用次模分解求一般集函数的最大值
Maximizing General Set Functions by Submodular Decomposition
摘要：
本文提出了一种分支定界法，用于极大化任意集函数h的2 ^V到R的映射.通过将h分解为f-g，其中f是一个次模函数，g是一个顶点集为V的（简单无向）图G的割函数，我们原来的问题被归结为一系列次模最大化问题。我们刻画了一类次模函数，当它在子问题中被最大化时，使得算法收敛到f-g的全局最大值。两个“自然”的成员这个类进行了分析，第一个产生多项式可解的子问题，第二个，这需要较少的分支，产生NP-困难的子问题，但服从多项式时间近似算法。这些结果被扩展到的解决方案被约束为一个子集系统的成员的问题。证明了f-g极大化子的结构性质。
英文摘要：
We present a branch and bound method for maximizing an arbitrary set function h mapping 2^V to R. By decomposing h as f-g, where f is a submodular function and g is the cut function of a (simple, undirected) graph G with vertex set V, our original problem is reduced to a sequence of submodular maximization problems. We characterize a class of submodular functions, which when maximized in the subproblems, lead the algorithm to converge to a global maximizer of f-g. Two "natural" members of this class are analyzed; the first yields polynomially-solvable subproblems, the second, which requires less branching, yields NP-hard subproblems but is amenable to a polynomial-time approximation algorithm. These results are extended to problems where the solution is constrained to be a member of a subset system. Structural properties of the maximizer of f-g are also proved.
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