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

Budget-Feasible Mechanism Design for Non-Monotone Submodular Objectives: Offline and Online

The framework of budget-feasible mechanism design studies procurement auctions where the auctioneer (buyer) aims to maximize his valuation function subject to a hard budget constraint. We study the problem of designing truthful mechanisms that have good approximation guarantees and never pay the participating agents (sellers) more than the budget. We focus on the case of general (non-monotone) submodular valuation functions and derive the first truthful, budget-feasible and $O(1)$-approximate mechanisms that run in polynomial time in the value query model, for both offline and online auctions. Prior to our work, the only $O(1)$-approximation mechanism known for non-monotone submodular objectives required an exponential number of value queries. At the heart of our approach lies a novel greedy algorithm for non-monotone submodular maximization under a knapsack constraint. Our algorithm builds two candidate solutions simultaneously (to achieve a good approximation), yet ensures that agents cannot jump from one solution to the other (to implicitly enforce truthfulness). Ours is the first mechanism for the problem where---crucially---the agents are not ordered with respect to their marginal value per cost. This allows us to appropriately adapt these ideas to the online setting as well. To further illustrate the applicability of our approach, we also consider the case where additional feasibility constraints are present. We obtain $O(p)$-approximation mechanisms for both monotone and non-monotone submodular objectives, when the feasible solutions are independent sets of a $p$-system. With the exception of additive valuation functions, no mechanisms were known for this setting prior to our work. Finally, we provide lower bounds suggesting that, when one cares about non-trivial approximation guarantees in polynomial time, our results are asymptotically best possible.Comment: Accepted to EC 201

Streaming Non-monotone Submodular Maximization: Personalized Video Summarization on the Fly

The need for real time analysis of rapidly producing data streams (e.g., video and image streams) motivated the design of streaming algorithms that can efficiently extract and summarize useful information from massive data "on the fly". Such problems can often be reduced to maximizing a submodular set function subject to various constraints. While efficient streaming methods have been recently developed for monotone submodular maximization, in a wide range of applications, such as video summarization, the underlying utility function is non-monotone, and there are often various constraints imposed on the optimization problem to consider privacy or personalization. We develop the first efficient single pass streaming algorithm, Streaming Local Search, that for any streaming monotone submodular maximization algorithm with approximation guarantee $\alpha$ under a collection of independence systems ${\cal I}$, provides a constant $1/\big(1+2/\sqrt{\alpha}+1/\alpha +2d(1+\sqrt{\alpha})\big)$ approximation guarantee for maximizing a non-monotone submodular function under the intersection of ${\cal I}$ and $d$ knapsack constraints. Our experiments show that for video summarization, our method runs more than 1700 times faster than previous work, while maintaining practically the same performance

Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints

We investigate two new optimization problems -- minimizing a submodular function subject to a submodular lower bound constraint (submodular cover) and maximizing a submodular function subject to a submodular upper bound constraint (submodular knapsack). We are motivated by a number of real-world applications in machine learning including sensor placement and data subset selection, which require maximizing a certain submodular function (like coverage or diversity) while simultaneously minimizing another (like cooperative cost). These problems are often posed as minimizing the difference between submodular functions [14, 35] which is in the worst case inapproximable. We show, however, that by phrasing these problems as constrained optimization, which is more natural for many applications, we achieve a number of bounded approximation guarantees. We also show that both these problems are closely related and an approximation algorithm solving one can be used to obtain an approximation guarantee for the other. We provide hardness results for both problems thus showing that our approximation factors are tight up to log-factors. Finally, we empirically demonstrate the performance and good scalability properties of our algorithms.Comment: 23 pages. A short version of this appeared in Advances of NIPS-201

Adversarially Robust Submodular Maximization under Knapsack Constraints

We propose the first adversarially robust algorithm for monotone submodular maximization under single and multiple knapsack constraints with scalable implementations in distributed and streaming settings. For a single knapsack constraint, our algorithm outputs a robust summary of almost optimal (up to polylogarithmic factors) size, from which a constant-factor approximation to the optimal solution can be constructed. For multiple knapsack constraints, our approximation is within a constant-factor of the best known non-robust solution. We evaluate the performance of our algorithms by comparison to natural robustifications of existing non-robust algorithms under two objectives: 1) dominating set for large social network graphs from Facebook and Twitter collected by the Stanford Network Analysis Project (SNAP), 2) movie recommendations on a dataset from MovieLens. Experimental results show that our algorithms give the best objective for a majority of the inputs and show strong performance even compared to offline algorithms that are given the set of removals in advance.Comment: To appear in KDD 201