Submodular functions are a broad class of set functions, which naturally arise in diverse areas. Many algorithms have been suggested for the maximization of these functions. Unfortunately, once the function deviates from submodularity, the known algorithms may perform arbitrarily poorly. Amending this issue, by obtaining approximation results for set functions generalizing submodular functions, has been the focus of recent works. One such class, known as weakly submodular functions, has received a lot of attention. A key result proved by Das and Kempe (2011) showed that the approximation ratio of the greedy algorithm for weakly submodular maximization subject to a cardinality constraint degrades smoothly with the distance from submodularity. However, no results have been obtained for maximization subject to constraints beyond cardinality. In particular, it is not known whether the greedy algorithm achieves any non-trivial approximation ratio for such constraints. In this paper, we prove that a randomized version of the greedy algorithm (previously used by (2014) for a different problem) achieves an approximation ratio of @math for the maximization of a weakly submodular function subject to a general matroid constraint, where @math is a parameter measuring the distance of the function from submodularity. Moreover, we also experimentally compare the performance of this version of the greedy algorithm on real world problems against natural benchmarks, and show that the algorithm we study performs well also in practice. To the best of our knowledge, this is the first algorithm with a non-trivial approximation guarantee for maximizing a weakly submodular function subject to a constraint other than the simple cardinality constraint. In particular, it is the first algorithm with such a guarantee for the important and broad class of matroid constraints.
The result of @cite_2 has motivated a long series of works on the maximization of non-monotone submodular functions subject to a matroid constraint @cite_32 @cite_13 @cite_9 @cite_4 @cite_43 . The currently best algorithm of this kind achieves an approximation ratio of @math @cite_26 for general matroid constraints, and no better approximation guarantee is known for uniform matroid constraints. On the inapproximability side, it is known that no polynomial time algorithm can achieve approximation ratios better than @math and @math for the maximization of non-monotone submodular functions subject to uniform and general matroid constraints, respectively. Hence, for non-monotone submodular functions there is still a large gap between the best known approximation and inapproximability results. This gap is somewhat bridged by a result of @cite_18 giving a @math -approximation algorithm for the maximization of a special class of non-monotone submodular functions, known as symmetric submodular functions, subject to a general matroid constraint.
@cite_18 @cite_26 @cite_4 @cite_9 @cite_32 @cite_43 @cite_2 @cite_13
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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 consider the problem of maximizing a nonnegative (possibly non-monotone) submodular set function with or without constraints. [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 ν is bounded away from 1 (a 1 4-approximation assuming that ν ≥ 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. 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. In this paper, we give constant approximation algorithms for the non-monotone case that work for p-independence systems (which generalize constraints given by the intersection of p matroids that had been studied previously), where the running time is poly(n, p). Our algorithms and analyses are simple, and essentially reduce non-monotone maximization to multiple runs of the greedy algorithm previously used in the monotone case. We extend these ideas to give a simple greedy-based constant factor algorithms for non-monotone submodular maximization subject to a knapsack constraint, and for (online) secretary setting (where elements arrive one at a time in random order and the algorithm must make irrevocable decisions) subject to uniform matroid or a partition matroid constraint. Finally, we give an O(log k) approximation in the secretary setting subject to a general matroid constraint of rank k. We consider the problem of maximizing a nonnegative submodular set function @math over a ground set @math 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 @math may be a nonmonotone function. Our algorithms are based on (approximately) maximizing the multilinear extension @math of @math over a polytope @math 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 @math over a downward-closed polytope @math described by an efficient separation oracle. Previously this was known only for monotone functions. For nonmonotone functions, a constant factor was ...