基于多线性松弛和竞争解决方案的次模函数最大化
Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes
摘要：
我们考虑的问题最大化的非负子模集函数$f:2^N \rightarrow \mathbb{R}_+$在地面集$N$受到各种包装类型的约束，包括（多）拟阵约束，背包约束，和它们的交集。在本文中，我们开发了一个通用的框架，使我们能够得出一些新的结果，特别是当$f$可能是一个非单调函数。我们的算法是基于（近似）最大化的多线性扩展$F$的$f$在一个多面体$P$表示的约束，然后有效地舍入的分数的解决方案。虽然这种方法已经得到了相当成功的使用，但它在某些重要方面受到了限制。我们克服这些限制如下。首先，我们给出了下闭多面体$P$上最大化$F$的常数因子近似算法，该算法由一个有效的分离预言机描述。以前，这是已知的只有单调函数。对于非单调函数，只有当多面体是固定数量的背包约束的交集或拟阵多面体时，才知道常数因子。第二，我们表明，竞争解决方案是一种有效的方法，以轮的分数的解决方案，即使当$f$是非单调的。特别地，可以组合用于不同多面体的竞争解决方案以处理不同约束的交集。通过LP对偶，我们表明，竞争解决方案的约束有关的相关间隙的加权秩函数的约束。这导致了一个最佳的竞争解决方案的拟阵多面体。我们的研究结果提供了一个广泛适用的框架，最大限度地提高线性和子模函数的独立性约束。我们给出几个说明性的例子。竞争解决方案可以找到其他应用。
英文摘要：
We consider the problem of maximizing a non-negative submodular set function $f:2^N \rightarrow \mathbb{R}_+$ 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) maximizing the multilinear extension $F$ of $f$ over a polytope $P$ 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 $F$ over a down-closed polytope $P$ described by an efficient separation oracle. Previously this was known only for monotone functions. For non-monotone functions, a constant factor was known only when the polytope was either the intersection of a fixed number of knapsack constraints or a matroid polytope. 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 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.
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