Adaptivity Gaps for Stochastic Probing: Submodular and XOS Functions

Abstract

Suppose we are given a submodular function f 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 only get value from active elements. Each element e is active independently with some known probability p e , but we don't know the element's status a priori : we find it out only when we probe the element e. Moreover, the sequence of elements we probe must satisfy a given prefix-closed constraint, e.g., matroid, orienteering, deadline, precedence, or any downward-closed constraint.In this paper we study the gap between adaptive and non-adaptive strategies for f 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 width. These bounds are nearly tight. Our techniques show new ways of arguing about the optimal adaptive decision tree for stochastic optimization problems.

The Price of Information in Combinatorial Optimization

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php similar problem in the Free-Info world. This latter problem turns out to be the same as the stochastic probing problem studied in [29,4,30,31]. By proving an extension of the adaptivity gap result of Gupta et al [31], we show that one can further simplify these Free-Info problems to non-adaptive utility-maximization problems (by losing a constant factor).…”

“…We prove Lemma 4.2 in the full version by generalizing a similar result for Bernoulli random variables of Gupta et al [31] to functions that are given by weighted matroid rank function. It tells us about the existence of a feasible set S ∈ J such that E[max i∈S {Y max i }] is at least 1/3 times the optimal adaptive strategy for the stochastic probing problem.…”

Pandora's Problem with Nonobligatory Inspection

“…Such a policy is almost non-adaptive; the only way in which it may adjust its behavior in response to information revealed during the search process is that it may terminate the search early. In this sense, questions about the ability of commi ing policies to approximate the optimal adaptive policy are akin to questions about adaptivity gaps in stochastic optimization [1,2,6,7]. e foregoing discussion inspires two interrelated questions.…”

Algorithms to Approximate Column-Sparse Packing Problems

“…The stochastic version of hypergraph matching can be viewed as a natural generalization of stochastic matching (e.g., Bansal et al [12]) to hypergraphs. The work of [15] gave an (k + + o(1))-approximation algorithm for any given > 0 asymptotically for large k. Other work on stochastic variants of PIPs includes [33,34,54,3,42,1,41].…”