For any ɛ > 0, we give a simple, deterministic (4+ɛ)-approximation algorithm for the Nash social welfare (NSW) problem under submodular valuations. We also consider the asymmetric variant of the problem, where the objective is to maximize the weighted geometric mean of agents’ valuations, and give an e(ω + 2 + ɛ)-approximation if the ratio between the largest weight and the average weight is at most ω. We also show that the 1/2-EFX envy-freeness property can be attained simultaneously with a constant-factor approximation. More precisely, we can find an allocation in polynomial time that is both 1/2-EFX and an (8+ɛ)-approximation to the symmetric NSW problem under submodular valuations.

We present a constant-factor approximation algorithm for the Nash Social Welfare (NSW) maximization problem with subadditive valuations accessible via demand queries. More generally, we propose a framework for NSW optimization which assumes two subroutines which (1) solve a configuration-type LP under certain additional conditions, and (2) round the fractional solution with respect to utilitarian social welfare. In particular, a constant-factor approximation for submodular valuations with value queries can also be derived from our framework.

Most of the literature on online algorithms and sequential decision-making focuses on settings with “irrevocable decisions” where the algorithm’s decision upon arrival of the new input is set in stone and can never change in the future. One canonical example is the classic prophet inequality problem, where realizations of a sequence of independent random variables X1, X2,… with known distributions are drawn one by one and a decision maker decides when to stop and accept the arriving random variable, with the goal of maximizing the expected value of their pick. We consider “prophet inequalities with recourse” in the linear buyback cost setting, where after accepting a variable Xi, we can still discard Xi later and accept another variable Xj, at a buyback cost of f × Xi. The goal is to maximize the expected net reward, which is the value of the final accepted variable minus the total buyback cost. Our first main result is an optimal prophet inequality in the regime of f ≥ 1, where we prove that we can achieve an expected reward 1+f/1+2f times the expected offline optimum. The problem is still open for 0<f<1 and we give some partial results in this regime. In particular, as our second main result, we characterize the asymptotic behavior of the competitive ratio for small f and provide almost matching upper and lower bounds that show a factor of 1−Θ(flog(1/f)). Our results are obtained by two fundamentally different approaches: One is inspired by various proofs of the classical prophet inequality, while the second is based on combinatorial optimization techniques involving LP duality, flows, and cuts.

Abstract. We revisit the Kahn–Kalai conjecture, recently proved in striking fashion by Park and Pham, and present a slightly reformulated simple proof which has a few advantages: (1) it works for nonuniform product measures, (2) it gives near-optimal bounds even for sampling probabilities close to 1, (3) it gives a clean bound of [Formula: see text] for every [Formula: see text]-bounded set system, [Formula: see text].

The maximization of submodular functions have found widespread application in areas such as machine learning, combinatorial optimization, and economics, where practitioners often wish to enforce various constraints; the matroid constraint has been investigated extensively due to its algorithmic properties and expressive power. Though tight approximation algorithms for general matroid constraints exist in theory, the running times of such algorithms typically scale quadratically, and are not practical for truly large scale settings. Recent progress has focused on fast algorithms for important classes of matroids given in explicit form. Currently, nearly-linear time algorithms only exist for graphic and partition matroids [Alina Ene and Huy L. Nguyen, 2019]. In this work, we develop algorithms for monotone submodular maximization constrained by graphic, transversal matroids, or laminar matroids in time near-linear in the size of their representation. Our algorithms achieve an optimal approximation of 1-1/e-ε and both generalize and accelerate the results of Ene and Nguyen [Alina Ene and Huy L. Nguyen, 2019]. In fact, the running time of our algorithm cannot be improved within the fast continuous greedy framework of Badanidiyuru and Vondrák [Ashwinkumar Badanidiyuru and Jan Vondrák, 2014]. To achieve near-linear running time, we make use of dynamic data structures that maintain bases with approximate maximum cardinality and weight under certain element updates. These data structures need to support a weight decrease operation and a novel Freeze operation that allows the algorithm to freeze elements (i.e. force to be contained) in its basis regardless of future data structure operations. For the laminar matroid, we present a new dynamic data structure using the top tree interface of Alstrup, Holm, de Lichtenberg, and Thorup [Stephen Alstrup et al., 2005] that maintains the maximum weight basis under insertions and deletions of elements in O(log n) time. This data structure needs to support certain subtree query and path update operations that are performed every insertion and deletion that are non-trivial to handle in conjunction. For the transversal matroid the Freeze operation corresponds to requiring the data structure to keep a certain set S of vertices matched, a property that we call S-stability. While there is a large body of work on dynamic matching algorithms, none are S-stable and maintain an approximate maximum weight matching under vertex updates. We give the first such algorithm for bipartite graphs with total running time linear (up to log factors) in the number of edges.

We study incentive-compatible mechanisms that maximize the Nash Social Welfare. Since traditional incentive-compatible mechanisms cannot maximize the Nash Social Welfare even approximately, we propose changing the traditional model. Inspired by a widely used charging method (e.g., royalties, a lawyer that charges some percentage of possible future compensation), we suggest charging the players some percentage of their value of the outcome. We call this model the percentage fee model. We show that there is a mechanism that maximizes exactly the Nash Social Welfare in every setting with non-negative valuations. Moreover, we prove an analog of Roberts theorem that essentially says that if the valuations are non-negative, then the only implementable social choice functions are those that maximize weighted variants of the Nash Social Welfare. We develop polynomial time incentive compatible approximation algorithms for the Nash Social Welfare with subadditive valuations and prove some hardness results. 26 pages. This is the TheoretiCS journal version

The independence polynomial of a graph is the generating polynomial of all its independent sets. Formally, given a graph G, its independence polynomial ZG (λ) is given by ΣIλ|I|, where the sum is over all independent sets I of G. The independence polynomial has been an important object of study in both combinatorics and computer science. In particular, the algorithmic problem of estimating ZG(λ) for a fixed positive λ on an input graph G is a natural generalization of the problem of counting independent sets, and its study has led to some of the most striking connections between computational complexity and the theory of phase transitions. More surprisingly, the independence polynomial for negative and complex values of λ also turns out to be related to problems in statistical physics and combinatorics. In particular, the locations of the complex roots of the independence polynomial of bounded degree graphs turn out to be very closely related to the Lovász local lemma, and also to the questions in the computational complexity of counting. Consequently, the locations of such zeros have been studied in many works. In this direction, it is known from the work of Shearer [29] and of Scott and Sokal [27] - inspired by the study of the Lovász local lemma - that the independence polynomial ZG (λ) of a graph G of maximum degree at most d + 1 does not vanish provided that . Significant extensions of this result have recently been given in the case when λ is in the right half-plane (i.e., when ℜλ ≥ 0) by Peters and Regts [26] and Bencs and Csikvári [9]. In this paper, our motivation is to further extend these results to find new zero free regions not only in the right half plane, but also in the left half-plane, that is, when ℜλ ≤ 0. We give new geometric criterions for establishing zero-free regions as well as for carrying out semi-rigorous numerical explorations. We then provide two examples of the (rigorous) use of these criterions, by establishing two new zero-free regions in the left-half plane. We also extend the results of Bencs and Csikvári [9] for the right half-plane using our framework. By a direct application of the interpolation method of Barvinok [5], combined with extensions due to Patel and Regts [25], our results also imply deterministic polynomial time approximation algorithms for the independence polynomial of bounded degree graphs in the new zero-free regions. * The arXiv version of the paper can be accessed at https://arxiv.org/abs/2204.04868.

In this paper, we investigate two variants of the secretary problem. In these variants, we are presented with a sequence of numbers Xi that come from distributions Di, and that arrive in either random or adversarial order. We do not know what the distributions are, but we have access to a single sample Yi from each distribution Di. After observing each number, we have to make an irrevocable decision about whether we would like to accept it or not with the goal of maximizing the probability of selecting the largest number. The random order version of this problem was first studied by Correa et al. [SODA 2020] who managed to construct an algorithm that achieves a probability of 0.4529. In this paper, we improve this probability to 0.5009, almost matching an upper bound of ≃ 0.5024 which we show follows from earlier work. We also show that there is an algorithm which achieves the probability of ≃ 0.5024 asymptotically if no particular distribution is especially likely to yield the largest number. For the adversarial order version of the problem, we show that we can select the maximum number with a probability of 1/4, and that this is best possible. Our work demonstrates that unlike in the case of the expected value objective studied by Rubinstein et al. [ITCS 2020], knowledge of a single sample is not enough to recover the factor of success guaranteed by full knowledge of the distribution.

For any >0, we give a simple, deterministic (4+)-approximation algorithm for the Nash social welfare (NSW) problem under submodular valuations. The previous best approximation factor was 380 via a randomized algorithm. We also consider the asymmetric variant of the problem, where the objective is to maximize the weighted geometric mean of agents’ valuations, and give an (ω + 2 + ) -approximation if the ratio between the largest weight and the average weight is at most ω.

We consider the bilateral trade problem, in which two agents trade a single indivisible item. It is known that the only dominant-strategy truthful mechanism is the fixed-price mechanism: given commonly known distributions of the buyer's value B and the seller's value S, a price p is offered to both agents and trade occurs if S ≤ p ≤ B. The objective is to maximize either expected welfare or expected gains from trade . We improve the approximation ratios for several welfare maximization variants of this problem. When the agents' distributions are identical, we show that the optimal approximation ratio for welfare is . With just one prior sample from the common distribution, we show that a 3/4-approximation to welfare is achievable. When agents' distributions are not required to be identical, we show that a previously best-known (1–1/e)-approximation can be strictly improved, but 1–1/e is optimal if only the seller's distribution is known.

We study the communication complexity of dominant strategy implementations of combinatorial auctions. We start with two domains that are generally considered “easy”: multi-unit auctions with decreasing marginal values and combinatorial auctions with gross substitutes valuations. For both domains we have fast algorithms that find the welfare-maximizing allocation with communication complexity that is poly-logarithmic in the input size. This immediately implies that welfare maximization can be achieved in ex-post equilibrium with no significant communication cost, by using VCG payments. In contrast, we show that in both domains the communication complexity of any dominant strategy implementation that achieves the optimal welfare is polynomial in the input size.

We present a 380-approximation algorithm for the Nash Social Welfare problem with submodular valuations. Our algorithm builds on and extends a recent constant-factor approximation for Rado valuations [15].

We consider the problem of maximizing submodular functions in single-pass streaming and secretaries-with-shortlists models, both with random arrival order.For cardinality constrained monotone functions, Agrawal, Shadravan, and Stein~\cite{SMC19} gave a single-pass $(1-1/e-\varepsilon)$-approximation algorithm using only linear memory, but their exponential dependence on $\varepsilon$ makes it impractical even for $\varepsilon=0.1$.We simplify both the algorithm and the analysis, obtaining an exponential improvement in the $\varepsilon$-dependence (in particular, $O(k/\varepsilon)$ memory).Extending these techniques, we also give a simple $(1/e-\varepsilon)$-approximation for non-monotone functions in $O(k/\varepsilon)$ memory. For the monotone case, we also give a corresponding unconditional hardness barrier of $1-1/e+\varepsilon$ for single-pass algorithms in randomly ordered streams, even assuming unlimited computation. Finally, we show that the algorithms are simple to implement and work well on real world datasets.

We study the Nash Social Welfare problem: Given n agents with valuation functions vi : 2[m] → ℝ+, partition [m] into S1, …, Sn so as to maximize . The problem has been shown to admit a constant-factor approximation for additive, budget-additive, and piecewise linear concave separable valuations; the case of submodular valuations is open. We provide a -approximation of the optimal value for several classes of submodular valuations: coverage, sums of matroid rank functions, and certain matching-based valuations.

In this paper, we introduce a novel technique for constrained submodular maximization, inspired by barrier functions in continuous optimization. This connection not only improves the running time for constrained submodular maximization but also provides the state of the art guarantee. More precisely, for maximizing a monotone submodular function subject to the combination of a $k$-matchoid and $\ell$-knapsack constraints (for $\ell\leq k$), we propose a potential function that can be approximately minimized. Once we minimize the potential function up to an $\epsilon$ error, it is guaranteed that we have found a feasible set with a $2(k+1+\epsilon)$-approximation factor which can indeed be further improved to $(k+1+\epsilon)$ by an enumeration technique. We extensively evaluate the performance of our proposed algorithm over several real-world applications, including a movie recommendation system, summarization tasks for YouTube videos, Twitter feeds and Yelp business locations, and a set cover problem.

In large-data applications, it is desirable to design algorithms with a high degree of parallelization. In the context of submodular optimization, adaptive complexity has become a widely-used measure of an algorithm’s “sequentiality”. Algorithms in the adaptive model proceed in rounds, and can issue polynomially many queries to a function f in each round. The queries in each round must be independent, produced by a computation that depends only on query results obtained in previous rounds.

Algorithmic stability is a classical approach to understanding and analysis of the generalization error of learning algorithms. A notable weakness of most stability-based generalization bounds is that they hold only in expectation. Generalization with high probability has been established in a landmark paper of Bousquet and Elisseeff (2001) albeit at the expense of an additional $\sqrt{n}$ factor in the bound. Specifically, their bound on the estimation error of any $\gamma$-uniformly stable learning algorithm on $n$ samples and range in $[0,1]$ is $O(\gamma \sqrt{n \log(1/\delta)} + \sqrt{\log(1/\delta)/n})$ with probability $\geq 1-\delta$. The $\sqrt{n}$ overhead makes the bound vacuous in the common settings where $\gamma \geq 1/\sqrt{n}$. A stronger bound was recently proved by the authors (Feldman and Vondrak, 2018) that reduces the overhead to at most $O(n^{1/4})$. Still, both of these results give optimal generalization bounds only when $\gamma = O(1/n)$. We prove a nearly tight bound of $O(\gamma \log(n)\log(n/\delta) + \sqrt{\log(1/\delta)/n})$ on the estimation error of any $\gamma$-uniformly stable algorithm. It implies that for algorithms that are uniformly stable with $\gamma = O(1/\sqrt{n})$, estimation error is essentially the same as the sampling error. Our result leads to the first high-probability generalization bounds for multi-pass stochastic gradient descent and regularized ERM for stochastic convex problems with nearly optimal rate — resolving open problems in prior work. Our proof technique is new and we introduce several analysis tools that might find additional applications.

Uniform stability of a learning algorithm is a classical notion of algorithmic stability introduced to derive high-probability bounds on the generalization error (Bousquet and Elisseeff, 2002). Specifically, for a loss function with range bounded in $[0,1]$, the generalization error of $\gamma$-uniformly stable learning algorithm on $n$ samples is known to be at most $O((\gamma +1/n) \sqrt{n \log(1/\delta)})$ with probability at least $1-\delta$. Unfortunately, this bound does not lead to meaningful generalization bounds in many common settings where $\gamma \geq 1/\sqrt{n}$. At the same time the bound is known to be tight only when $\gamma = O(1/n)$. Here we prove substantially stronger generalization bounds for uniformly stable algorithms without any additional assumptions. First, we show that the generalization error in this setting is at most $O(\sqrt{(\gamma + 1/n) \log(1/\delta)})$ with probability at least $1-\delta$. In addition, we prove a tight bound of $O(\gamma^2 + 1/n)$ on the second moment of the generalization error. The best previous bound on the second moment of the generalization error is $O(\gamma + 1/n)$. Our proofs are based on new analysis techniques and our results imply substantially stronger generalization guarantees for several well-studied algorithms.

We study an algorithm for approximating the multivariate independence polynomial Z(z), with negative and complex arguments. While the focus so far has been mostly on computing combinatorial polynomials restricted to the univariate positive setting (with seminal results for the independence polynomial by Weitz (2006) and Sly (2010)), the independence polynomial with negative or complex arguments has strong connections to combinatorics and to statistical physics. The independence polynomial with negative arguments, Z(–p), determines the Shearer region, the maximal region of probabilities to which the Lovász Local Lemma (LLL) can be extended (Shearer 1985). In statistical physics, complex zeros of the independence polynomial relate to existence of phase transitions. Our main result is a deterministic algorithm to compute approximately the independence polynomial in any root-free complex polydisc centered at the origin. More precisely, we can (1 + ε)-approximate the independence polynomial Z(z) for an n-vertex graph of degree at most d, for any complex vector z such that Z(z′) ≠ 0 for |z′i| ≤ (1 + α)|zi|, in running time . Our result also extends to graphs of unbounded degree that have a bounded connective constant. Our algorithm is essentially the same as Weitz's algorithm for positive parameters up to the tree uniqueness threshold. The core of the analysis is a novel multivariate form of the correlation decay technique, which can handle non-uniform complex parameters. In summary, we provide a unifying algorithm for all known regions where Z(z) is approximately computable. In particular, in the univariate real setting our work implies that Weitz's algorithm works in an interval between two critical points (−λ′c(d), λc(d)), and outside of this interval an approximation of Z(λ) is known to be NP-hard. As an application, we provide an algorithm to test membership in Shearer's region within a multiplicative error of 1 + α, in running time . We also give a deterministic algorithm for Shearer's lemma (extending the LLL) with n events on m independent variables under slack α, with running time . On the hardness side, we prove that evaluating Z(z) at an arbitrary point in Shearer's region, and testing membership in Shearer's region, are #P-hard problems. For Weitz's correlation decay technique in the negative regime, we show that the dependence in the exponent is optimal.

Computational and economic results suggest that social welfare maximization and combinatorial auction design are much easier when bidders' valuations satisfy the "gross substitutes" condition. The goal of this paper is to evaluate rigorously the folklore belief that the main take-aways from these results remain valid in settings where the gross substitutes condition holds only approximately. We show that for valuations that pointwise approximate a gross substitutes valuation (in fact even a linear valuation), optimal social welfare cannot be approximated to within a subpolynomial factor and demand oracles cannot be simulated using a subexponential number of value queries. We then provide several positive results by imposing additional structure on the valuations (beyond gross substitutes), using a more stringent notion of approximation, and/or using more powerful oracle access to the valuations. For example, we prove that the performance of the greedy algorithm degrades gracefully for near-linear valuations with approximately decreasing marginal values; that with demand queries, approximate welfare guarantees for XOS valuations degrade gracefully for valuations that are pointwise close to XOS; and that the performance of the Kelso-Crawford auction degrades gracefully for valuations that are close to various subclasses of gross substitutes valuations.

Two genres of heuristics that are frequently reported to perform much better on "real-world" instances than in the worst case are greedy algorithms and local search algorithms. In this paper, we systematically study these two types of algorithms for the problem of maximizing a monotone submodular set function subject to downward-closed feasibility constraints. We consider perturbation-stable instances, in the sense of Bilu and Linial [11], and precisely identify the stability threshold beyond which these algorithms are guaranteed to recover the optimal solution. Byproducts of our work include the first definition of perturbation-stability for non-additive objective functions, and a resolution of the worst-case approximation guarantee of local search in p-extendible systems.

A long-standing open question in algorithmic mechanism design is whether there exist computationally efficient truthful mechanisms for combinatorial auctions, with performance guarantees close to those possible without considerations of truthfulness. In this article, we answer this question negatively: the requirement of truthfulness can impact dramatically the ability of a mechanism to achieve a good approximation ratio for combinatorial auctions. More precisely, we show that every universally truthful randomized mechanism for combinatorial auctions with submodular valuations that approximates optimal social welfare within a factor of m 1/2−ϵ must use exponentially many value queries, where m is the number of items. Furthermore, we show that there exists a class of succinctly represented submodular valuation functions, for which the existence of a universally truthful polynomial-time mechanism that provides an m 1/2−ϵ -approximation would imply NP = RP . In contrast, ignoring truthfulness, there exist constant-factor approximation algorithms for this problem, and ignoring computational efficiency, the VCG mechanism is truthful and provides optimal social welfare. These are the first hardness results for truthful polynomial-time mechanisms for any type of combinatorial auctions, even for deterministic mechanisms. Our approach is based on a novel direct hardness technique that completely skips the notoriously hard step of characterizing truthful mechanisms. The characterization step was the main obstacle for proving impossibility results in algorithmic mechanism design so far.

We investigate the approximability of several classes of real-valued functions by functions of a small number of variables (juntas). Our main results are tight bounds on the number of variables required to approximate a function $f:\{0,1\}^n \rightarrow [0,1]$ within $\ell_2$-error $\epsilon$ over the uniform distribution: (a) If $f$ is submodular, then it is $\epsilon$-close to a function of $O(\frac{1}{\epsilon^2} \log \frac{1}{\epsilon})$ variables. This is an exponential improvement over previously known results [V. Feldman, P. Kothari, and J. Vondrák, JMLR Workshop Conf. Proc., 35 (2013), pp. 711--740]. We note that $\Omega(\frac{1}{\epsilon^2})$ variables are necessary even for linear functions. (b) If $f$ is fractionally subadditive (XOS) it is $\epsilon$-close to a function of $2^{O(1/\epsilon^2)}$ variables. This result holds for all functions with low total $\ell_1$-influence and is a real-valued generalization of Friedgut's theorem for Boolean functions. We show that $2^{\Omega(1/\epsilon)}$ variables are necessary even for XOS functions. As applications of these results, we provide learning algorithms over the uniform distribution. For XOS functions, we give a probably approximately correct learning algorithm that runs in time $2^{1/\mathrm{poly}(\epsilon)} \mathrm{poly}(n)$. For submodular functions we give an algorithm in the more demanding probably mostly approximately correct (PMAC) learning model [M. F. Balcan and N. Harvey, preprint, arXiv:1008.2159, 2012] which requires a multiplicative $(1+\gamma)$ factor approximation with probability at least $1-\epsilon$ over the target distribution. Our uniform distribution algorithm runs in time $2^{1/\mathrm{poly}(\gamma\epsilon)} \mathrm{poly}(n)$. This is the first algorithm in the PMAC model that can achieve a constant approximation factor arbitrarily close to 1 for all submodular functions (even over the uniform distribution). It relies crucially on our bounds for approximation by juntas. As follows from the lower bounds in the above paper by Feldman, Kothari, and Vondrák both of these algorithms are close to optimal. We also give applications for proper learning, testing, and agnostic learning of these classes.

Is it possible to maximize a monotone submodular function faster than the widely used lazy greedy algorithm (also known as accelerated greedy), both in theory and practice? In this paper, we develop the first linear-time algorithm for maximizing a general monotone submodular function subject to a cardinality constraint. We show that our randomized algorithm, STOCHASTIC-GREEDY, can achieve a (1 − 1/e − ε) approximation guarantee, in expectation, to the optimum solution in time linear in the size of the data and independent of the cardinality constraint. We empirically demonstrate the effectiveness of our algorithm on submodular functions arising in data summarization, including training large-scale kernel methods, exemplar-based clustering, and sensor placement. We observe that STOCHASTIC-GREEDY practically achieves the same utility value as lazy greedy but runs much faster. More surprisingly, we observe that in many practical scenarios STOCHASTIC-GREEDY does not evaluate the whole fraction of data points even once and still achieves indistinguishable results compared to lazy greedy.

Submodular and fractionally subadditive (or equivalently XOS) functions play a fundamental role in combinatorial optimization, algorithmic game theory and machine learning. Motivated by learnability of these classes of functions from random examples, we consider the question of how well such functions can be approximated by low-degree polynomials in ℓ2norm over the uniform distribution. This question is equivalent to understanding the concentration of Fourier weight on low-degree coefficients, a central concept in Fourier analysis. Denoting the smallest degree sufficient to approximate f in ℓ2norm within ∈ by deg∈(ℓ2)(f), we show that : For any submodular function f : {0, 1}n→ [0, 1], deg∈(ℓ2)(f) = O(log(1/∈)/∈4/5) and there is a submodular function that requires degree Ω(1/∈4/5). : For any XOS function f : {0, 1} → [0, 1], deg∈(ℓ2) (f) = O(1/∈) and there exists an XOS function that requires degree Ω(1/∈). This improves on previous approaches that all showed an upper bound of O(1/∈2) for submodular [CKKL12], [FKV13], [FV13] and XOS [FV13] functions. The best previous lower bound was Ω(1/∈2/3) for monotone submodular functions [FKV13]. Our techniques reveal new structural properties of submodular and XOS functions and the upper bounds lead to nearly optimal PAC learning algorithms for these classes of functions.

The Lovász local lemma is a seminal result in probabilistic combinatorics. It gives a sufficient condition on a probability space and a collection of events for the existence of an outcome that simultaneously avoids all of those events. Finding such an outcome by an efficient algorithm has been an active research topic for decades. The breakthrough work of Moser [ A constructive proof of the Lovász local lemma, in Proceedings of the ACM International Symposium on Theory of Computing, 2009, pp. 343--350] and Moser and Tardos [ J. ACM, 57 (2010), 11] presented an efficient algorithm for a general setting primarily characterized by a product structure on the probability space. In this work we present an efficient algorithm for a much more general setting. Our main assumption is that there exist certain functions, called resampling oracles, that can be invoked to address the undesired occurrence of the events. We show that, in all scenarios to which the original Lovász local lemma applies, there exist resampling oracles, although they are not necessarily efficient. Nevertheless, for essentially all known applications of the Lovász local lemma and its generalizations, we have designed efficient resampling oracles. As an application of these techniques, we present a new result on packings of rainbow spanning trees.

We develop a continuous-time framework based on multiplicative weight updates to approximately solve continuous optimization problems. The framework allows for a simple and modular analysis for a variety of problems involving convex constraints and concave or submodular objective functions. The continuous-time framework avoids the cumbersome technical details that are typically necessary in actual algorithms. We also show that the continuous-time algorithms can be converted into implementable algorithms via a straightforward discretization process. Using our framework and additional ideas we obtain significantly faster algorithms compared to previously known algorithms to maximize the multilinear relaxation of a monotone or non-monotone submodular set function subject to linear packing constraints.

We consider the problem of optimizing convex and concave functions with access to an erroneous zeroth-order oracle. In particular, for a given function $x \to f(x)$ we consider optimization when one is given access to absolute error oracles that return values in [f(x) - \epsilon,f(x)+\epsilon] or relative error oracles that return value in [(1+\epsilon)f(x), (1 +\epsilon)f (x)], for some \epsilon larger than 0. We show stark information theoretic impossibility results for minimizing convex functions and maximizing concave functions over polytopes in this model.

We prove three results about colorings of the simplex reminiscent of Sperner's Lemma, with applications in hardness of approximation and fair division. First, we prove a coloring lemma conjectured by [5]: Let Vk,q = {v ∊ ℤk+: ∑ki=1 vi = q} and Ek,q = {{a + e1,a + e2, …, a + ek}: a ∊ ℤk+, ∑ki = 1 ai = q — 1}· Then for every Sperner-admissible labeling {ℓ: Vk,q → [k] such that uℓ(v) > 0 for each v ∊ Vk,q), there are at least non-monochromatic hyperedges in Ek,q. This implies an optimal Unique-Games hardness of (k – 1 – ∊)-approximation for the Hypergraph Labeling with Color Lists problem [2]: Given a k-uniform hypergraph H = (V, E) with color lists L(v) ⊆ [k] ∀v ∊ V, find a labeling ℓ(v) ∊ L(v) that minimizes the number of non-monochromatic hyperedges. We also show that a (k — l)-approximation can be achieved. Second, we show that in contrast to Sperner's Lemma, there is a Sperner-admissible labeling of Vk,q such that every hyperedge in Ek,q contains at most 4 colors. We present an interpretation of this statement in the context of fair division: There is a preference function on Δk,q = {x ∊ ℝk+: ∑ki=1 xi = q} such that for any division of q units of a resource, (x1, x2, …, xk) ∊ δk,q such that ∑ki=1 ⌊xi⌋ = q – 1, at most 4 players out of k are satisfied. Third, we prove that there are subdivisions of the simplex with a fractional labeling (analogous to a fractional solution for Min-CSP problems) such that every hyperedge in the subdivision uses only labelings with 1 or 2 colors. This means that a natural LP cannot distinguish instances of Hypergraph Labeling with Color Lists that can be labeled so that every hyperedge uses at most 2 colors, and instances that must have a rainbow hyperedge. We prove that this problem is indeed NP-hard for k = 3.

We design new approximation algorithms for the problems of optimizing submodular and supermodular functions subject to a single matroid constraint. Specifically, we consider the case in which we wish to maximize a nondecreasing submodular function or minimize a nonincreasing supermodular function in the setting of bounded total curvature c. In the case of submodular maximization with curvature c, we obtain a (1 — c/e)-approximation — the first improvement over the greedy (1 — e−c)/c-approximation of Conforti and Cornuejols from 1984, which holds for a cardinality constraint, as well as recent approaches that hold for an arbitrary matroid constraint. Our approach is based on modifications of the continuous greedy algorithm and non-oblivious local search, and allows us to approximately maximize the sum of a nonnegative, nondecreasing submodular function and a (possibly negative) linear function. We show how to reduce both submodular maximization and supermodular minimization to this general problem when the objective function has bounded total curvature. We prove that the approximation results we obtain are the best possible in the value oracle model, even in the case of a cardinality constraint. Finally, we give two concrete applications of our results in the settings of maximum entropy sampling, and the column-subset selection problem.

We consider the Minimum Submodular Cost Allocation (MSCA) problem. In this problem, we are given k submodular cost functions f_1, ... , f_k: 2^V -> R_+ and the goal is to partition V into k sets A_1, ..., A_k so as to minimize the total cost sum_{i = 1}^k f_i(A_i). We show that MSCA is inapproximable within any multiplicative factor even in very restricted settings; prior to our work, only Set Cover hardness was known. In light of this negative result, we turn our attention to special cases of the problem. We consider the setting in which each function f_i satisfies f_i = g_i + h, where each g_i is monotone submodular and h is (possibly non-monotone) submodular. We give an O(k log |V|) approximation for this problem. We provide some evidence that a factor of k may be necessary, even in the special case of HyperLabel. In particular, we formulate a simplex-coloring conjecture that implies a Unique-Games-hardness of (k - 1 - epsilon) for k-uniform HyperLabel and label set [k]. We provide a proof of the simplex-coloring conjecture for k=3.

A classic result in probability theory known as de Finetti's theorem states that exchangeable random variables are equivalent to a mixture of distributions where each distribution is determined by an i.i.d. sequence of random variables (an "i.i.d. mix"). Motivated by a recent application and more generally by the relationship of local vs. global correlation in randomized rounding, we study weaker notions of exchangeability that still imply the conclusion of de Finetti's theorem. We say that a bivariate distribution rho is G-realizable for a graph G if there exists a joint distribution of random variables on the vertices such that the marginal distribution on each edge equals rho. We first characterize completely the G-realizable distributions for all symmetric/arc-transitive graphs G. Our main results are forms of de Finetti's theorem for general graphs, based on spectral properties. Let lambda_1(G) >= ... >= lambda_n(G) denote the eigenvalues of the adjacency matrix of G. 1. We prove that if rho is G_n-realizable for a sequence of graphs such that lambda_n(G_n) / lambda_1(G_n) tends to 0, then rho is described by a probability matrix that is positive-semidefinite. For random variables on domains of size |D| <= 4, this implies that rho must be an i.i.d. mix. 2. If rho is G_n-realizable for a sequence of (n,d,lambda)-graphs G_n (d-regular with all eigenvalues except for one bounded by lambda in absolute value) such that lambda(G_n) / d(G_n) tends to 0, then rho is an i.i.d. mix. 3. If rho is G_n-realizable for a sequence of directed graphs such that each of them is an arbitrary orientation of an (n,d,lambda)-graph G_n, and lambda(G_n) / d(G_n) tends to 0, then rho is an i.i.d. mix.

There has been much progress recently on improved approximations for problems involving submodular objective functions, and many interesting techniques have been developed. However, the resulting algorithms are often slow and impractical. In this paper we develop algorithms that match the best known approximation guarantees, but with significantly improved running times, for maximizing a monotone submodular function f : 2[n] → ℝ+ subject to various constraints. As in previous work, we measure the number of oracle calls to the objective function which is the dominating term in the running time. Our first result is a simple algorithm that gives a (1 − 1/∊ − ∊)-approximation for a cardinality constraint using queries, and a 1/(p + 2ℓ + 1 + ∊)-approximation for the intersection of a p-system and ℓ knapsack (linear) constraints using queries. This is the first approximation for a p-system combined with linear constraints. (We also show that the factor of p cannot be improved for maximizing over a p-system.) The main idea behind these algorithms serves as a building block in our more sophisticated algorithms. Our main result is a new variant of the continuous greedy algorithm, which interpolates between the classical greedy algorithm and a truly continuous algorithm. We show how this algorithm can be implemented for matroid and knapsack constraints using Õ(n2) oracle calls to the objective function. (Previous variants and alternative techniques were known to use at least Õ(n4) oracle calls.) This leads to an -time (1 − 1/∊ − ∊)-approximation for a matroid constraint. For a knapsack constraint, we develop a more involved (1 − 1/∊ − ∊)-approximation algorithm that runs in time .

We design new approximation algorithms for the Multiway Cut problem, improving the previously known factor of 1.32388 [Buchbinder et al., 2013].

We initiate the study of property testing of submodularity on the boolean hypercube. Submodular functions come up in a variety of applications in combinatorial optimization. For a vast range of algorithms, the existence of an oracle to a submodular function is assumed. But how does one check if this oracle indeed represents a submodular function? Consider a function f:{0,1} n →ℝ. The distance to submodularity is the minimum fraction of values of f that need to be modified to make f submodular. If this distance is more than ϵ>0, then we say that f is ϵ-far from being submodular. The aim is to have an efficient procedure that, given input f that is ϵ-far from being submodular, certifies that f is not submodular. We analyze a natural tester for this problem, and prove that it runs in subexponential time. This gives the first non-trivial tester for submodularity. On the other hand, we prove an interesting lower bound (that is, unfortunately, quite far from the upper bound) suggesting that this tester cannot be efficient in terms of ϵ. This involves non-trivial examples of functions which are far from submodular and yet do not exhibit too many local violations. We also provide some constructions indicating the difficulty in designing a tester for submodularity. We construct a partial function defined on exponentially many points that cannot be extended to a submodular function, but any strict subset of these values can be extended to a submodular function.

We consider the problem of maximizing a nonnegative submodular set function $f:2^N \rightarrow {\mathbb R}_+$ over a ground set $N$ subject to a variety of packing-type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that allows us to derive a number of new results, in particular, when $f$ may be a nonmonotone function. Our algorithms are based on (approximately) maximizing the multilinear extension $F$ of $f$ over a polytope $P$ that represents the constraints, and then effectively rounding the fractional solution. Although this approach has been used quite successfully, it has been limited in some important ways. We overcome these limitations as follows. First, we give constant factor approximation algorithms to maximize $F$ over a downward-closed polytope $P$ described by an efficient separation oracle. Previously this was known only for monotone functions. For nonmonotone functions, a constant factor was known only when the polytope was either the intersection of a fixed number of knapsack constraints or a matroid polytope. Second, we show that contention resolution schemes are an effective way to round a fractional solution, even when $f$ is nonmonotone. In particular, contention resolution schemes for different polytopes can be combined to handle the intersection of different constraints. Via linear programming duality we show that a contention resolution scheme for a constraint is related to the correlation gap of weighted rank functions of the constraint. This leads to an optimal contention resolution scheme for the matroid polytope. Our results provide a broadly applicable framework for maximizing linear and submodular functions subject to independence constraints. We give several illustrative examples. Contention resolution schemes may find other applications.

We study the complexity of approximate representation and learning of submodular functions over the uniform distribution on the Boolean hypercube {0,1}^n. Our main result is the following structural theorem: any submodular function is ε-close in \ell_2 to a real-valued decision tree (DT) of depth O(1/ε^2). This immediately implies that any submodular function is ε-close to a function of at most 2^O(1/ε^2) variables and has a spectral \ell_1 norm of 2^O(1/ε^2). It also implies the closest previous result that states that submodular functions can be approximated by polynomials of degree O(1/ε^2) (Cheraghchi et al., 2012). Our result is proved by constructing an approximation of a submodular function by a DT of rank 4/ε^2 and a proof that any rank-r DT can be ε-approximated by a DT of depth \frac52(r+\log(1/ε)). We show that these structural results can be exploited to give an attribute-efficient PAC learning algorithm for submodular functions running in time \tildeO(n^2) ⋅2^O(1/ε^4). The best previous algorithm for the problem requires n^O(1/ε^2) time and examples (Cheraghchi et al., 2012) but works also in the agnostic setting. In addition, we give improved learning algorithms for a number of related settings. We also prove that our PAC and agnostic learning algorithms are essentially optimal via two lower bounds: (1) an information-theoretic lower bound of 2^Ω(1/ε^2/3) on the complexity of learning monotone submodular functions in any reasonable model (including learning with value queries); (2) computational lower bound of n^Ω(1/ε^2/3) based on a reduction to learning of sparse parities with noise, widely-believed to be intractable. These are the first lower bounds for learning of submodular functions over the uniform distribution.

An increasingly important analytics scenario for Hadoop involves multiple (often ad hoc) grouping and aggregation queries with selection predicates over a slowly changing dataset. These queries are typically expressed via high-level query languages such as Jaql, Pig, and Hive, and are used either directly for business-intelligence applications or to prepare the data for statistical model building and machine learning. In such scenarios it has been increasingly recognized that, as in classical databases, techniques for avoiding access to irrelevant data can dramatically improve query performance. Prior work on Hadoop, however, has simply ported classical techniques to the MapReduce setting, focusing on record-level indexing and key-based partition elimination. Unfortunately, record-level indexing only slightly improves overall query performance, because it does not minimize the number of mapper "waves", which is determined by the number of processed splits. Moreover, key-based partitioning requires data reorganization, which is usually impractical in Hadoop settings. We therefore need to re-envision how data access mechanisms are defined and implemented. To this end, we introduce the Eagle-Eyed Elephant (E3) framework for boosting the efficiency of query processing in Hadoop by avoiding accesses of data splits that are irrelevant to the query at hand. Using novel techniques involving inverted indexes over splits, domain segmentation, materialized views, and adaptive caching, E3 avoids accessing irrelevant splits even in the face of evolving workloads and data. Our experiments show that E3 can achieve up to 20x cost savings with small to moderate storage overheads.

We investigate the approximability of several classes of real-valued functions by functions of a small number of variables (juntas). Our main results are tight bounds on the number of variables required to approximate a function f:{0, 1}n→ [0,1] within ℓ2-error ϵ over the uniform distribution: If f is sub modular, then it is ϵ-close to a function of O(1/ϵ2log 1/ϵ) variables. This is an exponential improvement over previously known results FeldmanKV:13. We note that Ω(1/ϵ2) variables are necessary even for linear functions. If f is fractionally sub additive (XOS) it is ε-close to a function of 2O(1/ϵ2)variables. This result holds for all functions with low total ℓ1-influence and is a real-valued analogue of Fried gut's theorem for boolean functions. We show that 2Ω(1/ϵ)variables are necessary even for XOS functions. As applications of these results, we provide learning algorithms over the uniform distribution. For XOS functions, we give a PAC learning algorithm that runs in time 21/poly(ϵ)poly(n). For sub modular functions we give an algorithm in the more demanding PMAC learning model BalcanHarvey:[12] which requires a multiplicative (1 + γ) factor approximation with probability at least 1 - ϵ over the target distribution. Our uniform distribution algorithm runs in time 21/poly(γϵ)poly(n). This is the first algorithm in the PMAC model that can achieve a constant approximation factor arbitrarily close to 1 for all sub modular functions (even over the uniform distribution). It relies crucially on our approximation by junta result. As follows from the lower bounds in FeldmanKV:13 both of these algorithms are close to optimal. We also give applications for proper learning, testing and agnostic learning with value queries of these classes.

We prove the first communication complexity lower bound for constant-factor approximation of the submodular welfare problem. More precisely, we show that a -approximation (≃ 0.816) for welfare maximization in combinatorial auctions with submodular valuations would require exponential communication. We also show NP-hardness of -approximation in a computational model where each valuation is given explicitly by a table of constant size. Both results rule out better than (1 − )-approximations in every oracle model with a separate oracle for each player, such as the demand oracle model. Our main tool is a new construction of monotone submodular functions that we call multi-peak submodular functions. Roughly speaking, given a family of sets , we construct a monotone submodular function f with a high value f(S) for every set S ∊ (a “peak”), and a low value on every set that does not intersect significantly any set in . We also study two other related problems: max-min allocation (for which we also get hardness of -approximation, in both models), and combinatorial public projects (for which we prove hardness of -approximation in the communication model, and hardness of -approximation in the computational model, using constant size valuations).

We study the approximability of multiway partitioning problems, examples of which include Multiway Cut, Node-weighted Multiway Cut, and Hypergraph Multiway Cut. We investigate these problems from the point of view of two possible generalizations: as Min-CSPs, and as Submodular Multiway Partition problems. These two generalizations lead to two natural relaxations that we call respectively the Local Distribution LP, and the Lovász relaxation. The Local Distribution LP is generally stronger than the Lovász relaxation, but applicable only to Min-CSP with predicates of constant size. The relaxations coincide in some cases such as Multiway Cut where they are both equivalent to the CKR relaxation. We show that the Lovász relaxation gives a (2 − 2/k)-approximation for Submodular Multiway Partition with k terminals, improving a recent 2-approximation [2]. We prove that this factor is optimal in two senses: (1) A (2 − 2/k − ∊)-approximation for Submodular Multiway Partition with k terminals would require exponentially many value queries (in the oracle model), or imply NP = RP (for certain explicit submodular functions). (2) For Hypergraph Multiway Cut and Node-weighted Multiway Cut with k terminals, both special cases of Submodular Multiway Partition, we prove that a (2 − 2/k − ∊)-approximation is NP-hard, assuming the Unique Games Conjecture. Both our hardness results are more general: (1) We show that the notion of symmetry gap, previously used for submodular maximization problems [19, 6], also implies hardness results for submodular minimization problems. (2) Assuming the Unique Games Conjecture, we show that the Local Distribution LP gives an optimal approximation for every Min-CSP that includes the Not-Equal predicate. Finally, we connect the two hardness techniques by proving that the integrality gap of the Local Distribution LP coincides with the symmetry gap of the multilinear relaxation (for a related instance). This shows that the appearance of the same hardness threshold for a Min-CSP and the related submodular minimization problem is not a coincidence.

We prove that no online algorithm (even randomized, against an oblivious adversary) is better than 1/2-competitive for welfare maximization with coverage valuations, unless NP = RP. Since the Greedy algorithm is known to be 1/2-competitive for monotone submodular valuations, of which coverage is a special case, this proves that Greedy provides the optimal competitive ratio. On the other hand, we prove that Greedy in a stochastic setting with i.i.d. items and valuations satisfying diminishing returns is (1 − 1/e)-competitive, which is optimal even for coverage valuations, unless NP = RP. For online budget-additive allocation, we prove that no algorithm can be 0.612-competitive with respect to a natural LP which has been used previously for this problem.

This paper studies the computational complexity of the classic problem of deletion propagation in a relational database, where tuples are deleted from the base relations in order to realize a desired deletion of tuples from the view. Such an operation may result in a (sometimes unavoidable) side effect: deletion of additional tuples from the view, besides the intentionally deleted ones. The goal is to minimize the side effect. The complexity of this problem has been well studied in the case where only a single tuple is deleted from the view. However, only little is known within the more realistic scenario of multi-tuple deletion, which is the topic of this paper. The class of conjunctive queries (CQs) is among the most well studied in the literature, and we focus here on views defined by CQs that are self-join free (sjf-CQs). Our main result is a trichotomy in complexity, classifying all sjf-CQs into three categories: those for which the problem is in polynomial time, those for which the problem is NP-hard but polynomial-time approximable (by a constant-factor), and those for which even an approximation (by any factor) is NP-hard to obtain. A corollary of this trichotomy is a dichotomy in the complexity of deciding whether a side-effect-free solution exists, in the multi-tuple case. We further extend the full classification to accommodate the presence of a constant upper bound on the number of view tuples to delete, and the presence of functional dependencies. Finally, we establish (positive and negative) complexity results on approximability for the dual problem of maximizing the number of view tuples surviving (rather than minimizing the side effect incurred in) the deletion propagation.

We consider the classical matroid matching problem. Unweighted matroid matching for linearly represented matroids was solved by Lovász, and the problem is known to be intractable for general matroids. We present a polynomial-time approximation scheme for unweighted matroid matching for general matroids. In contrast, we show that natural linear-programming relaxations that have been studied have an $\Omega(n)$ integrality gap, and, moreover, $\Omega(n)$ rounds of the Sherali--Adams hierarchy are necessary to bring the gap down to a constant. More generally, for any fixed $k \geq 2$ and $\epsilon>0$, we obtain a $(k/2+\epsilon)$-approximation for matroid matching in $k$-uniform hypergraphs, also known as the matroid $k$-parity problem. As a consequence, we obtain a $(k/2+\epsilon)$-approximation for the problem of finding the maximum-cardinality set in the intersection of $k$ matroids. We also give a $3/2$-approximation for the weighted version of a special case of matroid matching, the matchoid problem.

A number of recent results on optimization problems involving submodular functions have made use of the multilinear relaxation of the problem. These results hold typically in the value oracle model, where the objective function is accessible via a black box returning $f(S)$ for a given $S$. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of symmetry gap. Our main result is that for any fixed instance that exhibits a certain symmetry gap in its multilinear relaxation, there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries. This unifies several known hardness results for submodular maximization, e.g., the optimality of $(1-1/e)$-approximation for monotone submodular maximization under a cardinality constraint and the impossibility of $(\frac12+\epsilon)$-approximation for unconstrained (nonmonotone) submodular maximization. As a new application, we consider the problem of maximizing a nonmonotone submodular function over the bases of a matroid. A $(\frac16-o(1))$-approximation has been developed for this problem, assuming that the matroid contains two disjoint bases. We show that the best approximation one can achieve is indeed related to packings of bases in the matroid. Specifically, for any $k \geq 2$, there is a class of matroids of fractional base packing number $\nu = \frac{k}{k-1}$ such that any algorithm achieving a better than $(1-\frac{1}{\nu}) = \frac{1}{k}$-approximation for this class would require exponentially many value queries. In particular, there is no constant factor approximation for maximizing a nonmonotone submodular function over the bases of a general matroid. On the positive side, we present a $\frac12 (1-\frac{1}{\nu}-o(1))$-approximation algorithm assuming fractional base packing number at least $\nu$, where $\nu \in (1,2]$. We also present an improved $0.309$-approximation for maximization of a nonmonotone submodular function subject to a matroid independence constraint, improving the previously known factor of $\frac14-\epsilon$. For this problem, we obtain a hardness of $(\frac12 + \epsilon)$-approximation for any fixed $\epsilon>0$.

One of the fundamental questions of Algorithmic Mechanism Design is whether there exists an inherent clash between truthfulness and computational tractability: in particular, whether polynomial-time truthful mechanisms for combinatorial auctions are provably weaker in terms of approximation ratio than non-truthful ones. This question was very recently answered for universally truthful mechanisms for combinatorial auctions [4], and even for truthful-in-expectation mechanisms [12]. However, both of these results are based on information-theoretic arguments for valuations given by a value oracle, and leave open the possibility of polynomial-time truthful mechanisms for succinctly described classes of valuations.

We consider submodular optimization problems, and provide a general way of translating oracle inapproximability results arising from the symmetry gap technique to computational complexity inapproximability results, where the submodular function is given explicitly (under the assumption that NP ≠ RP). Applications of our technique include an optimal computational hardness of (1/2 + ε)-approximation for maximizing a symmetric nonnegative submodular function, an optimal hardness of (1-(1-1/k)k + ε)-approximation for welfare maximization in combinatorial auctions with k submodular bidders (for constant k), super-constant hardness for maximizing a nonnegative submodular function over matroid bases, and tighter bounds for maximizing a monotone submodular function subject to a cardinality constraint. Unlike the vast majority of computational inapproximability results, our approach does not use the PCP machinery or the Unique Games Conjecture, but relies instead on a direct reduction from Unique-SAT using list-decodable codes.

In deletion propagation, tuples from the database are deleted in order to reflect the deletion of a tuple from the view. Such an operation may result in the (often necessary) deletion of additional tuples from the view, besides the intentionally deleted one. The article studies the complexity of deletion propagation, where the view is defined by a conjunctive query (CQ), and the goal is to maximize the number of tuples that remain in the view. Buneman et al. showed that for some simple CQs, this problem can be solved by a straightforward algorithm, which is called here the unidimensional algorithm. The article identifies additional cases of CQs where the unidimensional algorithm succeeds, and in contrast, shows that for some other CQs the problem is NP-hard to approximate better than some constant ratio. In fact, it is shown here that among the CQs without self joins, the hard CQs are exactly the ones that the unidimensional algorithm fails on. In other words, the following dichotomy result is proved: for every CQ without self joins, deletion propagation is either APX-hard or solvable (in polynomial time) by the unidimensional algorithm. The article then presents approximation algorithms for certain CQs where deletion propagation is APX-hard. Specifically, two constant-ratio (and polynomial-time) approximation algorithms are given for the class of sunflower CQs (i.e., CQs having a sunflower hypergraph) without self joins. The first algorithm, providing the approximation ratio 1 − 1/ e , is obtained by formulating the problem at hand as that of maximizing a monotone submodular function subject to a matroid constraint, and then using a known algorithm for such maximization. The second algorithm gives a smaller approximation ratio, 1/2, yet in polynomial time even under combined complexity. Finally, it is shown that self joins can significantly harden approximation in deletion propagation.

The design of computationally efficient and incentive compatible mechanisms that solve or approximate fundamental resource allocation problems is the main goal of algorithmic mechanism design. A central example in both theory and practice is welfare-maximization in combinatorial auctions. Recently, a randomized mechanism has been discovered for combinatorial auctions that is truthful in expectation and guarantees a (1-1/e)-approximation to the optimal social welfare when players have coverage valuations [DRY11]. This approximation ratio is the best possible even for non-truthful algorithms, assuming P does not equal NP. Given the recent sequence of negative results for combinatorial auctions under more restrictive notions of incentive compatibility, this development raises a natural question: Are truthful-in-expectation mechanisms compatible with polynomial-time approximation in a way that deterministic or universally truthful mechanisms are not? In particular, can polynomial-time truthful-in-expectation mechanisms guarantee a near-optimal approximation ratio for more general variants of combinatorial auctions? We prove that this is not the case. Specifically, the result of [DRY11] cannot be extended to combinatorial auctions with sub modular valuations in the value oracle model. (Absent strategic considerations, a (1-1/e)-approximation is still achievable in this setting.) More precisely, we prove that there is a constant \gamma>0 such that there is no randomized mechanism that is truthful-in-expectation -- or even approximately truthful-in-expectation -- and guarantees an m^{-\gamma}-approximation to the optimal social welfare for combinatorial auctions with sub modular valuations in the value oracle model. We also prove an analogous result for the flexible combinatorial public projects (CPP) problem, where a truthful-in-expectation $(1-1/e)$-approximation for coverage valuations has been recently developed [Dughmi11]. We show that there is no truthful-in-expectation -- or even approximately truthful-in-expectation -- mechanism that achieves an m^{-\gamma}-approximation to the optimal social welfare for combinatorial public projects with sub modular valuations in the value oracle model. Both our results present an unexpected separation between coverage functions and sub modular functions, which does not occur for these problems without strategic considerations.

In deletion propagation, tuples from the database are deleted in order to reflect the deletion of a tuple from the view. Such an operation may result in the (often necessary) deletion of additional tuples from the view, besides the intentionally deleted one. The complexity of deletion propagation is studied, where the view is defined by a conjunctive query (CQ), and the goal is to maximize the number of tuples that remain in the view. Buneman et al. showed that for some simple CQs, this problem can be solved by a trivial algorithm. This paper identifies additional cases of CQs where the trivial algorithm succeeds, and in contrast, it proves that for some other CQs the problem is NP-hard to approximate better than some constant ratio. In fact, this paper shows that among the CQs without self joins, the hard CQs are exactly the ones that the trivial algorithm fails on. In other words, for every CQ without self joins, deletion propagation is either APX-hard or solvable by the trivial algorithm.

Motivated by multi-budgeted optimization and other applications, we consider the problem of randomly rounding a fractional solution x in the (non-bipartite graph) matching and matroid intersection polytopes. We show that for any fixed δ > 0, a given point x can be rounded to a random solution R such that E[1R] = (1 − δ)x and any linear function of x satisfies dimension-free Chernoff-Hoeffding concentration bounds (the bounds depend on S and the expectation μ). We build on and adapt the swap rounding scheme in our recent work [9] to achieve this result. Our main contribution is a non-trivial martingale based analysis framework to prove the desired concentration bounds. In this paper we describe two applications. We give a randomized PTAS for matroid intersection and matchings with any fixed number of budget constraints. We also give a deterministic PTAS for the case of matchings. The concentration bounds also yield related results when the number of budget constraints is not fixed. As a second application we obtain an algorithm to compute in polynomial time an ε-approximate Pareto-optimal set for the multi-objective variants of these problems, when the number of objectives is a fixed constant. We rely on a result of Papadimitriou and Yannakakis [26].

We consider the problem of maximizing a non-negative (possibly non-monotone) submodular set function with or without constraints. Feige et al. [9] showed a 2/5-approximation for the unconstrained problem and also proved that no approximation better than 1/2 is possible in the value oracle model. Constant-factor approximation has been also known for submodular maximization subject to a matroid independence constraint (a factor of 0.309 [33]) and for submodular maximization subject to a matroid base constraint, provided that the fractional base packing number v is bounded away from 1 (a 1/4-approximation assuming that v ≥ 2 [33]). In this paper, we propose a new algorithm for submodular maximization which is based on the idea of simulated annealing. We prove that this algorithm achieves improved approximation for two problems: a 0.41-approximation for unconstrained submodular maximization, and a 0.325-approximation for submodular maximization subject to a matroid independence constraint. On the hardness side, we show that in the value oracle model it is impossible to achieve a 0.478-approximation for submodular maximization subject to a matroid independence constraint, or a 0.394-approximation subject to a matroid base constraint in matroids with two disjoint bases. Even for the special case of cardinality constraint, we prove it is impossible to achieve a 0.491-approximation. (Previously it was conceivable that a 1/2-approximation exists for these problems.) It is still an open question whether a 1/2-approximation is possible for unconstrained submodular maximization.

We consider the problem of maximizing a non-negative submodular set function f:2N -> RR+ over a ground set N subject to a variety of packing type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that allows us to derive a number of new results, in particular when f may be a non-monotone function. Our algorithms are based on (approximately) solving the multilinear extension F of f [5] over a polytope P that represents the constraints, and then effectively rounding the fractional solution. Although this approach has been used quite successfully in some settings [6, 22, 24, 13, 3], it has been limited in some important ways. We overcome these limitations as follows.

Let $f:2^X \rightarrow \cal R_+$ be a monotone submodular set function, and let $(X,\cal I)$ be a matroid. We consider the problem ${\rm max}_{S \in \cal I} f(S)$. It is known that the greedy algorithm yields a $1/2$-approximation [M. L. Fisher, G. L. Nemhauser, and L. A. Wolsey, Math. Programming Stud., no. 8 (1978), pp. 73–87] for this problem. For certain special cases, e.g., ${\rm max}_{|S| \leq k} f(S)$, the greedy algorithm yields a $(1-1/e)$-approximation. It is known that this is optimal both in the value oracle model (where the only access to f is through a black box returning $f(S)$ for a given set S) [G. L. Nemhauser and L. A. Wolsey, Math. Oper. Res., 3 (1978), pp. 177–188] and for explicitly posed instances assuming $P \neq NP$ [U. Feige, J. ACM, 45 (1998), pp. 634–652]. In this paper, we provide a randomized $(1-1/e)$-approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [J. Combin. Optim., 8 (2004), pp. 307–328], and a continuous greedy process that may be of independent interest. As a special case, our algorithm implies an optimal approximation for the submodular welfare problem in the value oracle model [J. Vondrák, Proceedings of the $38$th ACM Symposium on Theory of Computing, 2008, pp. 67–74]. As a second application, we show that the generalized assignment problem (GAP) is also a special case; although the reduction requires $|X|$ to be exponential in the original problem size, we are able to achieve a $(1-1/e-o(1))$-approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.

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.

We consider the problem of randomly rounding a fractional solution x in an integer polytope P ⊆ [0,1]nto a vertex X of P, so that E[X] = x. Our goal is to achieve concentration properties for linear and submodular functions of the rounded solution. Such dependent rounding techniques, with concentration bounds for linear functions, have been developed in the past for two poly topes: the assignment poly tope (that is, bipartite matchings and 6-matchings) [32], [19], [23], and more recently for the spanning tree poly tope [2]. These schemes have led to a number of new algorithmic results. In this paper we describe a new swap rounding technique which can be applied in a variety of settings including matroids and matroid intersection, while providing Chernoff-type concentration bounds for linear and submodular functions of the rounded solution. In addition to existing techniques based on negative correlation, we use a martingale argument to obtain an exponential tail estimate for monotone submodular functions. The rounding scheme explicitly exploits exchange properties of the underlying combinatorial structures, and highlights these properties as the basis for concentration bounds. Matroids and matroid intersection provide a unifying framework for several known applications [19], [23], [7], [22], [2] as well as new ones, and their generality allows a richer set of constraints to be incorporated easily. We give some illustrative examples, with a more comprehensive discussion deferred to a later version of the paper.

We consider the classical matroid matching problem. Unweighted matroid matching for linear matroids was solved by Lovasz, and the problem is known to be intractable for general matroids. We present a PTAS for unweighted matroid matching for general matroids. In contrast, we show that natural LP relaxations have an Ω(n) integrality gap and moreover, Ω(n) rounds of the Sherali-Adams hierarchy are necessary to bring the gap down to a constant. More generally, for any fixed k>=2 and ε>0, we obtain a (k/2+ε)-approximation for matroid matching in k-uniform hypergraphs, also known as the matroid k-parity problem. As a consequence, we obtain a (k/2+ε)-approximation for the problem of finding the maximum-cardinality set in the intersection of k matroids. We have also designed a 3/2-approximation for the weighted version of a special case of matroid matching, the matchoid problem.

A number of recent results on optimization problems involving submodular functions have made use of the "multilinear relaxation" of the problem. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of "symmetry gap". Our main result is that for any fixed instance that exhibits a certain "symmetry gap" in its multilinear relaxation, there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries. This unifies several known hardness results for submodular maximization, e.g. the optimality of (1-1/e)-approximation for monotone submodular maximization under a cardinality constraint, and the impossibility of (1/2+epsilon)-approximation for unconstrained (non-monotone) submodular maximization. It follows from our result that (1/2+epsilon)-approximation is also impossible for non-monotone submodular maximization subject to a (non-trivial) matroid constraint. On the algorithmic side, we present a 0.309-approximation for this problem, improving the previously known factor of 1/4-o(1).As another application, we consider the problem of maximizing a non-monotone submodular function over the bases of a matroid. A (1/6-o(1))-approximation has been developed for this problem, assuming that the matroid contains two disjoint bases. We show that the best approximation one can achieve is indeed related to packings of bases in the matroid. Specifically, for any k≫=2, there is a class of matroids of fractional base packing number nu = k/(k-1), such that any algorithm achieving a better than (1-1/nu)-approximation for this class would require exponentially many value queries. On the positive side, we present a 1/2 (1-1/nu-o(1))-approximation algorithm for the same problem. Our hardness results hold in fact for very special symmetric instances. For such symmetric instances, we show that the approximation factors of 1/2 (for submodular maximization subject to a matroid constraint) and 1-1/nu (for a matroid base constraint) can be achieved algorithmically and hence are optimal.

We provide tight information-theoretic lower bounds for the welfare maximization problem in combinatorial auctions. In this problem, the goal is to partition m items among k bidders in a way that maximizes the sum of bidders' values for their allocated items. Bidders have complex preferences over items expressed by valuation functions that assign values to all subsets of items.

In the Submodular Welfare Problem, m items are to be distributed among n players with utility functions wi: 2[m] → R+. The utility functions are assumed to be monotone and submodular. Assuming that player i receives a set of items Si, we wish to maximize the total utility ∑i=1n wi(Si). In this paper, we work in the value oracle model where the only access to the utility functions is through a black box returning wi(S) for a given set S. Submodular Welfare is in fact a special case of the more general problem of submodular maximization subject to a matroid constraint: max{f(S): S ∈ I}, where f is monotone submodular and I is the collection of independent sets in some matroid.

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.

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 a stochastic variant of the NP-hard 0/1 knapsack problem in which item values are deterministic and item sizes are independent random variables with known, arbitrary distributions. Items are placed in the knapsack sequentially, and the act of placing an item in the knapsack instantiates its size. Our goal is to compute a solution "policy" that maximizes the expected value of items placed in the knapsack, and we consider both non-adaptive policies (that designate a priori a fixed sequence of items to insert) and adaptive policies (that can make dynamic choices based on the instantiated sizes of items placed in the knapsack thus far). We show that adaptivity provides only a constant-factor improvement by demonstrating a greedy non-adaptive algorithm that approximates the optimal adaptive policy within a factor of 7. We also design an adaptive polynomial-time algorithm which approximates the optimal adaptive policy within a factor of 5 + /spl epsiv/, for any constant /spl epsiv/ > 0.