Constrained Non-monotone Submodular Maximization: Offline and Secretary Algorithms

Abstract

Constrained submodular maximization problems have long been studied, most recently in the context of auctions and computational advertising, 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 well understood: the first approximation algorithms even for the unconstrained 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 Ω(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). Both our algorithms and analyses are simple: 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.

Streaming Algorithms for Submodular Function Maximization

“…The (offline) problem max S∈I f (S) for various constraints has been extensively explored starting with the early work of Fisher, Nemhauser, Wolsey on greedy and local search algorithms [NWF78,FNW78]. Recent work has obtained many new and powerful results based on a variety of methods including variants of greedy [GRST10,BFNS14,BFNS12], local search [LMNS10, LSV10,FW14], and the multilinear relaxation [CCPV11, KST13, BKNS12, CVZ11]. Monotone submodular functions admit better bounds than non-monotone functions (see Table 1).…”

Online Submodular Maximization with Free Disposal: Randomization Beats ¼ for Partition Matroids

“…For maximizing a non-monotone submodular function with multiple matroid constraints, Lee et Our work is also closely related to the literature of submodular matroid secretary problem, which can be formulated as online submodular maximization without free disposal but assuming the elements arrive in random order. The submodular secretary problem has been widely studied recently [BUCM12,FNS11,GRST10,MTW13], and constant-competitive algorithms have been found on some special cases, such as on a uniform matroid constraint [BHZ13], or when the objective function is to maximize the largest weighted element in the set [Fre83,JPG66]. However, there is no constant ratio for the general submodular matroid secretary problem till now.…”

On Variants of the Matroid Secretary Problem

“…In the model where the input ordering of weights is adversarial (AA-CK), it is easy to see that no algorithm achieves probability better than 1/n [5]. We remark that variants of the secretary problem with other objective functions have been also proposed, such as discounted profits [2], and submodular objective functions [4,13]. We do not discuss these variants here.…”