关于钻石上次模函数极小化的复杂性
On the Complexity of Submodular Function Minimisation on Diamonds
摘要：
设$(L; \sqcap, \sqcup)$是有限格，$n$是正整数.如果对所有$\tup{a}, \tup{b} \in L^n$都有$f(\tup{a} \sqcap \tup{b}) + f(\tup{a} \sqcup \tup{b}) \leq f(\tup{a}) + f(\tup{b})$，则称函数$f : L^n \to \mathbb{R}$是次模的。本文研究$L$为菱形时的次模函数。给定Oracle对$f$的访问，我们感兴趣的是找到$\tup{x} \in L^n$，使得$f(\tup{x}) = \min_{\tup{y} \in L^n} f(\tup{y})$尽可能高效。我们建立了一个最小-最大定理，该定理指出子模函数的最小值等于定义在某个多面体上的某个函数的最大值;以及极小化问题的一个很好的特征，即，我们表明，给定用于计算子模$f : L^n \to \mathbb{Z}$和整数$m$使得$\min_{\tup{x} \in L^n} f(\tup{x}) = m$的预言，存在可以在$n$和$\max_{\tup{t} \in L^n} \log |f(\tup{t})|$中的时间多项式中验证的该事实的证明;以及用于最小化问题的伪多项式时间算法，即，给定用于计算子模$f : L^n \to \mathbb{Z}$的预言，可以在时间上找到由$n$和$\max_{\tup{t} \in L^n} |f(\tup{t})|$中的多项式限定的$\min_{\tup{t} \in L^n} f(\tup{t})$。
英文摘要：
Let $(L; \sqcap, \sqcup)$ be a finite lattice and let $n$ be a positive integer. A function $f : L^n \to \mathbb{R}$ is said to be submodular if $f(\tup{a} \sqcap \tup{b}) + f(\tup{a} \sqcup \tup{b}) \leq f(\tup{a}) + f(\tup{b})$ for all $\tup{a}, \tup{b} \in L^n$. In this paper we study submodular functions when $L$ is a diamond. Given oracle access to $f$ we are interested in finding $\tup{x} \in L^n$ such that $f(\tup{x}) = \min_{\tup{y} \in L^n} f(\tup{y})$ as efficiently as possible. We establish a min--max theorem, which states that the minimum of the submodular function is equal to the maximum of a certain function defined over a certain polyhedron; and a good characterisation of the minimisation problem, i.e., we show that given an oracle for computing a submodular $f : L^n \to \mathbb{Z}$ and an integer $m$ such that $\min_{\tup{x} \in L^n} f(\tup{x}) = m$, there is a proof of this fact which can be verified in time polynomial in $n$ and $\max_{\tup{t} \in L^n} \log |f(\tup{t})|$; and a pseudo-polynomial time algorithm for the minimisation problem, i.e., given an oracle for computing a submodular $f : L^n \to \mathbb{Z}$ one can find $\min_{\tup{t} \in L^n} f(\tup{t})$ in time bounded by a polynomial in $n$ and $\max_{\tup{t} \in L^n} |f(\tup{t})|$.
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