Suppose we are given a submodular function @math over a set of elements, and we want to maximize its value subject to certain constraints. Good approximation algorithms are known for such problems under both monotone and non-monotone submodular functions. We consider these problems in a stochastic setting, where elements are not all active and we can only get value from active elements. Each element @math is active independently with some known probability @math , but we don't know the element's status . We find it out only when we the element @math ---probing reveals whether it's active or not, whereafter we can use this information to decide which other elements to probe. Eventually, if we have a probed set @math and a subset @math of active elements in @math , we can pick any @math and get value @math . Moreover, the sequence of elements we probe must satisfy a given ---e.g., these may be given by a matroid, or an orienteering constraint, or deadline, or precedence constraint, or an arbitrary downward-closed constraint---if we can probe some sequence of elements we can probe any prefix of it. What is a good strategy to probe elements to maximize the expected value? In this paper we study the gap between adaptive and non-adaptive strategies for @math being a submodular or a fractionally subadditive (XOS) function. If this gap is small, we can focus on finding good non-adaptive strategies instead, which are easier to find as well as to represent. We show that the adaptivity gap is a constant for monotone and non-monotone submodular functions, and logarithmic for XOS functions of small . These bounds are nearly tight. Our techniques show new ways of arguing about the optimal adaptive decision tree for stochastic problems.
The previous work of the authors @cite_2 gave results for the case where @math was the rank function of matroids (or their intersections). That work bounded the adaptivity gap by logarithmic factors, and gave better results for special cases like uniform and partition matroids. This work both improves the quantitative bounds (down to small constants), generalizes it to all submodular functions with the hope of getting to all subadditive functions, and arguably also makes the proof simpler.
@cite_2
2089729462
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.