊−-kln 2, F becomes whp unsatisfiable after smoothing. We show that a lower density that behaves roughly like ∊−-k+1 suffices for this purpose. We also show that our bound on d is nearly best possible in the sense that there are k-CNF formulas F of slightly lower density that whp remain satisfiable after smoothing. One consequence of our proof is a new lower bound of Ω(2k/k2) on the density up to which Walksat solves random k-CNFs in polynomial time whp. We are not aware of any previous rigorous analysis showing that Walksat is successful at densities that are increasing as a function of k.

We consider buffer management of unit packets with deadlines for a multi-port device with reconfiguration overhead. The goal is to maximize the throughput of the device, i.e., the number of packets delivered by their deadline. For a single port or with free reconfiguration, the problem reduces to the well-known packets scheduling problem, where the celebrated earliest-deadline-first (EDF) strategy is optimal 1-competitive. However, EDF is not 1-competitive when there is a reconfiguration overhead. We design an online algorithm that achieves a competitive ratio of 1 - o(1) when the ratio between the minimum laxity of the packets and the number of ports tends to infinity. This is one of the rare cases where one can design an almost 1-competitive algorithm. One ingredient of our analysis, which may be interesting on its own right, is a perturbation theorem on EDF for the classical packets scheduling problem. Specifically, we show that a small perturbation in the release and deadline times cannot significantly degrade the optimal throughput. This implies that EDF is robust in the sense that its throughput is close to the optimum even when the deadlines are not precisely known.

We consider the problem of maximizing welfare when allocating m items to n players with subadditive utility functions. Our main result is a way of rounding any fractional solution to a linear programming relaxation to this problem so as to give a feasible solution of welfare at least $1/2$ that of the value of the fractional solution. This approximation ratio of $1/2$ is an improvement over an $\Omega(1/\log m)$ ratio of Dobzinski, Nisan, and Schapira [Proceedings of the 37th Annual ACM Symposium on Theory of Computing (Baltimore, MD), ACM, New York, 2005, pp. 610–618]. We also show an approximation ratio of $1-1/e$ when utility functions are fractionally subadditive. A result similar to this last result was previously obtained by Dobzinski and Schapira [Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (Miami, FL), SIAM, Philadelphia, 2006, pp. 1064–1073], but via a different rounding technique that requires the use of a so-called “XOS oracle.” The randomized rounding techniques that we use are oblivious in the sense that they only use the primal solution to the linear program relaxation, but have no access to the actual utility functions of the players.

In a combinatorial optimization problem, when given an input instance, one seeks a feasible solution that optimizes the value of the objective function. Many combinatorial optimization problems are NP-hard. A way of coping with NP-hardness is by considering approximation algorithms. These algorithms run in polynomial time, and their performance is measured by their approximation ratio: the worst case ratio between the value of the solution produced and the value of the (unknown) optimal solution. In some cases the design of approximation algorithms includes a nonconstructive component. As a result, the algorithms become estimation algorithms rather than approximation algorithms: they allow one to estimate the value of the optimal solution, without actually producing a solution whose value is close to optimal. We shall present a few such examples, and discuss some open questions.

High-quality, personalized recommendations are a key feature in many online systems. Since these systems often have explicit knowledge of social network structures, the recommendations may incorporate this information. This paper focuses on networks that represent trust and recommendation systems that incorporate these trust relationships. The goal of a trust-based recommendation system is to generate personalized recommendations by aggregating the opinions of other users in the trust network.In analogy to prior work on voting and ranking systems, we use the axiomatic approach from the theory of social choice. We develop a set of five natural axioms that a trust-based recommendation system might be expected to satisfy. Then, we show that no system can simultaneously satisfy all the axioms. However, for any subset of four of the five axioms we exhibit a recommendation system that satisfies those axioms. Next we consider various ways of weakening the axioms, one of which leads to a unique recommendation system based on random walks. We consider other recommendation systems, including systems based on personalized PageRank, majority of majorities, and minimum cuts, and search for alternative axiomatizations that uniquely characterize these systems.Finally, we determine which of these systems are incentive compatible, meaning that groups of agents interested in manipulating recommendations can not induce others to share their opinion by lying about their votes or modifying their trust links. This is an important property for systems deployed in a monetized environment.

Most current banner advertising is sold through negotiation thereby incurring large transaction costs and possibly suboptimal allocations. We propose a new automated system for selling banner advertising. In this system, each advertiser specifies a collection of host webpages which are relevant to his product, a desired total quantity of impressions on these pages, and a maximum per-impression price. The system selects a subset of advertisers as 'winners' and maps each winner to a set of impressions on pages within his desired collection. The distinguishing feature of our system as opposed to current combinatorial allocation mechanisms is that, mimicking the current negotiation system, we guarantee that winners receive at least as many advertising opportunities as they requested or else receive ample compensation in the form of a monetary payment by the host. Such guarantees are essential in markets like banner advertising where a major goal of the advertising campaign is developing brand recognition.

We prove semilogarithmic inapproximability for a maximization problem called unique coverage: given a collection of sets, find a subcollection that maximizes the number of elements covered exactly once. Specifically, assuming that $\mathrm{NP}\not\subseteq\operatorname{BPTIME}(2^{n^\varepsilon})$ for an arbitrary $\varepsilon>0$, we prove $O(1/\log^{\sigma}n)$ inapproximability for some constant $\sigma=\sigma(\varepsilon)$. We also prove $O(1/\log^{1/3-\varepsilon}n)$ inapproximability for any $\varepsilon>0$, assuming that refuting random instances of 3SAT is hard on average; and we prove $O(1/\log n)$ inapproximability under a plausible hypothesis concerning the hardness of another problem, balanced bipartite independent set. We establish an $\Omega(1/\log n)$-approximation algorithm, even for a more general (budgeted) setting, and obtain an $\Omega(1/\log B)$-approximation algorithm when every set has at most B elements. We also show that our inapproximability results extend to envy-free pricing, an important problem in computational economics. We describe how the (budgeted) unique coverage problem, motivated by real-world applications, has close connections to other theoretical problems, including max cut, maximum coverage, and radio broadcasting.

We develop the algorithmic theory of vertex separators and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into $L_1$ (and even Euclidean embeddings) are insufficient but that the additional structure provided by many embedding theorems does suffice for our purposes. We obtain an $O(\sqrt{\log n})$ approximation for minimum ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be $\Theta(\sqrt{\log n})$. We also prove an optimal $O(\log k)$-approximate max-flow/min-vertex-cut theorem for arbitrary vertex-capacitated multicommodity flow instances on k terminals. For uniform instances on any excluded-minor family of graphs, we improve this to $O(1)$, and this yields a constant-factor approximation for minimum ratio vertex cuts in such graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best known ratio was $O(\log n)$. These results have a number of applications. We exhibit an $O(\sqrt{\log n})$ pseudoapproximation for finding balanced vertex separators in general graphs. In fact, we achieve an approximation ratio of $O(\sqrt{\log {opt}})$, where ${opt}$ is the size of an optimal separator, improving over the previous best bound of $O(\log {opt})$. Likewise, we obtain improved approximation ratios for treewidth: in any graph of treewidth k, we show how to find a tree decomposition of width at most $O(k \sqrt{\log k})$, whereas previous algorithms yielded $O(k \log k)$. For graphs excluding a fixed graph as a minor (which includes, e.g., bounded genus graphs), we give a constant-factor approximation for the treewidth. This in turn can be used to obtain polynomial-time approximation schemes for several problems in such graphs.

Motivated by the study of parallel repetition and also by the unique games conjecture, we investigate the value of the "odd cycle games" under parallel repetition. Using tools from discrete harmonic analysis, we show that after d rounds on the cycle of length m, the value of the game is at most 1-(1/m)ldrOmega macr(radicd) (for dlesm2, say). This beats the natural barrier of 1-Theta(1/m)2ldrd for Raz-style proofs and also the SDP bound of Feige-Lovasz; however, it just barely fails to have implications for unique games. On the other hand, we also show that improving our bound would require proving nontrivial lower bounds on the surface area of high-dimensional foams. Specifically, one would need to answer: what is the least surface area of a cell that tiles Rdby the lattice Zd?

We introduce the following model for generating .semi-random 3CNF formulas. First, an adversary is allowed to pick an arbitrary formula with n varialdes and in clauses. Then, the formula is slightly perturbed at random. Namely, the smoothing operation leaves the variables of the formula unchanged, but flips the polarity of every variable occurrence in the formula independently with probability a. If the density m/n of a 3CNF formula exceeds a certain threshold value (say, 5epsiv-3) then the smoothing operation almost surely results in a non-satisfiable formula. We present a randomized polynomial time refutation algorithm that for every sufficiently dense 3CNF formula manages to refute most of its smoothed instantiations. The density requirement for our refutation algorithm is roughly epsiv-2radic(n log log n), which almost matches the density Omega( radicn) required bv known algorithms for refuting 3CNF formulas that are completely random.

Submodular maximization generalizes many important problems including Max Cut in directed/undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is NP-hard.

We consider random 3CNF formulas with n variables and m clauses. It is well known that when m > cn (for a sufficiently large constant c), most formulas are not satisfiable. However, it is not known whether such formulas are likely to have polynomial size witnesses that certify that they are not satisfiable. A value of m sime n3/2was the forefront of our knowledge in this respect. When m > cn3/2, such witnesses are known to exist, based on spectral techniques. When m3/2-epsi, it is known that resolution (which is a common approach for refutation) cannot produce witnesses of size smaller than 2nepsiv. Likewise, it is known that certain variants of the spectral techniques do not work in this range. In the current paper we show that when m > cn7/5, almost all 3CNF formulas have polynomial size witnesses for non-satisfiability. We also show that such a witness can be found in time 2(O(n0.2 log n)), whenever it exists. Our approach is based on an extension of the known spectral techniques, and involves analyzing a certain fractional packing problem for random 3-uniform hypergraphs

Combinatorial allocation problems require allocating items to players in a way that maximizes the total utility. Two such problems received attention recently, and were addressed using the same linear programming (LP) relaxation. In the maximum submodular welfare (SMW) problem, utility functions of players are submodular, and for this case Dobzinski and Schapira [SODA 2006] showed an approximation ratio of 1 - 1/e. In the generalized assignment problem (GAP) utility functions are linear but players also have capacity constraints. GAP admits a (1 - 1/e)-approximation as well, as shown by Fleischer, Goemans, Mirrokni and Sviridenko [SODA 2006]. In both cases, the approximation ratio was in fact shown for a more general version of the problem, for which improving 1 - 1/e is NP-hard. In this paper, we show how to improve the 1 - 1/e approximation ratio, both for SMW and for GAP. A common theme in both improvements is the use of a new and optimal fair contention resolution technique. However, each of the improvements involves a different rounding procedure for the above mentioned LP. In addition, we prove APX-hardness results for SMW (such results were known for GAP). An important feature of our hardness results is that they apply even in very restricted settings, e.g. when every player has nonzero utility only for a constant number of items

We consider the problem of maximizing welfare when allocating m items to n players with subadditive utility functions. Our main result is a way of rounding any fractional solution to a linear programming relaxation to this problem so as to give a feasible solution of welfare at least half that of the value of the fractional solution. This approximation ratio of 1/2 improves over an Ω(1/log m) ratio of Dobzinski, Nisan and Schapira [STOC 2005]. We also show an approximation ratio of 1 - 1/e when utility functions are fractionally subadditive. A result similar to this last result was previously obtained by Dobzinski and Schapira [Soda 2006], but via a different rounding technique that requires the use of a so called "XOS oracle".The randomized rounding techniques that we use are oblivious in the sense that they only use the primal solution to the linear program relaxation, but have no access to the actual utility functions of the players. This allows us to suggest new incentive compatible mechanisms for combinatorial auctions, extending previous work of Lavi and Swamy [FOCS 2005].

Let G be an n-vertex graph that has a vertex separator of size k that partitions the graph into connected components of size smaller than α n, for some fixed 2/3 ≤ α < 1. Such a separator is called an α-separator. Finding an α-separator of size at most k is NP-hard. Moreover, under reasonable complexity theoretic assumptions, it is shown that this problem is not polynomially solvable even when k=O(log n). In this paper, we give a randomized algorithm that finds an α-separator of size k in the given graph, unless the graph contains an (α+ε)-separator of size strictly less than k, in which case our algorithm finds one such separator. For fixed ε, the running time of our algorithm is nO(1)2O(k), which is polynomial for k = O(log n). For bounded degree graphs (as well as for the case of finding balanced edge separators), we present a deterministic algorithm with similar running time.Our algorithm involves (among other things) a new concept that we call (ε,k)-samples. This is related to the notion of detection sets for network failures, introduced by Kleinberg [FOCS 2000]. Our proofs adapt and simplify techniques that were introduced by Kleinberg. As a by-product, our proof improves the known bounds on the size of detection sets. We also show applications of (ε,k)-samples to problems in approximation algorithms and rigorous analysis of heuristics.

We prove the following inequality: for every positive integer n and every collection $X_1, \ldots, X_n$ of nonnegative independent random variables, each with expectation 1, the probability that their sum remains below $n+1$ is at least $\alpha > 0$. Our proof produces a value of $\alpha = 1/13 \simeq 0.077$, but we conjecture that the inequality also holds with $\alpha = 1/e \simeq 0.368$. As an example for the use of the new inequality, we consider the problem of estimating the average degree of a graph by querying the degrees of some of its vertices. We show the following threshold behavior: approximation factors above 2 require far fewer queries than approximation factors below 2. The new inequality is used in order to get tight (up to multiplicative constant factors) relations between the number of queries and the quality of the approximation. We show how the degree approximation algorithm can be used in order to quickly find those edges in a network that belong to many shortest paths.

We consider the problem of finding a maximum independent set in a random graph. The random graph G is modelled as follows. Every edge is included independently with probability $\frac{d}{n}$ , where d is some sufficiently large constant. Thereafter, for some constant α, a subset I of αn vertices is chosen at random, and all edges within this subset are removed. In this model, the planted independent set I is a good approximation for the maximum independent set I max , but both I ∖ I max and I max ∖ I are likely to be nonempty. We present a polynomial time algorithms that with high probability (over the random choice of random graph G, and without being given the planted independent set I) finds a maximum independent set in G when $\alpha \geq \sqrt{c_0 \log d /d}$ , where c 0 is some sufficiently large constant independent of d.

We develop the algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into L1 (and even Euclidean embeddings) are insufficient, but that the additional structure provided by many embedding theorems does suffice for our purposes.We obtain an O(√log n) approximation for min-ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be Θ(√log n). We also prove various approximate max-flow/min-vertex-cut theorems, which in particular give a constant-factor approximation for min-ratio vertex cuts in any excluded-minor family of graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best-known ratio was O(log n).These results have a number of applications. We exhibit an O(√log n) pseudo-approximation for finding balanced vertex separators in general graphs. In fact, we achieve an approximation ratio of O(√log opt) where opt is the size of an optimal separator, improving over the previous best bound of O(log opt). Likewise, we obtain improved approximation ratios for treewidth: In any graph of treewidth k, we show how to find a tree decomposition of width at most O(k √log k), whereas previous algorithms yielded O(k log k). For graphs excluding a fixed graph as a minor (which includes, e.g., bounded genus graphs), we give a constant-factor approximation for the treewidth; this can be used to obtain the first polynomial-time approximation schemes for problems like minimum feedback vertex set and minimum connected dominating set in such graphs.

What is the lineage relation among the cells of an organism? The answer is sought by developmental biology, immunology, stem cell research, brain research, and cancer research, yet complete cell lineage trees have been reconstructed only for simple organisms such as Caenorhabditis elegans. We discovered that somatic mutations accumulated during normal development of a higher organism implicitly encode its entire cell lineage tree with very high precision. Our mathematical analysis of known mutation rates in microsatellites (MSs) shows that the entire cell lineage tree of a human embryo, or a mouse, in which no cell is a descendent of more than 40 divisions, can be reconstructed from information on somatic MS mutations alone with no errors, with probability greater than 99.95%. Analyzing all approximately 1.5 million MSs of each cell of an organism may not be practical at present, but we also show that in a genetically unstable organism, analyzing only a few hundred MSs may suffice to reconstruct portions of its cell lineage tree. We demonstrate the utility of the approach by reconstructing cell lineage trees from DNA samples of a human cell line displaying MS instability. Our discovery and its associated procedure, which we have automated, may point the way to a future "Human Cell Lineage Project" that would aim to resolve fundamental open questions in biology and medicine by reconstructing ever larger portions of the human cell lineage tree.

We prove the following inequality: for every positive integer n and every collection X1,..., Xn of nonnegative independent random variables that each has expectation 1, the probability that their sum remains below n+1 is at least α > 0. Our proof produces a value of α = 1/13 ≅ 0.077, but we conjecture that the inequality also holds with α = 1/e ≅ 0.368.As an example for the use of the new inequality, we consider the problem of estimating the average degree of a graph by querying the degrees of some of its vertices. We show the following threshold behavior: approximation factors above 2 require far less queries than approximation factors below 2. The new inequality is used in order to get tight (up to multiplicative constant factors) relations between the number of queries and the quality of the approximation. We show how the degree approximation algorithm can be used in order to quickly find those edges in a network that belong to many shortest paths.

Karger, Motwani, and Sudan [J. ACM, 45 (1998), pp. 246--265] introduced the notion of a vector coloring of a graph. In particular, they showed that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly $\Delta^{1 - 2/k}$ colors. Here $\Delta$ is the maximum degree in the graph and is assumed to be of the order of $n^{\delta}$ for some $0 < \delta < 1$. Their results play a major role in the best approximation algorithms used for coloring and for maximum independent sets. We show that for every positive integer k there are graphs that are vector k-colorable but do not have independent sets significantly larger than $n/\Delta^{1 - 2/k}$ (and hence cannot be colored with significantly fewer than $\Delta^{1 - 2/k}$ colors). For $k = O(\log n/\log\log n)$ we show vector k-colorable graphs that do not have independent sets of size (log n) c , for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylog n. As part of our proof, we analyze "property testing" algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are "far" from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser, and Ron [J. ACM, 45 (1998), pp. 653--750] for this problem.

We show an algorithm that finds cliques of size (log n/log log n) 2 whenever a graph has a clique of size at least n/(log n) b for an arbitrary constant b. This leads to an algorithm that approximates max clique within a factor of O(n(log log n) 2 /(log n) 3 ), which matches the best approximation ratio known for the chromatic number. The previously best approximation ratio known for max clique was O(n/(log n) 2 ).

Lov{ász and Schrijver [SIAM J. Optim., 1 (1991), pp. 166--190] devised a lift-and-project method that produces a sequence of convex relaxations for the problem of finding in a graph an independent set (or a clique) of maximum size. Each relaxation in the sequence is tighter than the one before it, while the first relaxation is already at least as strong as the Lov{ász theta function [IEEE Trans. Inform. Theory, 25 (1979), pp. 1--7]. We show that on a random graph G n,1/2 , the value of the rth relaxation in the sequence is roughly \rule{0pt}{7pt}$\smash{\sqrt{\rule{0pt}{7pt}\smash{n/2^r}}}$, almost surely. It follows that for those relaxations known to be efficiently computable, namely, for r=O(1), the value of the relaxation is comparable to the theta function. Furthermore, a perfectly tight relaxation is almost surely obtained only at the $r=\Theta(\log n)$ relaxation in the sequence.

We prove that the closest vector problem with preprocessing (CVPP) is NP-hard to approximate within any factor less than /spl radic/5/3. More specifically, we show that there exists a reduction from an NP-hard problem to the approximate closest vector problem such that the lattice depends only on the size of the original problem, and the specific instance is encoded solely, in the target vector. It follows that there are lattices for which the closest vector problem cannot be approximated within factors /spl gamma/ < /spl radic/5/3 in polynomial time, no matter how the lattice is represented, unless NP is equal to P (or NP is contained in P/poly, in case of nonuniform sequences of lattices). The result easily extends to any lp norm, for p /spl ges/ 1, showing that CVPP in the lp norm is hard to approximate within any factor /spl gamma/ < /sup p//spl radic/5/3. As an intermediate step, we establish analogous results for the nearest codeword problem with preprocessing (NCPP), proving that for any finite field GF(q), NCPP over GF(q) is NP-hard to approximate within any factor less than 5/3.

Karger Motwani and Sudan (1998) introduced the notion of a vector coloring of a graph. In particular they show that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly /spl Delta//sup 1-2/k/ colors. Here /spl Delta/ is the maximum degree in the graph. Their results play a major role in the best approximation algorithms for coloring and for maximal independent set. We show that for every positive integer k there are graphs that are vector k-colorable but do not have independent sets significantly larger than n//spl Delta//sup 1-2/k/ (and hence cannot be colored with significantly less that /spl Delta//sup 1-2/k/ colors). For k = O(log n/log log n) we show vector k-colorable graphs that do not have independent sets of size (log n)/sup c/, for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylogn. As part of our proof, we analyze "property testing" algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are "far" from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser and Ron (1998) for this problem.

We investigate relations between average case complexity and the complexity of approximation. Our preliminary findings indicate that this is a research direction that leads to interesting insights. Under the assumption that refuting 3SAT is hard on average on a natural distribution, we derive hardness of approximation results for min bisection, dense k-subgraph, max bipartite clique and the 2-catalog segmentation problem. No NP-hardness of approximation results are currently known for these problems.

A set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number of vertices, $\delta$ the minimum degree, and $\Delta$ the maximum degree. We show that every graph has a domatic partition with $(1 - o(1))(\delta + 1)/\ln n$ dominating sets and, moreover, that such a domatic partition can be found in polynomial-time. This implies a $(1 + o(1))\ln n$-approximation algorithm for domatic number, since the domatic number is always at most $\delta + 1$. We also show this to be essentially best possible. Namely, extending the approximation hardness of set cover by combining multiprover protocols with zero-knowledge techniques, we show that for every $\epsilon > 0$, a $(1 - \epsilon)\ln n$-approximation implies that $NP \subseteq DTIME(n^{O(\log\log n)})$. This makes domatic number the first natural maximization problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better. We also show that every graph has a domatic partition with $(1 - o(1))(\delta + 1)/\ln \Delta$ dominating sets, where the "o(1)" term goes to zero as $\Delta$ increases. This can be turned into an efficient algorithm that produces a domatic partition of $\Omega(\delta/\ln \Delta)$ sets.

A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size n/2. The bisection cost is the number of edges connecting the two sets. It is known that finding a bisection of minimum cost is NP-hard. We present an algorithm that finds a bisection whose cost is within ratio of O(log 2 n ) from the minimum. For graphs excluding any fixed graph as a minor (e.g., planar graphs) we obtain an improved approximation ratio of O(log n). The previously known approximation ratio for bisection was roughly $\sqrt{n}$.

MAX CUT is the problem of partitioning the vertices of a graph into two sets, maximizing the number of edges joining these sets. This problem is NP-hard. Goemans and Williamson proposed an algorithm that first uses a semidefinite programming relaxation of MAX CUT to embed the vertices of the graph on the surface of an n dimensional sphere, and then uses a random hyperplane to cut the sphere in two, giving a cut of the graph. They show that the expected number of edges in the random cut is at least α \cdot sdp, where α \simeq 0.87856 and sdp is the value of the semidefinite program.This manuscript shows the following results:1. The integrality ratio of the semidefinite program is α. The previously known bound on theintegrality ratio was roughly 0.8845.2. In the presence of the so called “triangle constraints”, the integrality ratio is no better than roughly 0.891. The previously known bound was above 0.95.

A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size n/2. The bisection cost is the number of edges connecting the two sets. Finding the bisection of minimum cost is NP-hard. We present an algorithm that finds a bisection whose cost is within ratio of O(log/sup 2/ n) from the optimal. For graphs excluding any fixed graph as a minor (e.g. planar graphs) we obtain an improved approximation ratio of O(log n). The previously known approximation ratio for bisection was roughly /spl radic/n.

A set of vertices in a graph is a dominating set if every vertex outside the set has aneighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number ofvertices, ffi the minimum degree, and \\Delta the maximum degree.We show that every graph has a domatic partition with (1-o(1))(ffi + 1) / ln n dominatingsets, and moreover, that such a domatic partition can be found in polynomial time. This implies a (1 + o(1)) ln n approximation algorithm for domatic number, since the domaticnumber is always at most ffi + 1. We also show this to be essentially best possible. Namely,extending the approximation hardness of set cover by combining multi-prover protocols with zero-knowledge techniques, we show that for every ffl> 0, a (1- ffl) ln n-approximation impliesthat N P ` DT IM E(nO(log log n)). This makes domatic number the first natural maximiza-tion problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better.We also show that every graph has a domatic partition with (1-o(1))(ffi + 1) / ln \\Delta dominating sets, where the " o(1) " term goes to zero as \\Delta increases. This can be turned intoan efficient algorithm that produces a domatic partition of \\Omega ( ffi / ln \\Delta) sets.

) Uriel Feige Robert Krauthgamer Kobbi Nissim Deptartment of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot 76100, Israel ffeige,robi,[email protected] February 22, 2000 Abstract A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size n=2. The bisection size is the number of edges connecting the two sets. Finding the bisection of minimum size is NP-hard. We present an algorithm that finds a bisection that is within O( p n log n) of optimal. No sublinear approximation ratio for bisection was previously known. 1 Introduction Let G(V; E) be a graph with n vertices and m edges, where n is even. A bisection of G is a set of vertices S ae V with cardinality jSj = n=2. The size of the bisection S is the number of edges connecting S to its complement V nS. The minimum size of the bisection of a graph is denoted by b. Computing b is NP-hard, cf. [8, 6]. We address the problem of approximating b....

We introduce the miss-match form for two-prover one-round proof systems. Any two-prover one-round proof system can be easily modified so as to be in miss-match form. Proof systems in miss-match form have the "projection" property that is important for deriving hardness of approximation results for NP-hard combinatorial optimization problems. Our main result is an upper bound on the number of parallel repetitions that suffice in order to reduce the error of miss-match proof systems from p to $\epsilon$. This upper bound depends only on p and on $\epsilon$ (polynomial in 1/(1-p) and in $1/\epsilon$). Based on previous work, it follows that for any $\epsilon >0,$ NP has two-prover one-round proof systems with logarithmic-sized questions, constant-sized answers, and error at most $\epsilon$. As part of our proof we prove upper bounds on the influence of random variables on multivariate functions, which may be of independent interest.

Selection tasks generalize some well studied problems, such as collective coin flipping and leader election. We present new selection protocols in the full information model, and new negative results. In particular when there are (1+/spl delta/)n/2 good players, we show a protocol that chooses a good leader with probability /spl Omega/(/spl delta//sup 1.65/), and show that every leader election protocol has success probability O(/spl delta//sup 1-/spl epsiv//), for every /spl epsiv/>0. Previously known protocols for this problem have success probability that is exponentially small in 1//spl delta/, and no nontrivial upper bounds on the success probability were known.

Article Free Access Share on Nonmonotonic phenomena in packet routing Author: Uriel Feige Department of Applied Mathematics and Computer Science, Weizmann Institute, Rehovot 76100, Israel Department of Applied Mathematics and Computer Science, Weizmann Institute, Rehovot 76100, IsraelView Profile Authors Info & Claims STOC '99: Proceedings of the thirty-first annual ACM symposium on Theory of ComputingMay 1999 Pages 583–591https://doi.org/10.1145/301250.301409Published:01 May 1999Publication History 5citation280DownloadsMetricsTotal Citations5Total Downloads280Last 12 Months2Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

In this paper we show how to construct noninteractive zero knowledge proofs for any NP statement under general (rather than number theoretic) assumptions, and how to enable polynomially many provers to give polynomially many such proofs based on a single random string. Our constructions can be used in cryptographic applications in which the prover is restricted to polynomial time.

We study a semi-random graph model for finding independent sets. For /spl alpha/>0, an n-vertex graph with an independent set S of site /spl alpha/n is constructed by blending random and adversarial decisions. Randomly and independently with probability p, each pair of vertices, such that one is in S and the other is not, is connected by an edge. An adversary can then add edges arbitrarily (provided that S remains an independent set). The smaller p is, the larger the control the adversary has over the semi-random graph. We design heuristics that with high probability recover S when p>(1+/spl epsiv/)ln n/|S|, for any constant /spl epsiv/>0. We show that when p<(1-/spl epsiv/) In n/|S|, an independent set of size |S| cannot be recovered, unless NP/spl sube/BPP. We use our remits to obtain greatly improved coloring algorithms for the model of k-colorable semi-random graphs introduced by A. Blum and J. Spencer (1995).

A linear arrangement of an n-vertex graph is a one-tc+one mapping of its vertices to the integers (1,. . ., n}.The bandwidt,h of a linear arrangement is the maximum difference between mapped values of adjacent vertices.The problem of finding a linear arrangement wit,h smallest possible bandwidt,h in NP-hard.We present a randomized algorithm that runs in nearly linear time and outputs a linear arrangement whose band&dt,h is within a polylogarithmic multiplicative factor of optimal.Our algorithm is based on a new notion, called volume respecting em&tidings, which is a natural extension of small distortion embeddings of Bourgain and of Linial, London and Ra~movich.

Article Free Access Share on Improved bounds for acyclic job shop scheduling (extended abstract) Authors: Uriel Feige Dept. of Appl. Math. and Comp. Sci. Weizmann Institute, 76100 Rehovot, Israel Dept. of Appl. Math. and Comp. Sci. Weizmann Institute, 76100 Rehovot, IsraelView Profile , Christian Scheideler Dept. of Math. and Comp. Sci., Paderborn University, 33095 Paderborn, Germany Dept. of Math. and Comp. Sci., Paderborn University, 33095 Paderborn, GermanyView Profile Authors Info & Claims STOC '98: Proceedings of the thirtieth annual ACM symposium on Theory of computingMay 1998Pages 624–633https://doi.org/10.1145/276698.276878Published:23 May 1998Publication History 13citation330DownloadsMetricsTotal Citations13Total Downloads330Last 12 Months13Last 6 weeks3 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Given a collection ℱ of subsets of S = {1,…, n }, set cover is the problem of selecting as few as possible subsets from ℱ such that their union covers S, , and max k-cover is the problem of selecting k subsets from ℱ such that their union has maximum cardinality. Both these problems are NP-hard. We prove that (1 - o (1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low-order terms) between the ratio of approximation achievable by the greedy alogorithm (which is (1 - o (1)) ln n), and provious results of Lund and Yanakakis, that showed hardness of approximation within a ratio of (log 2 n ) / 2 ≃0.72 ln n . For max k -cover, we show an approximation threshold of (1 - 1/ e )(up to low-order terms), under assumption that P ≠ NP .

We study the complexity of refereed games, in which two computationally unlimited players play against each other, and a polynomial time referee monitors the game and announces the winner.The players may exchange messages with the referee in private, resulting in a game of perfect recall but incomplete information.We show that any EXPTIME statement can be efficiently transformed into a refereed game in which if the statement is true, the first player wins with overwhelming probability y, and if the statement is false, the second player wins with overwhelming probability.We also prove matching PSPACE upper and lower bounds on the complexity of statements that have refereed games that take one round of communication.

We present a new technique, inspired by zero-knowledge proof systems, for proving lower bounds on approximating the chromatic number of a graph. To illustrate this technique we present simple reductions from max-3-coloring and max-3-sat, showing that it is hard to approximate the chromatic number within /spl Omega/(N/sup /spl delta//), for some /spl delta/>0. We then apply our technique in conjunction with the probabilistically checkable proofs of Bellare, Goldreich and Sudan (1995), and of Hastad (1996), and show that it is hard to approximate the chromatic number to within /spl Omega/(N/sup 1-/spl epsiv//) for any E>0, assuming NP/spl sub/ ZPP. Here, ZPP denotes the class of languages decidable by a random expected polynomial-time algorithm that makes no errors. Our result matches (up to low order terms) the known gap for approximating the size of the largest independent set. Previous 0(N/sup /spl delta//) gaps for approximating the chromatic number (such as those by Lund and Yannakakis (1994), and by Furer (1995)) did not match the gap for independent set, and do not extend beyond /spl Omega/(N/sup 1/2-/spl epsiv//).

We show that no fixed number of parallel repetitions suffices in order to reduce the error in two-prover one-round proof systems from one constant to another. Our results imply that the recent bounds proven by Ran Raz (1995), showing that the number of rounds that suffice is inversely proportional to the answer length, are nearly best possible.

A fundamental problem in designing secure multi-party protocols is how to deal with adaptive adversaries (i.e., adversaries that may choose the corrupted parties during the course of the computation), in a setting where the channels are insecure and secure communication is achieved by cryptographic primitives based on the computational limitations of the adversary.

We prove that (] -o(]))lnn is a threshold below which set, cover cannot be approximated efficiently, unless NP has slightly superpolynornial time algorithms.This closes tlw gap (up to low order terms) between the ratio of ap-prox&ation achievable by the greedy algorithm (which is (1 -O( 1 ) ) in n), and previous results of Lund and Yannakakis, that showed harclness of approximation within a ratio of (log2 7/)/2 E 0.7.?111 //,, 1

The contribution of this paper is two-fold. First, a connection is established between approximating the size of the largest clique in a graph and multi-prover interactive proofs. Second, an efficient multi-prover interactive proof for NP languages is constructed, where the verifier uses very few random bits and communication bits. Last, the connection between cliques and efficient multi-prover interaction proofs, is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph. Of independent interest is our proof of correctness for the multilinearity test of functions.

The short-term behavior of random walks on graphs is studied, in particular, the rate at which a random walk discovers new vertices and edges. A conjecture by Linial that the expected time to find $\mathcal{N}$ distinct vertices is $O(\mathcal{N}^3 )$ is proved. In addition, upper bounds of $O(\mathcal{M}^2 )$ on the expected time to traverse $\mathcal{M}$ edges and of $O(\mathcal{M}\mathcal{N})$ on the expected time to either visit $\mathcal{N}$ vertices or traverse $\mathcal{M}$ edges (whichever comes first) are proved.

For an undirected graph and an optimal cyclic list of all its vertices, the cyclic cover time is the expected time it takes a simple random walk to travel from vertex to vertex along the list until it completes a full cycle. The main result of this paper is a characterization of the cyclic cover time in terms of simple and easy-to-compute graph properties. Namely, for any connected graph, the cyclic cover time is $\Theta (n^2 d_{ave} (d^{ - 1} )_{ave} $), where n is the number of vertices in the graph, $d_{ave} $ is the average degree of its vertices, and $(d^{ - 1} )_{ave} $ is the average of the inverse of the degree of its vertices. Other results obtained in the processes of proving the main theorem are a similar characterization of minimum resistance spanning trees of graphs, improved bounds on the cover time of graphs, and a simplified proof that the maximum commute time in any connected graph is at most $4n^3 /27 + o(n^3 )$.

For a graphG, let α(G) denote the size of the largest independent set inG, and let ϑ(G) denote the Lovasz ϑ-function onG. We prove that for somec>0, there exists an infinite family of graphs such that\(\vartheta (G) > \alpha (G)n/2^{c\sqrt {\log n} }\), wheren denotes the number of vertices in a graph. this disproves a known conjecture regarding the ϑ function.

Recycling random bits (that is, replacing independent random bits by dependent random bits extracted from a pseudo-random bit generator) has been a successful enterprise in many scenarios, including cryptography, NC computations, space bounded computation, RP and BPP algorithms, and interactive proofs. A wide variety of techniques have been introduced for this purpose. Recycling random bits is highly motivated in the context of parallel repetition of two-prover one-round proof systems, where (if it were possible) it would lead to stronger hardness results for approximating NP-hard optimization problems. In this paper we show that the great success enjoyed by general techniques for recycling random bits in other contexts meets its limits when MIP(2,1) proof systems are concerned. That is, there are natural MIP(2,1) proof systems for 3-SAT for which parallel repetition using independent random bits can reduce the error to be polynomially small, but parallel repetition using pseudo-random bits cannot reduce the error below a constant, regardless of the nature of the pseudo-random source.

We introduce the miss-match form for two-prover one-round proof systems.Any two-prover one-round proof system can be easily modified so as to be in miss-match form.Proof systems in miss-match form have the projection property that is important for deriving hardness of approximation results for NP-hard combinatorial optimization problems. Our main result is an upper bound on the number of parallel repetitions that suffice in order to reduce the error of miss-match proof systems from p to � .This upper bound depends only on p and on � (polynomial in 1/(1 − p) and in 1/� ).Based on previous work, it follows that for any �> 0, NP has two-prover one-round proof systems with logarithmic-sized questions, constant-sized answers, and error at most � . As part of our proof we prove upper bounds on the influence of random variables on multivariate functions, which may be of independent interest.

We consider a minimal scenario for secure computation: Parties A and B have private inputs x and y and a shared random string r.A and B are each allowed to send a single message to a third party C, from which C is to learn the value of ~(z, y) for some function ~, but nothing else.We show that this model is surpris-Permission to copywithout fee all or part of this material is granted provided that the copies are not made or distributed for direct eommarcial advantaqe, tha ACM copyrioht notice a?d the title of the publicatiort 'and Its date appear, and notice is gwen that copying is by permission of the Association of Computing Machinery.

This paper studies the depth of noisy decision trees in which each node gives the wrong answer with some constant probability. In the noisy Boolean decision tree model, tight bounds are given on the number of queries to input variables required to compute threshold functions, the parity function and symmetric functions. In the noisy comparison tree model, tight bounds are given on the number of noisy comparisons for searching, sorting, selection and merging. The paper also studies parallel selection and sorting with noisy comparisons, giving tight bounds for several problems.

We present a randomized time space tradeoff of O/spl tilde/(mR/spl circ/) for undirected S-T-connectivity, where R/spl circ/ /spl Sigma//sub /spl upsi//spl epsiv/V/ 1/d/sub /spl upsi// is the virtual resistance of the graph. This solves an open question of Broder et al. (1989) (implicit also in Aleliunas et al. (1979)) who asked whether a tradeoff of O/spl tilde/(mn) is achievable, and also improves upon a tradeoff of O/spl tilde/(mn/d/sub min/) conjectured by Barnes and Feige (1993). Our algorithm is a modification of the Broder et al. algorithm. In passing, we also improve a result from Barnes and Feige regarding the rate at which a random walk discovers new vertices in a graph.>

Article Short random walks on graphs Share on Authors: Greg Barnes View Profile , Uriel Feige View Profile Authors Info & Claims STOC '93: Proceedings of the twenty-fifth annual ACM symposium on Theory of ComputingJune 1993 Pages 728–737https://doi.org/10.1145/167088.167275Online:01 June 1993Publication History 19citation659DownloadsMetricsTotal Citations19Total Downloads659Last 12 Months34Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

The authors consider a form of packet routing known as hot potato routing or deflection routing. Its striking feature is that there are no buffers at intermediate nodes. Thus packets are always moving (possibly in the 'wrong' direction), giving rise to the term 'hot potato'. They give a simple deterministic algorithm that on a n*n torus will route a random instance in 2n+O(log n) steps with high probability. They add random delays to this algorithm so that it solves the permutation routing problem on the torus in 9n steps with high probability, on every instance. On a hypercube with N=2/sup n/ nodes, they give a simple deterministic algorithm that will route a random instance in O(n) steps with high probability. Various other results are discussed.>

We study the complexity of computing the permanent on random inputs. We consider matrices drawn randomly from the space of n by n matrices with integer values between 0 and p–1, for any large enough prime p. We show that any polynomial time algorithm which computes the permanent correctly on even an exponentially small fraction of these matrices, implies the collapse of the polynomial-time hierarchy to its second level.

We characterize the power of two-prover one-round (MIP(2,1)) proof systems, showing that MIP(2,1)=NEXPTIME. However, the following intriguing question remains open: Does parallel repetition decrease the error probability of MIP(2,1) proof systems?.

The computational complexity of approximating omega (G), the size of the largest clique in a graph G, within a given factor is considered. It is shown that if certain approximation procedures exist, then EXPTIME=NEXPTIME and NP=P.>

The authors solve the two major open problems associated with noninteractive zero-knowledge proofs: how to enable polynomially many provers to prove in writing polynomially many theorems based on the basis of a single random string, and how to construct such proofs under general (rather than number-theoretic) assumptions. The constructions can be used in cryptographic applications in which the prover is restricted to polynomial time, and they are much simpler than earlier (and less capable) proposals.>

Article Free AccessComputing with unreliable information Authors: U. Feige The Weizmann Institute of Science, Rehovot, Israel and T.J. Watson and Almaden Research Centers The Weizmann Institute of Science, Rehovot, Israel and T.J. Watson and Almaden Research CentersView Profile , D. Peleg The Weizmann Institute of Science, Rehovot, Israel The Weizmann Institute of Science, Rehovot, IsraelView Profile , P. Raghavan IBM T.J. Watson Research Center, Yorktown Heights, NY IBM T.J. Watson Research Center, Yorktown Heights, NYView Profile , E. Upfal IBM Almaden Research Center, San Jose, CA, and The Weizmann Institute of Science, Rehovot, Israel IBM Almaden Research Center, San Jose, CA, and The Weizmann Institute of Science, Rehovot, IsraelView Profile Authors Info & Claims STOC '90: Proceedings of the twenty-second annual ACM symposium on Theory of ComputingApril 1990 Pages 128–137https://doi.org/10.1145/100216.100230Published:01 April 1990Publication History 46citation775DownloadsMetricsTotal Citations46Total Downloads775Last 12 Months69Last 6 weeks8 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

A two party protocol in which party A uses one of several secret witnesses to an NP assertion is witness indistinguishable if party B cannot tell which witness A is actually using.The protocol is witness hiding if by the end of the protocol B cannot compute any new witness which he did not know before the protocol began.Witness hiding is a natural security requirement, and can replace zero knowledge in many cryptographic protocols.We prove two central results: 1.Unlike zero knowledge protocols, witness indistinguishablity is preserved under arbitrary composition of protocols, including parallel execution.2. If a statement has at least two independent witnesses, then any witness indistinguishable protocol for this statement is also witness hiding.part of the paper is devoted to showing that if a protocol is WI, and if w(z) contains at least two independent witnesses, then the protocol must be WH.The WH property is a natural property which is sufficient to guarantee overall security of many cryptographic schemes (nontransitivity of proofs of knowledge, unforgeable proofs of identity etc.).It is natural to compare the concept of WH to that of zero knowledge (ZK [15]).ZK guarantees that no information whatsoever leaks during the execution of

In this paper we extend the notion of zero knowledge proofs of membership (which reveal one bit of information) to zero knowledge proofs of knowledge (which reveal no information whatsoever). After formally defining this notion, we show its relevance to identification schemes, in which parties prove their identity by demonstrating their knowledge rather than by proving the validity of assertions. We describe a novel scheme which is provably secure if factoring is difficult and whose practical implementations are about two orders of magnitude faster than RSA-based identification schemes. In the last part of the paper we consider the question of sequential versus parallel executions of zero knowledge protocols, define a new notion of “transferable information”, and prove that the parallel version of our identification scheme (which is not known to be zero knowledge) is secure since it reveals no transferable information.