On submodular prophet inequalities and correlation gap

We introduce a novel framework of Prophet Inequalities for combinatorial valuation functions. For a (non-monotone) submodular objective function over an arbitrary matroid feasibility constraint, we give an O(1)-competitive algorithm. For a monotone subadditive objective function over an arbitrary downward-closed feasibility constraint, we give an O(log n log2r)-competitive algorithm (where r is the cardinality of the largest feasible subset).Inspired by the proof of our subadditive prophet inequality, we also obtain an O(log n · log2r)-competitive algorithm for the Secretary Problem with a monotone subadditive objective function subject to an arbitrary downward-closed feasibility constraint. Even for the special case of a cardinality feasibility constraint, our algorithm circumvents an [EQUATION] lower bound by Bateni, Hajiaghayi, and Zadimoghaddam [10] in a restricted query model.En route to our submodular prophet inequality, we prove a technical result of independent interest: we show a variant of the Correlation Gap Lemma [14, 1] for non-monotone submodular functions.

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 study the problem of fairly dividing indivisible goods among a set of agents under the fairness notion of Any Price Share (APS). APS is known to dominate the widely studied Maximin share (MMS). Since an exact APS allocation may not exist, the focus has traditionally been on the computation of approximate APS allocations. [4] studied the problem under additive valuations, and asked (i) how large can the APS value be compared to the MMS value? and (ii) what guarantees can one achieve beyond additive functions. We partly answer these questions by considering valuations beyond additive, namely submodular and XOS functions, with binary marginals. For the submodular functions with binary marginals, also known as matroid rank functions (MRFs), we show that APS is exactly equal to MMS. Consequently, following [5] we show that an exact APS allocation exists and can be computed efficiently while maximizing the social welfare. Complementing this result, we show that it is NP-hard to compute the APS value within a factor of 5/6 for submodular valuations with three distinct marginals of {0, 1/2, 1.} We then consider binary XOS functions, which are immediate generalizations of binary submodular functions in the complement free hierarchy. In contrast to the MRFs setting, MMS and APS values are not equal under this case. Nevertheless, we can show that they are only a constant factor apart. In particular, we show that under binary XOS valuations, MMS ≤= APS ≤= 2 x MMS + 1. Further, we show that this is almost the tightest bound we can get using MMS, by giving an instance where APS ≥= 2 x MMS. The upper bound on APS, combined with [17], implies a 0.1222-approximation for APS under binary XOS valuations. And the lower bound implies the non-existence of better than 0.5-APS even when agents have identical valuations, which is in sharp contrast to the guaranteed existence of exact MMS allocation when agent valuations are identical.

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Tight Guarantees for Multi-unit Prophet Inequalities and Online Stochastic KnapsackJiashuo Jiang, Will Ma, and Jiawei ZhangJiashuo Jiang, Will Ma, and Jiawei Zhangpp.1221 - 1246Chapter DOI:https://doi.org/10.1137/1.9781611977073.51PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Prophet inequalities are a useful tool for designing online allocation procedures and comparing their performance to the optimal offline allocation. In the basic setting of k-unit prophet inequalities, the procedure of Alaei [2] with its celebrated performance guarantee of has found widespread adoption in mechanism design and general online allocation problems in online advertising, healthcare scheduling, and revenue management. Despite being commonly used for implementing a fractional allocation in an online fashion, the tightness of Alaei's procedure for a given k has remained unknown. In this paper we resolve this question, characterizing the tight bound by identifying the structure of the optimal online implementation, and consequently improving the best-known guarantee for k-unit prophet inequalities for all k > 1. We also consider the more general online stochastic knapsack problem where each individual allocation can consume an arbitrary fraction of the initial capacity. Here we introduce a new “best-fit” procedure for implementing a fractionally-feasible knapsack solution online, with a performance guarantee of ≈ 0.319, which we also show is tight with respect to the standard LP relaxation. This improves the previously best-known guarantee of 0.2 for online knapsack. Our analysis differs from existing ones by eschewing the need to split items into “large” or “small” based on capacity consumption, using instead an invariant for the overall utilization on different sample paths. All in all, our results imply tight (non-greedy) Online Contention Resolution Schemes for k-uniform matroids and the knapsack polytope, respectively, which has further implications. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771

We consider a ground set E and a submodular function acting on it. We first propose a “set multicovering” problem in which the value (price) of any is . We show that the linear program (LP) of this problem can be directly solved by applying a submodular function minimization (SFM) algorithm on the dual LP. However, the main results of this study concern prize‐collecting multicovering with submodular pricing, that is, a more general and more difficult “multicovering” problem in which elements can be left uncovered by paying a penalty. We formulate it as a large‐scale LP (with variables representing all subsets of E) that could be tackled by column generation (CG; for a CG algorithm for “set‐covering” problems with submodular pricing). However, we do not solve this large‐scale LP by CG, but we solve it in polynomial time by exploiting its integrality properties. More exactly, after appropriate restructuring, the dual LP can be transformed into an LP that has been thoroughly studied (as a primal) in the SFM theory. Solving this LP reduces to optimizing a strong map of submodular functions. For this, we use the Fleischer–Iwata framework that optimizes all these functions within the same asymptotic running time as solving a single SFM, that is, in , where and γ is the complexity of one submodular evaluation. Besides solving the problem, the proposed approach can be useful to (a) simultaneously find the best solution of up to functions under strong map relations in time, (b) perform sensitivity analysis in very short time (even at no extra cost), and (c) reveal combinatorial insight into the primal–dual optimal solutions. We present several potential applications throughout the paper, from production planning to combinatorial auctions.

Models for the processes by which ideas and influence propagate through a social network have been studied in a number of domains, including the diffusion of medical and technological innovations, the sudden and widespread adoption of various strategies in game-theoretic settings, and the effects of “word of mouth” in the promotion of new products. Motivated by the design of viral marketing strategies, Domingos and Richardson posed a fundamental algorithmic problem for such social network processes: if we can try to convince a subset of individuals to adopt a new product or innovation, and the goal is to trigger a large cascade of further adoptions, which set of individuals should we target? We consider this problem in several of the most widely studied models in social network analysis. The optimization problem of selecting the most influential nodes is NP-hard here. The two conference papers upon which this article is based (KDD 2003 and ICALP 2005) provide the first provable approximation guarantees for efficient algorithms. Using an analysis framework based on submodular functions, we show that a natural greedy strategy obtains a solution that is provably within 63% of optimal for several classes of models; our framework suggests a general approach for reasoning about the performance guarantees of algorithms for these types of influence problems in social networks. We also provide computational experiments on large collaboration networks, showing that in addition to their provable guarantees, our approximation algorithms significantly out-perform node-selection heuristics based on the well-studied notions of degree centrality and distance centrality from the field of social networks. The present article is an expanded version of two conference papers which appeared in KDD 2003 and ICALP 2005, respectively.

In this paper, we consider the problem of maximizing a non-negative submodular function f , defined on a (finite) ground set N , subject to matroid constraints. A function $f: 2^N \rightarrow {\mathbb R}$ is submodular if for all S , T *** N , f (S *** T ) + f (S *** T ) ≤ f (S ) + f (T ). Furthermore, all submodular functions that we deal with are assumed to be non-negative. Throughout, we assume that our submodular function f is given by a value oracle ; i.e., for a given set S *** N , an algorithm can query an oracle to find the value f (S ). Without loss of generality, we take the ground set N to be [n ] = {1,2,...,n }.

We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries. We obtain the first algorithms with low adaptivity for submodular maximization with a matroid constraint. Our algorithms achieve a $1-1/e-ε$ approximation for monotone functions and a $1/e-ε$ approximation for non-monotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is $O(\log^2{n}/ε^3)$, which is an exponential speedup over the existing algorithms. We obtain the first parallel algorithm for non-monotone submodular maximization subject to packing constraints. Our algorithm achieves a $1/e-ε$ approximation using $O(\log(n/ε) \log(1/ε) \log(n+m)/ ε^2)$ parallel rounds, which is again an exponential speedup in parallel time over the existing algorithms. For monotone functions, we obtain a $1-1/e-ε$ approximation in $O(\log(n/ε)\log(m)/ε^2)$ parallel rounds. The number of parallel rounds of our algorithm matches that of the state of the art algorithm for solving packing LPs with a linear objective. Our results apply more generally to the problem of maximizing a diminishing returns submodular (DR-submodular) function.

Given a set V , the problem of unconstrained submodular maximization with modular costs (USM-MC) asks for a subset S ⊆ V that maximizes f ( S ) - c ( S ), where f is a non-negative, monotone, and submodular function that gauges the utility of S , and c is a non-negative and modular function that measures the cost of S. This problem finds applications in numerous practical scenarios, such as profit maximization in viral marketing on social media. This paper presents ROI-Greedy, a polynomial time algorithm for USM-MC that returns a solution S satisfying [EQUATION], where S * is the optimal solution to USM-MC. To our knowledge, ROI-Greedy is the first algorithm that provides such a strong approximation guarantee. In addition, we show that this worst-case guarantee is tight , in the sense that no polynomial time algorithm can ensure [EQUATION], for any ϵ > 0. Further, we devise a non-trivial extension of ROI-Greedy to solve the profit maximization problem, where the precise value of f ( S ) for any set S is unknown and can only be approximated via sampling. Extensive experiments on benchmark datasets demonstrate that ROI-Greedy significantly outperforms competing methods in terms of the tradeoff between efficiency and solution quality.

This paper addresses the Minimum Linear Ordering Problem (MLOP): Given a nonnegative set function f on a finite set V, find a linear ordering on V such that the sum of the function values for all the suffixes is minimized. This problem generalizes well-known problems such as the Minimum Linear Arrangement, Min Sum Set Cover, Minimum Latency Set Cover, and Multiple Intents Ranking. Extending a result of Feige, Lovasz, and Tetali (2004) on Min Sum Set Cover, we show that the greedy algorithm provides a factor 4 approximate optimal solution when the cost function f is supermodular. We also present a factor 2 rounding algorithm for MLOP with a monotone submodular cost function, using the convexity of the Lovasz extension. These are among very few constant factor approximation algorithms for NP-hard minimization problems formulated in terms of submodular/supermodular functions. In contrast, when f is a symmetric submodular function, the problem has an information theoretic lower bound of 2 on the approximability.

Server deployment is a fundamental task in mobile edge computing: where to place the edge servers and what user cells to assign to them. To make this decision is context-specific, but common goals are (1) computing efficiency: maximize the amount of workload processed by the edge, and (2) communication efficiency: minimize the communication cost between the cells and their assigned servers. We focus on practical scenarios where the user workload in each cell is unknown and time-varying, and so are the effective capacities of the servers. Our research problem is to choose a subset of candidate servers and assign them to the user cells such that the above goals are sustainably achieved under the above uncertainties. We formulate this problem as a stochastic bilevel optimization, which is strongly NP-hard and unseen in the literature. By approximating the objective function with submodular functions, we can utilize state-of-the-art greedy algorithms for submodular maximization to effectively solve our problem. We evaluate the proposed algorithm using real-world data, showing its superiority to alternative methods; the improvement can be as high as 55%.

We consider the problem of budget feasible mechanism design proposed by Singer, but in a Bayesian setting. A principal has a public value for hiring a subset of the agents and a budget, while the agents have private costs for being hired. We consider both additive and submodular value functions of the principal. We show that there are simple, practical, ex post budget balanced posted pricing mechanisms that approximate the value obtained by the Bayesian optimal mechanism that is budget balanced only in expectation. A main motivating application for this work is crowdsourcing, e.g., on Mechanical Turk, where workers are drawn from a large population and posted pricing is standard. Our analysis methods relate to contention resolution schemes in submodular optimization of Vondrak et al. and the correlation gap analysis of Yan.

Deterministic approximation algorithm for submodular maximization subject to a matroid constraint

The emergence of applications in machine learning, social networks, and sensor placement has naturally introduced challenges in maximizing submodular or non-submodular functions over integer lattices under knapsack constraints. While efficient algorithms exist for maximizing monotone non-negative submodular functions over both finite sets and integer lattices, these methods do not generalize straightforwardly to non-submodular objectives. To address this gap, we propose a two-phase greedy algorithm with a provable approximation ratio of [Formula: see text], given that the objective function satisfies the Diminishing Return ratio (DR-ratio) [Formula: see text] and the generalized curvature [Formula: see text]. To our knowledge, this represents the first non-trivial approximation guarantee for such problem. Experimental evaluations demonstrate the algorithm’s practical effectiveness.

Many algorithms have been devised for minimizing the costs associated with obtaining the answer to a single, isolated query in a distributed database system. However, if more than one query may be processed by the system at the same time and if the arrival times of the queries are unknown, the determination of optimal query-processing strategies becomes a stochastic optimization problem. In order to cope with such problems, a theoretical state-transition model is presented that treats the system as one operating under a stochastic load. Query-processing strategies may then be distributed over the processors of a network as probability distributions, in a manner which accommodates many queries over time. It is then shown that the model leads to the determination of optimal query-processing strategies as the solution of mathematical programming problems, and analytical results for several examples are presented. Furthermore, a divide-and-conquer approach is introduced for decomposing stochastic query optimization problems into distinct subproblems for processing queries sequentially and in parallel.