关于k-次模松弛
On k-Submodular Relaxation
摘要：
$k$ -次模函数，由Huber和Kolmogorov引入，是定义在$\{0, 1, 2, \dots, k\}^n$上满足某些次模型不等式的函数。$k$ -次模函数通常作为NP难问题的松弛而出现，并且由$k$ -次模函数得到的松弛在高效、近似或固定参数的易处理算法的设计中起着关键作用。受此启发，我们考虑以下问题：给定一个函数$f : \{1, 2, \dots, k\}^n \rightarrow \mathbb{R} \cup \{\infty\}$，确定$f$是否扩展到一个$k$ -次模函数$g : \{0, 1, 2, \dots, k\}^n \rightarrow \mathbb{R} \cup \{\infty\}$，其中$g$被称为$f$的一个$k$ -次模松弛。我们给出了允许$k$ -次模松弛的函数的多态性刻画，并给出了一个组合$O((k^n)^2)$ -时间算法来寻找一个$k$ -次模松弛或建立一个$k$ -次模松弛不存在.我们的算法具有有趣的性质：（1）如果输入函数是整数值，那么我们的算法输出一个半整数松弛，（2）如果输入函数是二进制的，那么我们的算法输出唯一的最优松弛。我们介绍了我们的算法在有值约束满足问题中的应用。
英文摘要：
$k$-submodular functions, introduced by Huber and Kolmogorov, are functions defined on $\{0, 1, 2, \dots, k\}^n$ satisfying certain submodular-type inequalities. $k$-submodular functions typically arise as relaxations of NP-hard problems, and the relaxations by $k$-submodular functions play key roles in design of efficient, approximation, or fixed-parameter tractable algorithms. Motivated by this, we consider the following problem: Given a function $f : \{1, 2, \dots, k\}^n \rightarrow \mathbb{R} \cup \{\infty\}$, determine whether $f$ is extended to a $k$-submodular function $g : \{0, 1, 2, \dots, k\}^n \rightarrow \mathbb{R} \cup \{\infty\}$, where $g$ is called a $k$-submodular relaxation of $f$. We give a polymorphic characterization of those functions which admit a $k$-submodular relaxation, and also give a combinatorial $O((k^n)^2)$-time algorithm to find a $k$-submodular relaxation or establish that a $k$-submodular relaxation does not exist. Our algorithm has interesting properties: (1) If the input function is integer valued, then our algorithm outputs a half-integral relaxation, and (2) if the input function is binary, then our algorithm outputs the unique optimal relaxation. We present applications of our algorithm to valued constraint satisfaction problems.
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