Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes

Abstract

We consider the problem of maximizing a non-negative submodular set function f : 2 N → R+ over a ground set N 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 f may be a non-monotone function. Our algorithms are based on (approximately) solving the multilinear extension F of f [5] over a polytope P that represents the constraints, and then effectively rounding the fractional solution. Although this approach has been used quite successfully in some settings [6,22,24,13,3], it has been limited in some important ways. We overcome these limitations as follows.First, we give constant factor approximation algorithms to maximize F over an arbitrary down-closed polytope P that has an efficient separation oracle. Previously this was known only for monotone functions [36]. For non-monotone functions, a constant factor was known only when the polytope was either the intersection of a fixed number of knapsack constraints [24] or a matroid polytope [37,30]. Second, we show that contention resolution schemes are an effective way to round a fractional solution, even when f is non-monotone. In particular, contention resolution schemes for different polytopes can be combined to handle the intersection of different constraints. Via LP duality we show that a contention resolution scheme for a constraint is related to the correlation gap [1] of weighted rank functions of the constraint. This leads to an optimal contention resolution scheme for the matroid polytope.Our results provide a broadly applicable framework for maximizing linear and submodular functions subject to independence constraints. We give several illustrative examples. Contention resolution schemes may find other applications.

Learning with Submodular Functions: A Convex Optimization Perspective

“…Given the previous result on cardinality constraints, several extensions have been considered, such as knapsack constraints or matroid constraints (see [42] and references therein). Moreover, fast algorithms and improved online data-dependent bounds can be further derived [150].…”

“…When the function is known to be non-negative (i.e., with non-negative values F (A) for all A ⊆ V ), then simple local search algorithm have led to theoretical guarantees [67,42,33]. It has first been shown in [67] that a 1/2 relative bound could not be improved in general if a polynomial number of queries of the submodular function is used.…”

“…Interestingly, in the analysis of submodular function maximization, a new extension from {0, 1} p to [0, 1] p has emerged, the multi-linear extension [42], which is equal tõ It is equal to the expectation of F (B) where B is a random subset where the i-th element is selected with probability w i (note that this interpretation allows the computation of the extension through sampling), and this is to be contrasted with the Lovász extension, which is equal to the expectation of F ({w u}) for u a random variable with uniform distribution in [0, 1]. The multi-linear extension is neither convex nor concave but has marginal convexity properties that may be used for the design and analysis of algorithms for maximization problems [42,68].…”

Submodular Unsplittable Flow on Trees

“…In this section, we extend our results to submodular objectives by combining the results from the previous section with the framework from [15]. Let N be a finite ground set.…”

“…The problem of maximizing submodular functions subject to various constraints is very well-studied; we refer the reader to [15] for an overview. UFP with a linear objective is also extensively studied.…”