用围道积分的数值分析计算多面体的整点
Counting Integral Points in Polytopes via Numerical Analysis of Contour Integration
摘要：
本文研究了由$P(y) = \{ x \in \mathbb{R}^m \colon Ax = y, x \geq 0\}$描述的有理多面体中整数点的计数问题，其中$A$是$n \times m$整数矩阵，$y$是$n$维整数向量.本文从数值分析的角度研究了Brion-Vergne，Beck和Lasserre-Zeron提出的Z变换方法，得到了一个新的求解该问题的算法：若$A$非负，则$P(y)$中整数点的个数可以在$O(\mathrm{poly} (n,m, \|y\|_\infty) (\|y\|_\infty + 1)^n)$时间和$O(\mathrm{poly} (n,m, \|y\|_\infty))$空间中计算，这在空间复杂度上改进了一个使用$O((\|y\|_\infty + 1)^n)$大小DP表的朴素DP算法.我们的结果是基于逆Z变换的数值围道积分的标准误差分析，并建立了一种新的$P(y)$中整数点的容斥公式。我们把这个结果应用到超图的$b$ -匹配上，得到了一个计算具有$n$顶点和$m$超边的$k$ -部超图的$b$ -匹配的$O(\mathrm{poly}( n,m,\|b\|_\infty) (\|b\|_\infty +1)^{(1-1/k)n})$时间算法.这个结果被看作是Ryser关于$k=2$的经典结果和Bj{<$}rklund-Husfeldt关于它的多部推广的$b$匹配推广.
英文摘要：
In this paper, we address the problem of counting integer points in a rational polytope described by $P(y) = \{ x \in \mathbb{R}^m \colon Ax = y, x \geq 0\}$, where $A$ is an $n \times m$ integer matrix and $y$ is an $n$-dimensional integer vector. We study the Z-transformation approach initiated by Brion-Vergne, Beck, and Lasserre-Zeron from the numerical analysis point of view, and obtain a new algorithm on this problem: If $A$ is nonnegative, then the number of integer points in $P(y)$ can be computed in $O(\mathrm{poly} (n,m, \|y\|_\infty) (\|y\|_\infty + 1)^n)$ time and $O(\mathrm{poly} (n,m, \|y\|_\infty))$ space.This improves, in terms of space complexity, a naive DP algorithm with $O((\|y\|_\infty + 1)^n)$-size DP table. Our result is based on the standard error analysis to the numerical contour integration for the inverse Z-transform, and establish a new type of an inclusion-exclusion formula for integer points in $P(y)$. We apply our result to hypergraph $b$-matching, and obtain a $O(\mathrm{poly}( n,m,\|b\|_\infty) (\|b\|_\infty +1)^{(1-1/k)n})$ time algorithm for counting $b$-matchings in a $k$-partite hypergraph with $n$ vertices and $m$ hyperedges. This result is viewed as a $b$-matching generalization of the classical result by Ryser for $k=2$ and its multipartite extension by Bj{ö}rklund-Husfeldt.
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