Streaming Algorithms for Submodular Function Maximization

We consider the problem of maximizing a nonnegative submodular set function $f:2^{\mathcal{N}} \rightarrow \mathbb{R}^+$ subject to a $p$-matchoid constraint in the single-pass streaming setting. Previous work in this context has considered streaming algorithms for modular functions and monotone submodular functions. The main result is for submodular functions that are {\em non-monotone}. We describe deterministic and randomized algorithms that obtain a $\Omega(\frac{1}{p})$-approximation using $O(k \log k)$-space, where $k$ is an upper bound on the cardinality of the desired set. The model assumes value oracle access to $f$ and membership oracles for the matroids defining the $p$-matchoid constraint.Comment: 29 pages, 7 figures, extended abstract to appear in ICALP 201

Online Contention Resolution Schemes

We introduce a new rounding technique designed for online optimization problems, which is related to contention resolution schemes, a technique initially introduced in the context of submodular function maximization. Our rounding technique, which we call online contention resolution schemes (OCRSs), is applicable to many online selection problems, including Bayesian online selection, oblivious posted pricing mechanisms, and stochastic probing models. It allows for handling a wide set of constraints, and shares many strong properties of offline contention resolution schemes. In particular, OCRSs for different constraint families can be combined to obtain an OCRS for their intersection. Moreover, we can approximately maximize submodular functions in the online settings we consider. We, thus, get a broadly applicable framework for several online selection problems, which improves on previous approaches in terms of the types of constraints that can be handled, the objective functions that can be dealt with, and the assumptions on the strength of the adversary. Furthermore, we resolve two open problems from the literature; namely, we present the first constant-factor constrained oblivious posted price mechanism for matroid constraints, and the first constant-factor algorithm for weighted stochastic probing with deadlines.Comment: 33 pages. To appear in SODA 201

Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms

Constrained submodular maximization problems have long been studied, with near-optimal results known under a variety of constraints when the submodular function is monotone. The case of non-monotone submodular maximization is less understood: the first approximation algorithms even for the unconstrainted setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC '09, APPROX '09) show how to approximately maximize non-monotone submodular functions when the constraints are given by the intersection of p matroid constraints; their algorithm is based on local-search procedures that consider p-swaps, and hence the running time may be n^Omega(p), implying their algorithm is polynomial-time only for constantly many matroids. In this paper, we give algorithms that work for p-independence systems (which generalize constraints given by the intersection of p matroids), where the running time is poly(n,p). Our algorithm essentially reduces the non-monotone maximization problem to multiple runs of the greedy algorithm previously used in the monotone case. Our idea of using existing algorithms for monotone functions to solve the non-monotone case also works for maximizing a submodular function with respect to a knapsack constraint: we get a simple greedy-based constant-factor approximation for this problem. With these simpler algorithms, we are able to adapt our approach to constrained non-monotone submodular maximization to the (online) secretary setting, where elements arrive one at a time in random order, and the algorithm must make irrevocable decisions about whether or not to select each element as it arrives. We give constant approximations in this secretary setting when the algorithm is constrained subject to a uniform matroid or a partition matroid, and give an O(log k) approximation when it is constrained by a general matroid of rank k.Comment: In the Proceedings of WINE 201

Prophet Inequalities with Limited Information

In the classical prophet inequality, a gambler observes a sequence of stochastic rewards $V_1,...,V_n$ and must decide, for each reward $V_i$, whether to keep it and stop the game or to forfeit the reward forever and reveal the next value $V_i$. The gambler's goal is to obtain a constant fraction of the expected reward that the optimal offline algorithm would get. Recently, prophet inequalities have been generalized to settings where the gambler can choose $k$ items, and, more generally, where he can choose any independent set in a matroid. However, all the existing algorithms require the gambler to know the distribution from which the rewards $V_1,...,V_n$ are drawn. The assumption that the gambler knows the distribution from which $V_1,...,V_n$ are drawn is very strong. Instead, we work with the much simpler assumption that the gambler only knows a few samples from this distribution. We construct the first single-sample prophet inequalities for many settings of interest, whose guarantees all match the best possible asymptotically, \emph{even with full knowledge of the distribution}. Specifically, we provide a novel single-sample algorithm when the gambler can choose any $k$ elements whose analysis is based on random walks with limited correlation. In addition, we provide a black-box method for converting specific types of solutions to the related \emph{secretary problem} to single-sample prophet inequalities, and apply it to several existing algorithms. Finally, we provide a constant-sample prophet inequality for constant-degree bipartite matchings. We apply these results to design the first posted-price and multi-dimensional auction mechanisms with limited information in settings with asymmetric bidders

Prophet Secretary for Combinatorial Auctions and Matroids

The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and Feldman et al. \cite{feldman2015combinatorial} show that for adversarial arrival order of random variables the optimal prophet inequalities give a $1/2$-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the $1/2$-approximation and obtain $(1-1/e)$-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan \cite{yan2011mechanism} and Esfandiari et al. \cite{esfandiari2015prophet} who worked in the special cases where we can fully control the arrival order or when there is only a single item. Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.Comment: Preliminary version appeared in SODA 2018. This version improves the writeup on Fixed-Threshold algorithm