Maximizing Symmetric Submodular Functions
Maximizing Symmetric Submodular Functions
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最大化对称子模函数
DOI：
10.1145/3070685
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Moran Feldman
中科院分区：
文献类型：
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作者：
Moran Feldman
对称的下函数是一个重要的子函数，捕获了许多有趣的病例，包括剪切的图形和超图工作，我们确定了一些少量最大问题，可以获得更好的对称性近似值目的比一般子模函数的最新近似值。 N.对于此问题，我们描述了一种算法，该算法至少是0.432ċf（opt）的算法，其中OPT是最佳的积分解决方案。 Max {f（s）：| S |。至少0.432也可以应用于非阴性的非对称性下函数，在这种情况下，它会产生1/e-o（1）近似值，而不是对此问题的最大化结果改善。非负对称性下函数，我们描述了确定性线性时间1/2- approximation算法。负性下效用功能，并表明这是问题的最佳近似值。
Symmetric submodular functions are an important family of submodular functions capturing many interesting cases, including cut functions of graphs and hypergraphs. Maximization of such functions subject to various constraints receives little attention by current research, unlike similar minimization problems that have been widely studied. In this work, we identify a few submodular maximization problems for which one can get a better approximation for symmetric objectives than the state-of-the-art approximation for general submodular functions. We first consider the problem of maximizing a non-negative symmetric submodular function f:2N → R+ subject to a down-monotone solvable polytope P ⊆ [0, 1]N. For this problem, we describe an algorithm producing a fractional solution of value at least 0.432 ċ f(OPT), where OPT is the optimal integral solution. Our second result considers the problem max{f(S): |S| = k} for a non-negative symmetric submodular function f:2N → R+. For this problem, we give an approximation ratio that depends on the value k/|N| and is always at least 0.432. Our method can also be applied to non-negative non-symmetric submodular functions, in which case it produces 1/e − o(1) approximation, improving over the best-known result for this problem. For unconstrained maximization of a non-negative symmetric submodular function, we describe a deterministic linear-time 1/2-approximation algorithm. Finally, we give a [1 − (1 − 1/k)k − 1]-approximation algorithm for Submodular Welfare with k players having identical non-negative submodular utility functions and show that this is the best possible approximation ratio for the problem.