The 𝓁₂² min-sum k-clustering problem is to partition an input set into clusters C_1,…,C_k to minimize ∑_{i=1}^k ∑_{p,q ∈ C_i} ‖p-q‖₂². Although 𝓁₂² min-sum k-clustering is NP-hard, it is not known whether it is NP-hard to approximate 𝓁₂² min-sum k-clustering beyond a certain factor. In this paper, we give the first hardness-of-approximation result for the 𝓁₂² min-sum k-clustering problem. We show that it is NP-hard to approximate the objective to a factor better than 1.056 and moreover, assuming a balanced variant of the Johnson Coverage Hypothesis, it is NP-hard to approximate the objective to a factor better than 1.327. We then complement our hardness result by giving a fast PTAS for 𝓁₂² min-sum k-clustering. Specifically, our algorithm runs in time O(n^{1+o(1)}d⋅ 2^{(k/ε)^O(1)}), which is the first nearly linear time algorithm for this problem. We also consider a learning-augmented setting, where the algorithm has access to an oracle that outputs a label i ∈ [k] for input point, thereby implicitly partitioning the input dataset into k clusters that induce an approximately optimal solution, up to some amount of adversarial error α ∈ [0,1/2). We give a polynomial-time algorithm that outputs a (1+γα)/(1-α)²-approximation to 𝓁₂² min-sum k-clustering, for a fixed constant γ > 0.

Since this paper is under journal submission, we publish only an extended abstract here. A full version can be found at https://arxiv.org/abs/2406.03938.

Abstract. In a 1989 paper titled “shortest paths without a map,” Papadimitriou and Yannakakis introduced an online model of searching in a weighted layered graph for a target node, while attempting to minimize the total length of the path traversed by the searcher. This problem, later called layered graph traversal, is parametrized by the maximum cardinality [Formula: see text] of a layer of the input graph. It is an online setting for dynamic programming, and it is known to be a rather general and fundamental model of online computing, which includes as special cases other acclaimed models. The deterministic competitive ratio for this problem was soon discovered to be exponential in [Formula: see text], and it is now nearly resolved: it lies between [Formula: see text] and [Formula: see text]. Regarding the randomized competitive ratio, in 1993 Ramesh proved, surprisingly, that this ratio has to be at least [Formula: see text] (for any constant [Formula: see text]). In the same paper, Ramesh also gave an [Formula: see text]-competitive randomized online algorithm. Between 1993 and the results obtained in this paper, no progress has been reported on the randomized competitive ratio of layered graph traversal. In this work we show how to apply the mirror descent framework on a carefully selected evolving metric space, and obtain an [Formula: see text]-competitive randomized online algorithm. This matches asymptotically an improvement of the aforementioned lower bound [S. Bubeck, C. Coester, and Y. Rabani, ACM Symposium on the Theory of Computing, 2023], which we announced (among other results) after the initial publication of the results here.

We consider the problem of \emph{identifying,} from statistics, a distribution of discrete random variables $X_1 \ldots,X_n$ that is a mixture of $k$ product distributions. The best previous sample complexity for $n \in O(k)$ was $(1/\zeta)^{O(k^2 \log k)}$ (under a mild separation assumption parameterized by $\zeta$). The best known lower bound was $\exp(\Omega(k))$. It is known that $n\geq 2k-1$ is necessary and sufficient for identification. We show, for any $n\geq 2k-1$, how to achieve sample complexity and run-time complexity $(1/\zeta)^{O(k)}$. We also extend the known lower bound of $e^{\Omega(k)}$ to match our upper bound across a broad range of $\zeta$. Our results are obtained by combining (a) a classic method for robust tensor decomposition, (b) a novel way of bounding the condition number of key matrices called Hadamard extensions, by studying their action only on flattened rank-1 tensors.

We study the generalized load-balancing (GLB) problem, where we are given n jobs, each of which needs to be assigned to one of m unrelated machines with processing times {pij}. Under a job assignment σ, the load of each machine i is Ψi(pi[σ]) where ψi: ℝn → ℝ≥0 is a symmetric monotone norm and pi[σ] is the n- dimensional vector {pij ·

We prove a few new lower bounds on the randomized competitive ratio for the k-server problem and other related problems, resolving some long-standing conjectures. In particular, for metrical task systems (MTS) we asympotically settle the competitive ratio and obtain the first improvement to an existential lower bound since the introduction of the model 35 years ago (in 1987).

In a 1989 paper titled “shortest paths without a map”, Papadimitriou and Yannakakis introduced an online model of searching in a weighted layered graph for a target node, while attempting to minimize the total length of the path traversed by the searcher. This problem, later called layered graph traversal, is parametrized by the maximum cardinality k of a layer of the input graph. It is an online setting for dynamic programming, and it is known to be a rather general and fundamental model of online computing, which includes as special cases other acclaimed models. The deterministic competitive ratio for this problem was soon discovered to be exponential in k, and it is now nearly resolved: it lies between $\Omega(2^{k})$ and $O(k2^{k})$. Regarding the randomized competitive ratio, in 1993 Ramesh proved, surprisingly, that this ratio has to be at least $\Omega(k^{2}/log^{1+\varepsilon}k)$ (for any constant $\varepsilon\gt0)$. In the same paper, Ramesh also gave an $O(k^{13})$-competitive randomized online algorithm. Since 1993, no progress has been reported on the randomized competitive ratio of layered graph traversal. In this work we show how to apply the mirror descent framework on a carefully selected evolving metric space, and obtain an $O(k^{2})$ competitive randomized online algorithm, nearly matching the known lower bound on the randomized competitive ratio.

In the Euclidean k-Means problem we are given a collection of n points D in an Euclidean space and a positive integer k. Our goal is to identify a collection of k points in the same space (centers) so as to minimize the sum of the squared Euclidean distances between each point in D and the closest center. This problem is known to be APX-hard and the current best approximation ratio is a primal-dual 6.357 approximation based on a standard LP for the problem [Ahmadian et al. FOCS'17, SICOMP'20]. In this note we show how a minor modification of Ahmadian et al.'s analysis leads to a slightly improved 6.12903 approximation. As a related result, we also show that the mentioned LP has integrality gap at least 16+515>1.2157.

We study two generalizations of classic clustering problems called dynamic ordered k-median and dynamic k-supplier, where the points that need clustering evolve over time, and we are allowed to move the cluster centers between consecutive time steps. In these dynamic clustering problems, the general goal is to minimize certain combinations of the service cost of points and the movement cost of centers, or to minimize one subject to some constraints on the other. We obtain a constant-factor approximation algorithm for dynamic ordered k-median under mild assumptions on the input. We give a 3-approximation for dynamic k-supplier and a multi-criteria approximation for its outlier version where some points can be discarded, when the number of time steps is two. We complement the algorithms with almost matching hardness results.

Abstract. The purpose of this note is to correct mistakes and inaccuracies in technical claims in [Y. Rabani and A. Shpilka, Explicit Construction of a Small 𝜖 -net for Linear Threshold Functions, SIAM J. Comput., 39 (2010), pp. 3501–3520]. These have no effect on the main results in the paper.

We give a constant factor polynomial time pseudo-approximation algorithm for min-sum clustering with or without outliers. The algorithm is allowed to exclude an arbitrarily small constant fraction of the points. For instance, we show how to compute a solution that clusters 98% of the input data points and pays no more than a constant factor times the optimal solution that clusters 99% of the input data points. More generally, we give the following bicriteria approximation: For any ε > 0, for any instance with n input points and for any positive integer n' ≤ n, we compute in polynomial time a clustering of at least (1-ε) n' points of cost at most a constant factor greater than the optimal cost of clustering n' points. The approximation guarantee grows with 1/(ε). Our results apply to instances of points in real space endowed with squared Euclidean distance, as well as to points in a metric space, where the number of clusters, and also the dimension if relevant, is arbitrary (part of the input, not an absolute constant).

We give an algorithm for source identification of a mixture of k product distributions on n bits. This is a fundamental problem in machine learning with many applications. Our algorithm identifies the source parameters of an identifiable mixture, given, as input, approximate values of multilinear moments (derived, for instance, from a sufficiently large sample), using $2^{O(k^2)}n^{O(k)}$ arithmetic operations. Our result is the first explicit bound on the computational complexity of source identification of such mixtures. The running time improves previous results by Feldman, O’Donnell, and Servedio (FOCS 2005) and Chen and Moitra (STOC 2019) that guaranteed only learning the mixture (without parametric identification of the source). Our analysis gives a quantitative version of a qualitative characterization of identifiable sources that is due to Tahmasebi, Motahari, and Maddah-Ali (ISIT 2018).

We study the proportional dynamics in exchange economies, where each player starts with some amount of money and a good. Every day, players bring one unit of their good and submit bids on goods they like, each good gets allocated in proportion to the bid amounts, and each seller collects the bids received. Then every player updates their bids proportionally to the contribution of each good in their utility. This dynamic models a process of learning how to bid and has been studied in a series of papers on Fisher and production markets, but not in exchange economies. Our main results are as follows: 1). For all linear utilities, the dynamic converges to market equilibrium utilities and allocations, while the bids and prices may cycle. We give a combinatorial characterization of limit cycles for prices and bids. 2). We introduce a lazy version of the dynamic, where players may save money for later, and show this converges in everything: utilities, allocations, and prices. This answers an open question about exchange markets with linear utilities, where tatonnement does not converge to market equilibria, and no natural process leading to equilibria was known for all additive utilities. We also note this dynamics represents a process where the players exchange goods throughout time (in out-of-equilibrium states), while tatonnement only explains how exchange happens in the limit.

We introduce the problem of k-chasing of convex functions, a simultaneous generalization of both the famous k-server problem in ℝd, and of the problem of chasing convex bodies and functions. Aside from fundamental interest in this general form, it has natural applications to online k-clustering problems with objectives such as k-median or k-means. We show that this problem exhibits a rich landscape of behavior. In general, if both k > 1 and d > 1 there does not exist any online algorithm with bounded competitiveness. By contrast, we exhibit a class of nicely behaved functions (which include in particular the above-mentioned clustering problems), for which we show that competitive online algorithms exist, and moreover with dimension-free competitive ratio. We also introduce a parallel question of top-k action regret minimization in the realm of online convex optimization. There, too, a much rougher landscape emerges for k > 1. While it is possible to achieve vanishing regret, unlike the top-one action case the rate of vanishing does not speed up for strongly convex functions. Moreover, vanishing regret necessitates both intractable computations and randomness. Finally we leave open whether almost dimension-free regret is achievable for k > 1 and general convex losses. As evidence that it might be possible, we prove dimension-free regret for linear losses via an information-theoretic argument.

We consider parametrized versions of metrical task systems and metrical service systems, two fundamental models of online computing, where the constrained parameter is the number of possible distinct requests $m$. Such parametrization occurs naturally in a wide range of applications. Striking examples are certain power management problems, which are modeled as metrical task systems with $m=2$. We characterize the competitive ratio in terms of the parameter $m$ for both deterministic and randomized algorithms on hierarchically separated trees. Our findings uncover a rich and unexpected picture that differs substantially from what is known or conjectured about the unparametrized versions of these problems. For metrical task systems, we show that deterministic algorithms do not exhibit any asymptotic gain beyond one-level trees (namely, uniform metric spaces), whereas randomized algorithms do not exhibit any asymptotic gain even for one-level trees. In contrast, the special case of metrical service systems (subset chasing) behaves very differently. Both deterministic and randomized algorithms exhibit gain, for $m$ sufficiently small compared to $n$, for any number of levels. Most significantly, they exhibit a large gain for uniform metric spaces and a smaller gain for two-level trees. Moreover, it turns out that in these cases (as well as in the case of metrical task systems for uniform metric spaces with $m$ being an absolute constant), deterministic algorithms are essentially as powerful as randomized algorithms. This is surprising and runs counter to the ubiquitous intuition/conjecture that, for most problems that can be modeled as metrical task systems, the randomized competitive ratio is polylogarithmic in the deterministic competitive ratio.

In many classic clustering problems, we seek to sketch a massive data set of n points (a.k.a clients) in a metric space, by segmenting them into k categories or clusters, each cluster represented concisely by a single point in the metric space (a.k.a. the cluster’s center or its facility). The goal is to find such a sketch that minimizes some objective that depends on the distances between the clients and their respective facilities (the objective is a.k.a. the service cost). Two notable examples are the k-center/k-supplier problem where the objective is to minimize the maximum distance from any client to its facility, and the k-median problem where the objective is to minimize the sum over all clients of the distance from the client to its facility. In practical applications of clustering, the data set may evolve over time, reflecting an evolution of the underlying clustering model. Thus, in such applications, a good clustering must simultaneously represent the temporal data set well, but also not change too drastically between time steps. In this paper, we initiate the study of a dynamic version of clustering problems that aims to capture these considerations. In this version there are T time steps, and in each time step t ∈ {1,2,… ,T}, the set of clients needed to be clustered may change, and we can move the k facilities between time steps. The general goal is to minimize certain combinations of the service cost and the facility movement cost, or minimize one subject to some constraints on the other. More specifically, we study two concrete problems in this framework: the Dynamic Ordered k-Median and the Dynamic k-Supplier problem. Our technical contributions are as follows: - We consider the Dynamic Ordered k-Median problem, where the objective is to minimize the weighted sum of ordered distances over all time steps, plus the total cost of moving the facilities between time steps. We present one constant-factor approximation algorithm for T = 2 and another approximation algorithm for fixed T ≥ 3. - We consider the Dynamic k-Supplier problem, where the objective is to minimize the maximum distance from any client to its facility, subject to the constraint that between time steps the maximum distance moved by any facility is no more than a given threshold. When the number of time steps T is 2, we present a simple constant factor approximation algorithm and a bi-criteria constant factor approximation algorithm for the outlier version, where some of the clients can be discarded. We also show that it is NP-hard to approximate the problem with any factor for T ≥ 3.

Osborne's iteration is a method for balancing $n\times n$ matrices which is widely used in linear algebra packages, as balancing preserves eigenvalues and stabilizes their numeral computation. The iteration can be implemented in any norm over $\mathbb{R}^n$, but it is normally used in the $L_2$ norm. The choice of norm not only affects the desired balance condition, but also defines the iterated balancing step itself. In this paper we focus on Osborne's iteration in any $L_p$ norm, where $p < \infty$. We design a specific implementation of Osborne's iteration in any $L_p$ norm that converges to a strictly $ε$-balanced matrix in $\tilde{O}(ε^{-2}n^{9} K)$ iterations, where $K$ measures, roughly, the {\em number of bits} required to represent the entries of the input matrix. This is the first result that proves that Osborne's iteration in the $L_2$ norm (or any $L_p$ norm, $p < \infty$) strictly balances matrices in polynomial time. This is a substantial improvement over our recent result (in SODA 2017) that showed weak balancing in $L_p$ norms. Previously, Schulman and Sinclair (STOC 2015) showed strong balancing of Osborne's iteration in the $L_\infty$ norm. Their result does not imply any bounds on strict balancing in other norms.

We consider the problem of embedding a finite set of points $\{x_1, \ldots, x_n\} \in \mathbb{R}^d$ that satisfy $\ell_2^2$ triangle inequalities into $\ell_1$, when the points are approximately low-dimensional. Goemans (unpublished, appears in a work of [Magen and Moharammi, 2008]) showed that such points residing in \emph{exactly} $d$ dimensions can be embedded into $\ell_1$ with distortion at most $\sqrt{d}$. We prove the following robust analogue of this statement: if there exists a $r$-dimensional subspace $Π$ such that the projections onto this subspace satisfy $\sum_{i,j \in [n]}\Vert Πx_i - Πx_j \Vert _2^2 \geq Ω(1) \sum_{i,j \in [n]}\Vert x_i - x_j \Vert _2^2$, then there is an embedding of the points into $\ell_1$ with $O(\sqrt{r})$ average distortion. A consequence of this result is that the integrality gap of the well-known Goemans-Linial SDP relaxation for the Uniform Sparsest Cut problem is $O(\sqrt{r})$ on graphs $G$ whose $r$-th smallest normalized eigenvalue of the Laplacian satisfies $λ_r(G)/n \geq Ω(1)Φ_{SDP} (G)$. Our result improves upon the previously known bound of $O(r)$ on the average distortion, and the integrality gap of the Goemans-Linial SDP under the same preconditions, proven in the previous works of [Deshpande and Venkat, 2014] and [Deshpande, Harsha and Venkat, 2016].

In both general equilibrium theory and game theory, the dominant mathematical models rest on a fully rational solution concept in which every player's action is a best-response to the actions of the other players. In both theories there is less agreement on suitable out- of-equilibrium modeling, but one attractive approach is the level k model in which a level 0 player adopts a very simple response to current conditions, a level 1 player best-responds to a model in which others take level 0 actions, and so forth. (This is analogous to k-ply exploration of game trees in AI, and to receding-horizon control in control theory.) If players have deterministic mental models with this kind of finite-level response, there is obviously no way their mental models can all be consistent. Nevertheless, there is experimental evidence that people act this way in many situations, motivating the question of what the dynamics of such interactions lead to. We address the problem of out-of-equilibrium price dynamics in the setting of Fisher markets. We develop a general framework in which sellers have (a) a set of atomic price update rules which are simple responses to a price vector; (b) a belief-formation procedure that simulates actions of other sellers (themselves using the atomic price updates) to some finite horizon in the future. In this framework, sellers use an atomic price update rule to respond to a price vector they generate with the belief formation procedure. The framework is general and allows sellers to have inconsistent and time- varying beliefs about each other. Under certain assumptions on the atomic update rules, we show that despite the inconsistent and time-varying nature of beliefs, the market converges to a unique equilibrium. (If the price updates are driven by weak-gross substitutes demands, this is the same equilibrium point predicted by those demands.) This result holds for both synchronous and asynchronous discrete-time updates. Moreover, the result is computationally feasible in the sense that the convergence rate is linear, i.e., the distance to equilibrium decays exponentially fast. To the best of our knowledge, this is the first result that demonstrates, in Fisher markets, convergence at any rate for dynamics driven by a plausible model of seller incentives. We then specialize our results to Fisher markets with elastic demands (a further special case corresponds to demand generated by buyers with constant elasticity of substitution (CES) utilities, in the weak gross substitutes (WGS) regime) and show that the atomic update rule in which a seller uses the best-response (=profit- maximizing) update given the prices of all other sellers, satisfies the assumptions required on atomic price update rules in our framework. We can even characterize the convergence rate (as a function of elasticity parameters of the demand function). Our results apply also to settings where, to the best of our knowledge, there exists no previous demonstration of efficient convergence of any discrete dynamic of price updates. Even for the simple case of (level 0) best- response dynamics, our result is the first to demonstrate a linear rate of convergence.

We study an iterative matrix conditioning algorithm due to Osborne (1960). The goal of the algorithm is to convert a square matrix into a balanced matrix where every row and corresponding column have the same norm. The original algorithm was proposed for balancing rows and columns in the L2 norm, and it works by iterating over balancing a row-column pair in fixed round-robin order. Variants of the algorithm for other norms have been heavily studied and are implemented as standard preconditioners in many numerical linear algebra packages. Recently, Schulman and Sinclair (2015), in a first result of its kind for any norm, analyzed the rate of convergence of a variant of Osborne's algorithm that uses the L∞ norm and a different order of choosing row-column pairs. In this paper we study matrix balancing in the L1 norm and other Lp norms. We show the following results for any matrix , resolving in particular a main open problem mentioned by Schulman and Sinclair. 1. We analyze the iteration for the L1 norm under a greedy order of balancing. We show that it converges to an ∊-balanced matrix in K = O(min{ ∊−2 log w, ∊−1n3/2 log(w / ∊)}) iterations that cost a total of O(m + Kn log n) arithmetic operations over O(n log(w/∊))-bit numbers. Here m is the number of non-zero entries of A, and w =∑i,j |aij|/amin with amin = min{|aij| : aj ≠ 0}. 2. We show that the original round-robin implementation converges to an ∊ -balanced matrix in O(∊−2n2 log w) iterations totaling O(∊−2mn log w) arithmetic operations over O(nlog(w/∊))-bit numbers. 3. We show that a random implementation of the iteration converges to an ∊ -balanced matrix in O(∊−2 log w) iterations using O(m + ∊−2n log w) arithmetic operations over O(log(wn/∊))-bit numbers. 4. We demonstrate a lower bound of on the convergence rate of any implementation of the iteration. 5. We observe, through a known trivial reduction, that our results for L1 balancing apply to any Lp norm for all finite p, at the cost of increasing the number of iterations by only a factor of p. We note that our techniques are very different from those used by Schulman and Sinclair.

No abstract available.

We study the problem of learning from unlabeled samples very general statistical mixture models on large finite sets. Specifically, the model to be learned, mix, is a probability distribution over probability distributions p, where each such p is a probability distribution over [n] = {1,2,...,n}. When we sample from mix, we do not observe p directly, but only indirectly and in very noisy fashion, by sampling from [n] repeatedly, independently K times from the distribution p. The problem is to infer mix to high accuracy in transportation (earthmover) distance.

We design and analyze an online reordering buffer management algorithm with improved O (log k /log log k ) competitive ratio for nonuniform costs, where k is the buffer size. This improves on the best previous result (even for uniform costs) of Englert and Westermann (2005) giving O (log k ) competitive ratio, which was also the best (offline) polynomial time approximation guarantee for this problem. Our analysis is based on an intricate dual fitting argument using a linear programming relaxation for the problem that we introduce in this article.

We give an algorithm for learning a mixture of unstructured distributions. This problem arises in various unsupervised learning scenarios, for example in learning topic models from a corpus of documents spanning several topics. We show how to learn the constituents of a mixture of k arbitrary distributions over a large discrete domain [n]={1, 2, ...,n} and the mixture weights, using O(n polylog n) samples. (In the topic-model learning setting, the mixture constituents correspond to the topic distributions.)

We give an Õ(log log k)-competitive randomized online algorithm for reordering buffer management, where k is the buffer size. Our bound matches the lower bound of Adamaszek et al. (STOC 2011). Our algorithm has two stages which are executed online in parallel. The first stage computes deterministically a feasible fractional solution to an LP relaxation for reordering buffer management. The second stage "rounds" using randomness the fractional solution. The first stage is based on the online primal-dual schema, combined with a dual fitting charging scheme. As primal-dual steps and dual fitting steps are interleaved and in some sense conflicting, combining them is challenging. We also note that we apply the primal-dual schema to a relaxation with mixed packing and covering constraints. The first stage produces a fractional LP solution with cost within a factor of Õ(log log k) of the optimal LP cost. The second stage is an online algorithm that converts any LP solution to an integral solution, while increasing the cost by a constant factor. This stage generalizes recent results that gave a similar approximation guarantee using an offline rounding algorithm.

In the reordering buffer management problem (RBM) a sequence of n colored items enters a buffer with limited capacity k. When the buffer is full, one item is removed to the output sequence, making room for the next input item. This step is repeated until the input sequence is exhausted and the buffer is empty. The objective is to find a sequence of removals that minimizes the total number of color changes in the output sequence. The problem formalizes numerous applications in computer and production systems, and is known to be NP-hard. We give the first constant factor approximation guarantee for RBM. Our algorithm is based on an intricate “rounding” of the solution to an LP relaxation for RBM, so it also establishes a constant upper bound on the integrality gap of this relaxation. Our results improve upon the best previous bound of O(√log k) of Adamaszek et al. (STOC 2011) that used different methods and gave an online algorithm. Our constant factor approximation beats the super-constant lower bounds on the competitive ratio given by Adamaszek et al. This is the first demonstration of a polynomial time offline algorithm for RBM that is provably better than any online algorithm.

We discuss recent results giving algorithms for learning mixtures of unstructured distributions.

We investigate variants of Lloyd's heuristic for clustering high-dimensional data in an attempt to explain its popularity (a half century after its introduction) among practitioners, and in order to suggest improvements in its application. We propose and justify aclusterabilitycriterion for data sets. We present variants of Lloyd's heuristic that quickly lead to provably near-optimal clustering solutions when applied to well-clusterable instances. This is the first performance guarantee for a variant of Lloyd's heuristic. The provision of a guarantee on output quality does not come at the expense of speed: some of our algorithms are candidates for beingfaster in practicethan currently used variants of Lloyd's method. In addition, our other algorithms are faster on well-clusterable instances than recently proposed approximation algorithms, while maintaining similar guarantees on clustering quality. Our main algorithmic contribution is a novel probabilistic seeding process for the starting configuration of a Lloyd-type iteration.

Motivated by applications in combinatorial optimization, we study the extent to which the global properties of a metric space, and especially its embeddability into $\ell_1$ with low distortion, are determined by the properties of its small subspaces. We establish both upper and lower bounds on the distortion of embedding locally constrained metrics into various target spaces. Other aspects of locally constrained metrics are studied as well, in particular, how far are those metrics from general metrics.

We construct a small set of explicit linear transformations mapping $\mathbb{R}^n$ to $\mathbb{R}^t$, where $t=O(\log (\gamma^{-1}) \epsilon^{-2})$, such that the $L_2$ norm of any vector in $\mathbb{R}^n$ is distorted by at most $1\pm \epsilon$ in at least a fraction of $1 - \gamma$ of the transformations in the set. Albeit the tradeoff between the size of the set and the success probability is suboptimal compared with probabilistic arguments, we nevertheless are able to apply our construction to a number of problems. In particular, we use it to construct an $\epsilon$-sample (or pseudorandom generator) for linear threshold functions on $\mathbb{S}^{n-1}$ for $\epsilon = o(1)$. We also use it to construct an $\epsilon$-sample for spherical digons in $\mathbb{S}^{n-1}$ for $\epsilon = o(1)$. This construction leads to an efficient oblivious derandomization of the Goemans–Williamson Max-Cut algorithm and similar approximation algorithms (i.e., we construct a small set of hyperplanes such that for any instance we can choose one of them to generate a good solution). Our technique for constructing an $\epsilon$-sample for linear threshold functions on the sphere is considerably different than previous techniques that rely on k-wise independent sample spaces.

This paper studies the ``explanation problem'' for tree- and linearly-ordered array data, a problem motivated by database applications and recently solved for the one-dimensional tree-ordered case. In this paper, one is given a matrix A=(a_{ij}) whose rows and columns have semantics: special subsets of the rows and special subsets of the columns are meaningful, others are not. A submatrix in A is said to be meaningful if and only if it is the cross product of a meaningful row subset and a meaningful column subset, in which case we call it an ``allowed rectangle.'' The goal is to ``explain'' A as a sparse sum of weighted allowed rectangles. Specifically, we wish to find as few weighted allowed rectangles as possible such that, for all i,j, a_ij equals the sum of the weights of all rectangles which include cell (i,j). In this paper we consider the natural cases in which the matrix dimensions are tree-ordered or linearly-ordered. In the tree-ordered case, we are given a rooted tree $T_1$ whose leaves are the rows of $A$ and another, $T_2$, whose leaves are the columns. Nodes of the trees correspond in an obvious way to the sets of their leaf descendants. In the linearly-ordered case, a set of rows or columns is meaningful if and only if it is contiguous. For tree-ordered data, we prove the explanation problem NP-Hard and give a randomized $2$-approximation algorithm for it. For linearly-ordered data, we prove the explanation problem NP-Har and give a $2.56$-approximation algorithm. To our knowledge, these are the first results for the problem of sparsely and exactly representing matrices by weighted rectangles.

We study the problem of finding a most profitable subset of n given tasks, each with a given start and finish time as well as profit and resource requirement, that at no time exceeds the quantity B of available resource. We show that this NP-hard Resource Allocation problem can be (1/2 − ε)-approximated in randomized polynomial time, which improves upon earlier approximation results.

We design and analyze an on-line reordering buffer management algorithm with improved competitive ratio for non-uniform costs, where k is the buffer size. This improves on the best previous result (even for uniform costs) of Englert and Westermann (ICALP 2005) giving O(log k) competitive ratio, which was also the best (off-line) polynomial time approximation guarantee for this problem. Our analysis is based on an intricate dual fitting argument using a linear programming relaxation for the problem that we introduce in this paper.

We study bargaining networks, discussed in a recent paper of Kleinberg and Tardos [KT08], from the perspective of cooperative game theory. In particular we examine three solution concepts, the nucleolus, the core center and the core median. All solution concepts define unique solutions, so they provide testable predictions. We define a new monotonicity property that is a natural axiom of any bargaining game solution, and we prove that all three of them satisfy this monotonicity property. This is actually in contrast to the conventional wisdom for general cooperative games that monotonicity and the core condition (which is a basic property that all three of them satisfy) are incompatible with each other. Our proofs are based on a primal-dual argument (for the nucleolus) and on the FKG inequality (for the core center and the core median). We further observe some qualitative differences between the solution concepts. In particular, there are cases where a strict version of our monotonicity property is a natural axiom, but only the core center and the core median satisfy it. On the other hand, the nucleolus is easy to compute, whereas computing the core center or the core median is #P-hard (yet it can be approximated in polynomial time).

We give explicit constructions of $\epsilon$-nets for linear threshold functions on the binary cube and on the unit sphere. The size of the constructed nets is polynomial in the dimension n and in $\frac{1}{\epsilon}$. To the best of our knowledge no such constructions were previously known. Our results match, up to the exponent of the polynomial, the bounds that are achieved by probabilistic arguments. As a corollary we also construct subsets of the binary cube that have size polynomial in n and a covering radius of $\frac{n}{2}-c\sqrt{n\log n}$ for any constant c. This improves upon the well-known construction of dual BCH codes that guarantee only a covering radius of $\frac{n}{2}-c\sqrt{n}$.

We give explicit constructions of epsilon nets for linear threshold functions on the binary cube and on the unit sphere. The size of the constructed nets is polynomial in the dimension n and in 1/ε. To the best of our knowledge no such constructions were previously known. Our results match, up to the exponent of the polynomial, the bounds that are achieved by probabilistic arguments. As a corollary we also construct subsets of the binary cube that have size polynomial in n and covering radius of n/2 - c√{n log n}, for any constant c. This improves upon the well known construction of dual BCH codes that only guarantee covering radius of n/2 - c√n.

We study the fundamental classification problems 0-Extension and Metric Labeling. A generalization of Multiway Cut, 0-Extension is closely related to partitioning problems in graph theory and to Lipschitz extensions in Banach spaces; its generalization Metric Labeling is motivated by applications in computer vision. Researchers had proposed using earthmover metrics to get polynomial–time-solvable relaxations for these problems. A conjecture that has attracted much attention recently is that the integrality ratio for these relaxations is constant. We prove (1) that the integrality ratio of the earthmover relaxation for Metric Labeling is $\Omega(\log k)$ (which is asymptotically tight), k being the number of labels, whereas the best previous lower bound on the integrality ratio was only constant; (2) that the integrality ratio of the earthmover relaxation for 0-Extension is $\Omega(\sqrt{\log k})$, k being the number of terminals (it was known to be $O((\log k)/\log\log k)$), whereas the best previous lower bound was only constant; (3) that for no $\epsilon>0$ is there a polynomial-time $O((\log n)^{1/4-\epsilon})$-approximation algorithm for 0-Extension, n being the number of vertices, unless NP$\subseteq$DTIME$(n^{\mathrm{poly}(\log n)})$, whereas the strongest inapproximability result known before was only MAX SNP-hardness; and (4) that there is a polynomial-time approximation algorithm for 0-Extension with performance ratio $O(\sqrt{\mathrm{diam}(d)})$, where $\mathrm{diam}(d)$ is the ratio of the largest to smallest nonzero distances in the terminal metric.

We initiate the study of the minimum distortion problem: Given as input two n-point metric spaces, find a bijection between them with minimum distortion. This is an abstraction of certain geometric problems in shape and image matching and is also a natural variation and extension of the fundamental problems of graph isomorphism and bandwidth. Our focus is on algorithms that find an optimal (or near-optimal) bijection when the distortion is fairly small. We present a polynomial time algorithm that finds an optimal bijection between two line metrics, provided the distortion is less than $5+2\sqrt{6}\approx9.9$. We also give a parameterized polynomial time algorithm that finds an optimal bijection between an arbitrary unweighted graph metric and a bounded-degree tree metric.

We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into $L_1$. In particular, we show that for every $n\ge1$, there is an n-point metric space of negative type that requires a distortion of $\Omega(\log\log n)$ for such an embedding, implying the same lower bound on the integrality gap of a well-known semidefinite programming relaxation for sparsest cut. This result builds upon and improves the recent lower bound of $(\log\log n)^{1/6-o(1)}$ due to Khot and Vishnoi [The unique games conjecture, integrality gap for cut problems and the embeddability of negative type metrics into $l_1$, in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 2005, pp. 53–62]. We also show that embedding the edit distance metric on $\{0,1\}^n$ into $L_1$ requires a distortion of $\Omega(\log n)$. This result improves a very recent $(\log n)^{1/2-o(1)}$ lower bound by Khot and Naor [Nonembeddability theorems via Fourier analysis, in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 2005, pp. 101–112].

We consider an optimization problem consisting of an undirected graph, with cost and profit functions defined on all vertices. The goal is to find a connected subset of vertices with maximum total profit, whose total cost does not exceed a given budget. The best result known prior to this work guaranteed a (2, O (log n )) bicriteria approximation, that is, the solution's profit is at least a fraction of 1/ O (log n ) of an optimum solution respecting the budget, while its cost is at most twice the given budget. We improve these results and present a bicriteria tradeoff that, given any ε ∈ (0,1], guarantees a (1 + ϵ, O (1/ε log n ))-approximation.

Systems with automatic feedback control may consist of several remote devices, connected only by unreliable communication channels. It is necessary in these conditions to have a method for accurate, real-time state estimation in the presence of channel noise. This problem is addressed, for the case of polynomial-growth-rate state spaces, through a new type of error-correcting code that is online and computationally efficient. This solution establishes a constructive analog, for some applications in estimation and control, of the Shannon coding theorem.

We show that {0, 1} d endowed with edit distance embeds into ℓ 1 with distortion 2 O (√log d log log d ). We further show efficient implementation of the embedding that yield solutions to various computational problems involving edit distance. These include sketching, communication complexity, nearest neighbor search. For all these problems, we improve upon previous bounds.

We consider a class of optimization problems where the input is an undirected graph with two weight functions defined for each node, namely the node's profit and its cost. The goal is to find a connected set of nodes of low cost and high profit. We present approximation algorithms for three natural optimization criteria that arise in this context, all of which are NP-hard. The budget problem asks for maximizing the profit of the set subject to a budget constraint on its cost. The quota problem requires minimizing the cost of the set subject to a quota constraint on its profit. Finally, the prize collecting problem calls for minimizing the cost of the set plus the profit (here interpreted as a penalty) of the complement set. For all three problems, our algorithms give an approximation guarantee of $O(\log n)$, where n is the number of nodes. To the best of our knowledge, these are the first approximation results for the quota problem and for the prize collecting problem, both of which are at least as hard to approximate as the set cover. For the budget problem, our results improve on a previous $O(\log^2 n)$ result of Guha et al. Our methods involve new theorems relating tree packings to (node) cut conditions. We also show similar theorems (with better bounds) using edge cut conditions. These imply bounds for the analogous budget and quota problems with edge costs which are comparable to known (constant factor) bounds.

We investigate variants of Lloyd's heuristic for clustering high dimensional data in an attempt to explain its popularity (a half century after its introduction) among practitioners, and in order to suggest improvements in its application. We propose and justify a clusterability criterion for data sets. We present variants of Lloyd's heuristic that quickly lead to provably near-optimal clustering solutions when applied to well-clusterable instances. This is the first performance guarantee for a variant of Lloyd's heuristic. The provision of a guarantee on output quality does not come at the expense of speed: some of our algorithms are candidates for being faster in practice than currently used variants of Lloyd's method. In addition, our other algorithms are faster on well-clusterable instances than recently proposed approximation algorithms, while maintaining similar guarantees on clustering quality. Our main algorithmic contribution is a novel probabilistic seeding process for the starting configuration of a Lloyd-type iteration

We study the fundamental classification problems O-EXTENSION and METRIC LABELING. MINIMUM WEIGHT TRIANGULATION is closely related to partitioning problems in graph theory and to Lipschitz extensions in Banach spaces; its generalization METRIC LABELING is motivated by applications in computer vision. Researchers had proposed using earthmover metrics to get polynomial time-solvable relaxations for these problems. A conjecture that has attracted much attention recently is that the integrality ratio for these relaxations is constant.We prove

We show that the Multicut, Sparsest-Cut, and Min-2CNF ≡ Deletion problems are NP-hard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot (2002). A quantitatively stronger version of the conjecture implies an inapproximability factor of $$\Omega(\sqrt{\log \log n}).$$

We show that the MULTICUT, SPARSEST-CUT, and MIN-2CNF/spl equiv/DELETION problems are NP-hard to approximate within every constant factor, assuming the unique games conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of /spl Omega/(log log n).

In many control-theory applications one can classify all possible states of the device by an infinite state graph with polynomially-growing expansion. In order for a controller to control or estimate the state of such a device, it must receive reliable communications from its sensors; if there is channel noise, the encoding task is subject to a stringent real-time constraint. We show a constructive on-line error correcting code that works for this class of applications. Our code is computationally efficient and enables on-line estimation and control in the presence of channel noise. It establishes a constructive (and optimal-within-constants) analog, for control applications, of the Shannon coding theorem.

We show that 0,1d endowed with edit distance embeds into l1 with distortion 2O(√log dlog log d). We further show efficient implementations of the embedding that yield solutions to various computational problems involving edit distance. These include sketching, communication complexity, nearest neighbor search. For all these problems, we improve upon previous bounds.

We initiate the study of the minimum distortion problem: given as input two n-point metric spaces, find a bijection between them with minimum distortion. This is an abstraction of certain geometric problems in shape and image matching, and is also a natural variation and extension of the fundamental problems of graph isomorphism and bandwidth. Our focus is on algorithms that find an optimal (or near-optimal) bijection when the distortion is fairly small. We present a polynomial time algorithm that finds an optimal bijection between two line metrics, provided the distortion is less than 3+2√2. We also give a parameterized polynomial time algorithm that finds an optimal bijection between an arbitrary unweighted graph metric and a bounded-degree tree metric.

In the 0-extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge's weight and the distance between the terminals to which its endpoints are assigned. This problem generalizes the multiway cut problem of Dahlhaus et al. [SIAM J. Comput.}, 23 (1994), pp. 864--894] and is closely related to the metric labeling problem introduced by Kleinberg and Tardos [Proceedings of the 40th IEEE Annual Symposium on Foundations of Computer Science, New York, 1999, pp. 14--23]. We present approximation algorithms for {\sc 0-Extension}. In arbitrary graphs, we present a O(log k)-approximation algorithm, k being the number of terminals. We also give O(1)-approximation guarantees for weighted planar graphs. Our results are based on a natural metric relaxation of the problem previously considered by Karzanov [European J. Combin., 19 (1998), pp. 71--101]. It is similar in flavor to the linear programming relaxation of Garg, Vazirani, and Yannakakis [SIAM J. Comput.}, 25 (1996), pp. 235--251] for the multicut problem, and similar to relaxations for other graph partitioning problems. We prove that the integrality ratio of the metric relaxation is at least $c \sqrt{\lg k}$ for a positive c for infinitely many k. Our results improve some of the results of Kleinberg and Tardos, and they further our understanding on how to use metric relaxations.

Given a database of n points in (0,1)d, the partial match problem is: In response to a query x in (0, 1, *)d, find a database point y such that for every i whenever xi ≠ *, we have xi = yi. In this paper we show randomized lower bounds in the cell-probe model for this well-studied problem[18, 11, 19, 16, 4, 6 ].Our lower bounds follow from a two-party asymmetric randomized communication complexity near-optimal lower bound for this problem, where we show that either Alice has to send Ω(d log n) bits or Bob has to send Ω(n1 - o(1)) bits. When applied to the cell-probe model, it means that if the number of cells is restricted to be poly(n, d) where each cell is of size poly(log n, d), then Ω(d/log2 n) probes are needed. This is an exponential improvement over the previously known lower bounds for this problem[16, 4].

Let k be a fixed integer. We consider the problem of partitioning an input set of points endowed with a distance function into k clusters. We give polynomial time approximation schemes for the following three clustering problems: Metric k-Clustering, l 22k-Clustering, and l22k-Median. In the k-Clustering problem, the objective is to minimize the sum of all intra-cluster distances. In the k-Median problem, the goal is to minimize the sum of distances from points in a cluster to the (best choice of) cluster center. In metric instances, the input distance function is a metric. In l 22 instances, the points are in R d and the distance between two points x,y is measured by x−y22 (notice that (R d, ⋅ 22 is not a metric space). For the first two problems, our results are the first polynomial time approximation schemes. For the third problem, the running time of our algorithms is a vast improvement over previous work.

The Johnson--Lindenstrauss lemma states that n points in a high-dimensional Hilbert space can be embedded with small distortion of the distances into an O (log n ) dimensional space by applying a random linear transformation. We show that similar (though weaker) properties hold for certain random linear transformations over the Hamming cube. We use these transformations to solve NP-hard clustering problems in the cube as well as in geometric settings.More specifically, we address the following clustering problem. Given n points in a larger set (e.g., ℝ d ) endowed with a distance function (e.g., L 2 distance), we would like to partition the data set into k disjoint clusters, each with a "cluster center," so as to minimize the sum over all data points of the distance between the point and the center of the cluster containing the point. The problem is provably NP-hard in some high-dimensional geometric settings, even for k = 2. We give polynomial-time approximation schemes for this problem in several settings, including the binary cube {0,1} d with Hamming distance, and ℝ d either with L 1 distance, or with L 2 distance, or with the square of L 2 distance. In all these settings, the best previous results were constant factor approximation guarantees.We note that our problem is similar in flavor to the k -median problem (and the related facility location problem), which has been considered in graph-theoretic and fixed dimensional geometric settings, where it becomes hard when k is part of the input. In contrast, we study the problem when k is fixed, but the dimension is part of the input.

The seminal paper of F.T. Leighton and S. Rao (1988) and subsequent papers presented approximate min-max theorems relating multicommodity flow values and cut capacities in undirected networks, developed the divide-and-conquer method for designing approximation algorithms, and generated novel tools for utilizing linear programming relaxations. Yet, despite persistent research efforts, these achievements could not be extended to directed networks, excluding a few cases that are "symmetric" and therefore similar to undirected networks. The paper is an attempt to remedy the situation. We consider the problem of finding a minimum multicut in a directed multicommodity flow network, and give the first nontrivial upper bounds on the maxflow-to-min multicut ratio. Our results are algorithmic, demonstrating nontrivial approximation guarantees.

The authors consider the job interval selection problem (JISP), a simple scheduling model with a rich history and numerous applications. Special cases of this problem include the so-called real-time scheduling problem (also known as the throughput maximization problem) in single and multiple machine environments. In these special cases we have to maximize the number of jobs scheduled between their release date and deadline (preemption is not allowed). Even the single machine case is NP-hard. The unrelated machines case, as well as other special cases of JISP, are MAX SNP-hard. A simple greedy algorithm gives a 2-approximation for JISP. Despite many efforts, this was the best approximation guarantee known, even for throughput maximization on a single machine. The authors break this barrier and show an approximation guarantee of less than 1.582 for arbitrary instances of JISP. For some special cases, we show better results. Our methods can be used to give improved bounds for some related resource allocation problems that were considered recently in the literature.

We consider a class of optimization problems, where the input is an undirected graph with two weight functions defined for each node, namely the node's profit and its cost. The goal is to find a connected set of nodes of low cost and high profit. We present approximation algorithms for three natural optimization criteria that arise in this context, all of which are NP-hard. The budget problem asks for maximizing the profit of the set subject to a budget constraint on its cost. The quota problem requires minimizing the cost of the set subject to a quota constraint on its profit. Finally, the prize collecting problem calls for minimizing the cost of the set plus the profit (here interpreted as a penalty) of the complement set. For all three problems, our algorithms give an approximation guarantee of O(\log n), where n is the number of nodes. To the best of our knowledge, these are the first approximation results for the quota problem and for the prize collecting problem, both of which are at least as hard to approximate as set cover. For the budget problem, our results improve on a previous O(\log^2 n) result of Guha, Moss, Naor, and Schieber. Our methods involve new theorems relating tree packings to (node) cut conditions. We also show similar theorems (with better bounds) using edge cut conditions. These imply bounds for the analogous budget and quota problems with edge costs which are comparable to known (constant factor) bounds.

We deal with the problem of clustering data points. Given n points in a larger set (for example, R/sup d/) endowed with a distance function (for example, L/sup 2/ distance), we would like to partition the data set into k disjoint clusters, each with a "cluster center", so as to minimize the sum over all data points of the distance between the point and the center of the cluster containing the point. The problem is provably NP-hard in some high dimensional geometric settings, even for k=2. We give polynomial time approximation schemes for this problem in several settings, including the binary cube (0, 1)/sup d/ with Hamming distance, and R/sup d/ either with L/sup 1/ distance, or with L/sup 2/ distance, or with the square of L/sup 2/ distance. In all these settings, the best previous results were constant factor approximation guarantees. We note that our problem is similar in flavor to the k-median problem (and the related facility location problem), which has been considered in graph-theoretic and fixed dimensional geometric settings, where it becomes hard when k is part of the input. In contrast, we study the problem when k is fixed, but the dimension is part of the input. Our algorithms are based on a dimension reduction construction for the Hamming cube, which may be of independent interest.

We prove new lower bounds for nearest neighbor search in the Hamming cube.Our lower bounds are for randomized, two-sided error, algorithms in Yao's cell probe model.Our bounds are in the form of a tradeoff among the number of cells, the size of a cell, and the search time.For example, suppose we are searching among n points in the d dimensional cube, we use poly(n, d) cells, each containing poly(d, log n) bits.We get a lower bound of ~2(d/log n) on the search time, a significant improvement over the recent bound of ft(log d) of Borodin et al.This should be contrasted with the upper bound of O(loglogd) for approximate search (and O(1) for a decision version of the problem; our lower bounds hold in that case).By previous results, the bounds for the cube imply similar bounds for nearest neighbor search in high dimensional Euclidean space, and for other geometric problems.

A lower bound of $\Omega(\sqrt{\log k / \log \log k})$ is proved for the competitive ratio of randomized algorithms for the k-server problem against an oblivious adversary. The bound holds for arbitrary metric spaces (having at least k+1 points) and provides a new lower bound for the metrical task system problem as well. This improves the previous best lower bound of $\Omega(\log \log k)$ for arbitrary metric spaces [H.J. Karloff, Y. Rabani, and Y. Ravid, SIAM J. Comput., 23 (1994), pp. 293--312] and more closely approaches the conjectured lower bound of $\Omega(\log k)$. For the server problem on k+1 equally spaced points on a line, which corresponds to a natural motion-planning problem, a lower bound of $\Omega(\frac{\log k}{\log \log k})$ is obtained. The results are deduced from a general decomposition theorem for a simpler version of both the k-server and the metrical task system problems, called the "pursuit-evasion game." It is shown that if a metric space $\cal M$ can be decomposed into two spaces $\cal M_L$ and $\cal M_R$ such that the distance between them is sufficiently large compared to their diameter, then the competitive ratio for this game on $\cal M$ can be expressed nearly exactly in terms of the ratios on each of the two subspaces. This yields a divide-and-conquer approach to bounding the competitive ratio of a space.

In this paper, we undertake the first study of statistical multiplexing from the perspective of approximation algorithms. The basic issue underlying statistical multiplexing is the following: in high-speed networks, individual connections (i.e., communication sessions) are very bursty, with transmission rates that vary greatly over time. As such, the problem of packing multiple connections together on a link becomes more subtle than in the case when each connection is assumed to have a fixed demand. We consider one of the most commonly studied models in this domain: that of two communicating nodes connected by a set of parallel edges, where the rate of each connection between them is a random variable. We consider three related problems: (1) stochastic load balancing, (2) stochastic bin-packing, and (3) stochastic knapsack. In the first problem the number of links is given and we want to minimize the expected value of the maximum load. In the other two problems the link capacity and an allowed overflow probabilityp are given, and the objective is to assign connections to links, so that the probability that the load of a link exceeds the link capacity is at most p. In bin-packing we need to assign each connection to a link using as few links as possible. In the knapsack problem each connection has a value, and we have only one link. The problem is to accept as many connections as possible. For the stochastic load balancing problem we give an O(1)-approximation algorithm for arbitrary random variables. For the other two problems we have algorithms restricted to on-off sources (the most common special case studied in the statistical multiplexing literature), with a somewhat weaker range of performance guarantees. A standard approach that has emerged for dealing with probabilistic resource requirements is the notion of effective bandwidth---this is a means of associating a fixed demand with a bursty connection that "represents" its distribution as closely as possible. Our approximation algorithms make use of the standard definition of effective bandwidth and also a new one that we introduce; the performance guarantees are based on new results showing that a combination of these measures can be used to provide bounds on the optimal solution.

We address the problem of designing data structures that allow efficient search for approximate nearest neighbors. More specifically, given a database consisting of a set of vectors in some high dimensional Euclidean space, we want to construct a space-efficient data structure that would allow us to search, given a query vector, for the closest or nearly closest vector in the database. We also address this problem when distances are measured by the L 1 norm and in the Hamming cube. Significantly improving and extending recent results of Kleinberg, we construct data structures whose size is polynomial in the size of the database and search algorithms that run in time nearly linear or nearly quadratic in the dimension. (Depending on the case, the extra factors are polylogarithmic in the size of the database.)

We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria interacts in a complex fashion with the optimization of network utilization and throughput; in this work, we undertake an investigation of this relationship through the framework of approximation algorithms. In this work we consider the problem of selecting paths for routing so as to provide a bandwidth allocation that is as fair as possible (in the max-min sense). We obtain the first approximation algorithms for this basic optimization problem, for single-source unsplittable routings in an arbitrary directed graph. Special cases of our model include several fundamental load balancing problems, endowing them with a natural fairness criterion to which our approach can be applied. Our results form an interesting counterpart to the work of Megiddo (1974), who considered max-min fairness for single-source fractional flow. The optimization problems in our setting become NP-complete, and require the development of new techniques for relating fractional relaxations of routing to the equilibrium constraints imposed by the fairness criterion.

IntroductionThe cw.w of dimensionality describes the phenomenon whereby (in spite of extensive and continuing research) for various geometric search problems we only have algorithms with performance that grows exponentially in the dimension.Recent results [31,30, 331 show that in some sense it is possible to avoid the curse of dimensionality for the approximate nearest neighbor search problem.But must the exact nearest neighbor search problem suffer this curse?We provide some evidence in support of the curse.Specifically we investigate the exact nearest neighbor search problem and the related problem of exact partial match within the asymmetric communication model first used by Miltersen [36] to study data structure problems.We derive non-trivial asymptotic lower bounds for the exact problem that stand in contrast to known algorithms for approximate nearest neighbor search.Background.

One of the central problems in information retrieval, data mining, computational biology, statistical analysis, computer vision, geographic analysis, pattern recognition, distributed protocols is the question of classification of data according to some clustering rule. Often the data is noisy and even approximate classification is of extreme importance. The difficulty of such classification stems from the fact that usually the data has many incomparable attributes, and often results in the question of clustering problems in high dimensional spaces. Since they require measuring distance between every pair of data points, standard algorithms for computing the exact clustering solutions use quadratic or "nearly quadratic" running time; i.e., O(dn 2\\Gammaff(d) ) time where n is the number of data points, d is the dimen- Computer Science Department, University of Toronto. Part of this work was done while visiting Bell Communications Research. y Bell Communications Research, MCC-1C365...

We develop a general technique for the quantitative analysis of iterative distributed load balancing schemes. We illustrate the technique by studying two simple, intuitively appealing models that are prevalent in the literature: the diffusive paradigm, and periodic balancing circuits (or the dimension exchange paradigm). It is well known that such load balancing schemes can be roughly modeled by Markov chains, but also that this approximation can be quite inaccurate. Our main contribution is an effective way of characterizing the deviation between the actual loads and the distribution generated by a related Markov chain, in terms of a natural quantity which we call the local divergence. We apply this technique to obtain bounds on the number of rounds required to achieve coarse balancing in general networks, cycles and meshes in these models. For balancing circuits, we also present bounds for the stronger requirement of perfect balancing, or counting.

Given an undirected graph wit.h edge co&s and a subset of k nodes called terminals, a multiway cut is a subset of edges whose removal disconnects each terminal from the rest.~iULTIW.~yCUT is the problem of finding a multiway cut of minimum cost..Previously, a very simple combinatorial algorithm due to Dahlhaus, Johnson, Papadimitriou, Seymour, and %nnr-lkakis gave a performance guarantee of 2 (1 -$), In this paper, we present a new linear programming rslax-&ion for ~fULTIW&Y CUT and a new approximation dgorithm based on it.The algorithm breaks the threshold of 2 for approximating MULTIWAY CUT, achieving a performance ratio of at.most 1.5 -$.This improves the previous result for every value of k.In particular, for k = 3 we get a ratio ofZ

We address the problem of designing data structures that allow efficient search for approximate nearest neighbors.More specifically, given a database consisting of a set of vectors in some high dimensional Euclidean space, we want to construct a space-efficient data struct,ure t,hat would allow us to search, given a query vector, for t.he closest or nearly closest vector in the database.We also address t.hii problem when distances are measured by the L1 norm, and in t.he Hamming cube.Sign%cant.lyimproving and extending recent results of Kleinberg, we const,ruct data structures whose size is polynomial in the size of t,he database, and search algorithms t,hat run in time nearly linear or nearly quadratic in the dimension (depending on the case; the extra factors are polylogarit.hmicin the size of the database).

It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for k-commodity flow instances with arbitrary capacities and demands. This improves upon the previously best-known bound of O(log 2 k ) and is existentially tight, up to a constant factor. An algorithm for finding a cut with ratio within a factor of O(log k) of the maximum concurrent flow, and thus of the optimal min-cut ratio, is presented.

A layered graph is a connected graph whose vertices are partitioned into sets L 0 =s, L 1 , L 2 ,..., and whose edges, which have nonnegative integral weights, run between consecutive layers. Its width is $\max\{|L_i|\}$. In the on-line layered graph traversal problem, a searcher starts at s in a layered graph of unknown width and tries to reach a target vertex t; however, the vertices in layer i and the edges between layers i-1 and i are only revealed when the searcher reaches layer i-1. We give upper and lower bounds on the competitive ratio of layered graph traversal algorithms. We give a deterministic on-line algorithm which is O(9 w )-competitive on width-w graphs and prove that for no w can a deterministic on-line algorithm have a competitive ratio better than 2 w-2 on width-w graphs. We prove that for all w, w/2 is a lower bound on the competitive ratio of any randomized on-line layered graph traversal algorithm. For traversing layered graphs consisting of w disjoint paths tied together at a common source, we give a randomized on-line algorithm with a competitive ratio of O(log w) and prove that this is optimal up to a constant factor.

Abstract. In this paper, we undertake the first study of statistical multiplexing from the perspective of approximation algorithms. The basic issue underlying statistical multiplexing is the following: in high-speed networks, individual connections (i.e., communication sessions) are very bursty, with transmission rates that vary greatly over time. As such, the problem of packing multiple connections together on a link becomes more subtle than in the case when each connection is assumed to have a fixed demand. We consider one of the most commonly studied models in this domain: that of two communicating nodes connected by a set of parallel edges, where the rate of each connection between them is a random variable. We consider three related problems: (1) stochastic load balancing, (2) stochastic bin-packing, and (3) stochastic knapsack. In the first problem the number of links is given and we want to minimize the expected value of the maximum load. In the other two problems the link capacity and an allowed overflow probability p are given, and the objective is to assign connections to links, so that the probability that the load of a link exceeds the link capacity is at most p. In binpacking we need to assign each connection to a link using as few links as possible. In the knapsack problem each connection has a value, and we have only one link. The problem is to accept as many

Article Free Access Share on Universal O(congestion + dilation + log1+εN) local control packet switching algorithms Authors: Rafail Ostrovsky Bell Communications Research, MCC-1C365B, Morristown, NJ Bell Communications Research, MCC-1C365B, Morristown, NJView Profile , Yuval Rabani Computer Science Department, Technion IIT, Haifa 32000, Israel Computer Science Department, Technion IIT, Haifa 32000, IsraelView Profile Authors Info & Claims STOC '97: Proceedings of the twenty-ninth annual ACM symposium on Theory of computingMay 1997 Pages 644–653https://doi.org/10.1145/258533.258659Online:04 May 1997Publication History 42citation269DownloadsMetricsTotal Citations42Total Downloads269Last 12 Months4Last 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

We consider algorithms for many-to-many hot potato routing. In hot potato (deflection) routing, a packet cannot be buffered, and is therefore always moving until it reaches its destination. We give optimal and nearly optimal deterministic algorithms for many-to-many packet routing in commonly occurring networks such as the hypercube, meshes, and tori of various dimensions and sizes, trees, and hypercubic networks such as the butterfly. All these algorithms are analyzed using a charging scheme that may be applicable to other algorithms as well. Moreover, all bounds hold in a dynamic setting in which packets can be injected at arbitrary times.

In the minimum path coloring problem, we are given a list of pairs of vertices of a graph. We are asked to connect each pair by a colored path. Paths of the same color must be edge disjoint. Our objective is to minimize the number of colors used. This problem was raised by A. Aggarwal et al. (1994) and P. Raghavan and E. Upfal (1994) as a model for routing in all-optical networks. It is also related to questions in circuit routing. In this paper, we improve the O(ln N) approximation result of J. Kleinberg and E. Tardos (1995) for path coloring on the N/spl times/N mesh. We give an O(1) approximation algorithm to the number of colors needed, and a poly(ln ln N) approximation algorithm to the choice of paths and colors. To the best of our knowledge, these are the first sub-logarithmic bounds for any network other than trees, rings, or trees of rings. Our results are based on developing new techniques for randomized rounding. These techniques iteratively improve a fractional solution until it approaches integrality. They are motivated by the method used by F.T. Leighton, B.M. Maggs, and S.B. Rao (1994) for packet routing.

In a seminal paper Leighton, Maggs, and Rao consider the packet scheduling problem when a single packet has to traverse each path. They show that there exists a schedule where each packet reaches its destination in O(C + D) steps, where C is the congestion and D is the dilation. The proof relies on the Lov'asz Local Lemma, and hence is not algorithmic. In a followup paper Leighton and Maggs use an algorithmic version of the Local Lemma due to Beck to give centralized algorithms for the problem. Leighton, Maggs, and Rao also give a distributed randomized algorithm where all packets reach their destinations with high probability in O(C +D log n) steps. In this paper we develop techniques to guarantee the high probability of delivering packets without resorting to the Lov'asz Local Lemma. We improve the distributed algorithm for problems with relatively high dilation to O(C) + (log n) O(log n) D + poly(log n). We extend the techniques to handle the case of infinite streams of ...

We discuss settings where several “agents” combine efforts to solve problems. This is a well-known setting in distributed artificial intelligence. Our work addresses theoretical questions in this model which are motivated by the work of Deng and Papadimitriou [Proc. 12th IFIPS Congress, Madrid, 1992; Proc. World Economic Congress, Moscow, 1992]. We consider optimization problems, in particular load balancing and virtual circuit routing, in which the input is divided among the agents. An underlying directed graph, whose nodes are the agents, defines the constraints on the information each agent may have about the portion of the input held by other agents. The questions we discuss are as follows: Given a bound on the maximum out-degree in this graph, which is the best graph? What is the quality of the solution obtained as a function of the maximum out-degree?

Article Free Access Share on A computational view of population genetics Authors: Yuval Rabani Department of Computer Science, University of Toronto, Toronto, Ontario M5S 1A4, Canada Department of Computer Science, University of Toronto, Toronto, Ontario M5S 1A4, CanadaView Profile , Yuri Rabinovich Department of Computer Science, University of Toronto, Toronto, Ontario M5S 1A4, Canada Department of Computer Science, University of Toronto, Toronto, Ontario M5S 1A4, CanadaView Profile , Alistair Sinclair Computer Science Division, University of California, Berkeley, CA Computer Science Division, University of California, Berkeley, CAView Profile Authors Info & Claims STOC '95: Proceedings of the twenty-seventh annual ACM symposium on Theory of computingMay 1995 Pages 83–92https://doi.org/10.1145/225058.225088Online:29 May 1995Publication History 19citation457DownloadsMetricsTotal Citations19Total Downloads457Last 12 Months9Last 6 weeks2 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

This paper considers the problems of admission control and virtual circuit routing in high performance computing and communication systems. Admission control and virtual circuit routing problems arise in numerous applications, including video-servers, real-lime database servers, and the provision of permanent virtual channel in large-scale communications networks. The paper describes both upper and lower bounds on the competitive ratio of algorithms for admission control and virtual circuit routing in trees, arrays, and hypercubes (the networks most commonly used in conjunction with nigh performance computing and communication). Our results include optimal algorithms for admission control and virtual circuit routing in trees, as well as the first competitive algorithms for these problems on non-tree networks. A key result of our research is the development of on-line algorithms that substantially outperform the greedy-based approaches that are used in practice.>

Quadratic Dynamical Systems (QDS), whose definition extends that of Markov chains, are used to model phenomena in a variety of fields like statistical physics and natural evolution. Such systems also play a role in genetic algorithms, a widelyused class of heuristics that are notoriously hard to analyze. Recently Rabinovich et al. took an important step in the study of QDS’s by showing, under some technical assumptions, that such systems converge to a stationary distribution (similar theorems for Markov Chains are well-known). We show, however, that the following sampling problem for QDS’s is PSPACE-hard: Given an initial distribution, produce a random sample from the t’th generation. The hardness result continues to hold for very restricted classes of QDS’s with very simple initial distributions, thus suggesting that QDS’s are intrinsically more complicated than Markov chains. ∗Supported by an IBM Graduate Fellowship and partly under NSF grant CCR-9310214. Email: [email protected]. †Work done while at ICSI, Berkeley, and supported in part by a Rothschild postdoctoral fellowship. Email: [email protected]. ‡Supported by NSF grant CCR-9310214. Email: [email protected].

In this paper, the authors prove lower bounds on the competitive ratio of randomized algorithms for two on-line problems: the k-server problem, suggested by Manasse, McGeoch, and Sleator [Competitive lgorithms for on-line problems, J. Algorithms, 11 (1990), pp. 208–230], and an on-line motion-planning problem due to Papadimitriou and Yannakakis [Shortest paths without a map, Lecture Notes in Comput. Sci. 372, Springer-Verlag, New York, 1989, pp. 610–620]. The authors prove, against an oblivious adversary, 1. an $\Omega \log k$ lower bound on the competitive ratio of any randomized on-line k-server algorithm in any sufficiently large metric space, 2. an $\Omega (\log \log k)$ lower bound on the competitive ratio of any randomized on-line k-server algorithm in any metric space with at least $k + 1$ points, and 3. an $\Omega (\log \log n)$ lower bound on the competitive ratio of any on-line motion-planning algorithm for a scene with n obstacles. Previously, no superconstant lower bound on the competitive ratio of randomized on-line algorithms was known for any of these problems.

We discuss settings where several "agents" combine efforts to solve problems. This is a well-known setting in distributed artificial intelligence. Our work addresses theoretical questions in this model which are motivated by the work of X. Deng and C.H. Papadimitriou (1992). We consider optimization problems, in particular load balancing and virtual circuit routing, in which the input is divided among the agents. An underlying directed graph, whose nodes are the agents, defines the constraints on the information each agent may have about the portion of the input held by other agents. The questions we discuss are: Given a bound on the maximum out-degree in this graph, which is the best graph? What is the quality of the solution obtained as a function of the maximum out-degree?.>

The authors prove a lower bound of Omega ( square root logk/loglogk) for the competitive ratio of randomized algorithms for the k-server problem against an oblivious adversary. The bound holds for arbitrary metric spaces (of at least k+1 points) and provides a new lower bound for the metrical task system problem as well. This improves the previous best lower bound of Omega (loglogk) for arbitrary metric spaces, more closely approaching the conjectured lower bound of Omega (logk). They also prove a lower bound of Omega (/sup logk///sub loglogk/) for the server problem on k+1 equally-spaced points on a line, which corresponds to some natural motion-planning problems.>

We deal with the competitive analysis of algorithms for managing data in a distributed environment. We deal with the file allocation problem ([C], [DF], [ML]), where copies of a file may be stored in the local storage of some subset of processors, copies may be replicated and discarded over time so as to optimize communication costs, but multiple copies must be kept consistent and at least one copy must be stored somewhere in the network at all times. We deal with competitive algorithms for minimizing communication costs, over arbitrary sequences of reads and writes, and arbitrary network topologies. We define the constrained file allocation problem to be the solution of many individual file allocation problems simultaneously, subject to the constraints of local memory size. We give competitive algorithms for this prblem on uniform networks. We then introduce distributed competitive algorithms for on-line data tracking (a generalization of mobile user tracking [AP1, AP3] to transform our competitive distributed data management algorithms into distributed algorithms themselves.

A layered graph is a connected, weighted graph whose vertices are partitioned into sets L/sub 0/=(s), L/sub 1/, L/sub 2/, . . ., and whose edges run between consecutive layers. Its width is max( mod L/sub i/ mod ). In the online layered graph traversal problem, a searcher starts at s in a layered graph of unknown width and tries to reach a target vertex t; however, the vertices in layer i and the edges between layers i-1 and i are only revealed when the searcher reaches layer i-1. The authors give upper and lower bounds on the competitive ratio of layered graph traversal algorithms. They give a deterministic online algorithm that is O(9w)-competitive on width-w graphs and prove that for no w can a deterministic online algorithm have a competitive ratio better than 2w/sup -2/ on width-w graphs. They prove that for all w, w/2 is a lower bound on the competitive ratio of any randomized online layered graph traversal algorithm. For traversing layered graphs consisting of w disjoint paths tied together at a common source, they give a randomized online algorithm with a competitive ratio of O(log w) and prove that this is optimal up to a constant factor.>

No abstract available.

Deterministic competitive k-server algorithms are given for all k and all metric spaces. This settles the k-server conjecture of M.S. Manasse et al. (1988) up to the competitive ratio. The best previous result for general metric spaces was a three-server randomized competitive algorithm and a nonconstructive proof that a deterministic three-server competitive algorithm exists. The competitive ratio the present authors can prove is exponential in the number of servers. Thus, the question of the minimal competitive ratio for arbitrary metric spaces is still open. The methods set forth here also give competitive algorithms for a natural generalization of the k-server problem, called the k-taxicab problem.>