无圈竞赛图上原始和对偶最小费用流问题的对偶计算代数分析
Dualistic computational algebraic analyses of primal and dual minimum cost flow problems on acyclic tournament graphs
摘要：
对于整数规划问题，近年来，通过环面理想的离散性，研究了使用Grobner基或标准对的计算代数方法。虽然这些方法并没有得到改善的时间复杂度界相比，现有的方法解决整数规划问题，这些给出了代数分析其结构。在本文中，我们专注于系数矩阵是单模的情况下，特别是对原始和对偶最小费用流问题，其结构是相当有名的，但新的结构可以揭示我们的方法。研究了幺模规划的Grobner基和标准对，给出了对偶可行基的最大个数.对于最小费用流问题，我们用图的形式刻画了既约Grobner基，并给出了对偶（分别是）的个数的界.原始的）原始的可行基（或双重）问题：对于原始问题，最小值和最大值被证明是1和Catalan数$\frac{1}{d}\tbinom{2(d-1)}{d-1}$，而对于对偶问题，下限被证明是$Ω(2^{\lfloor d/6\rfloor})$。为了分析算术度，我们使用了两种方法：一种是既约Gr {<$} bner基与标准对之间的关系，其中最小费用流的对应关系--回路子集与对偶可行基之间的对应关系还不是很清楚;另一种是组合数学中与复曲面理想有关的结果。
英文摘要：
To integer programming problems, computational algebraic approaches using Grobner bases or standard pairs via the discreteness of toric ideals have been studied in recent years. Although these approaches have not given improved time complexity bound compared with existing methods for solving integer programming problems, these give algebraic analysis of their structures. In this paper, we focus on the case that the coefficient matrix is unimodular, especially on the primal and dual minimum cost flow problems, whose structure is rather well-known, but new structures can be revealed by our approach. We study the Grobner bases and standard pairs for unimodular programming, and give the maximum number of dual feasible bases in terms of the volume of polytopes. And for the minimum cost flow problems, we characterize reduced Grobner bases in terms of graphs, and give bounds for the number of dual (resp. primal) feasible bases of the primal (resp. dual) problems: for the primal problems the minimum and the maximum are shown to be 1 and the Catalan number $\frac{1}{d}\tbinom{2(d-1)}{d-1}$, while for the dual problems the lower bound is shown to be $Ω(2^{\lfloor d/6\rfloor})$. To analyze arithmetic degrees, we use two approaches: one is the relation between reduced Gr{ö}bner bases and standard pairs, where the corresponding relation on the minimum cost flow -- between a subset of circuits and dual feasible bases -- has not been so clear, the other is the results in combinatorics related with toric ideals.
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