AI 中文总结
研究与参数约束矩阵相关的离散优化问题，针对约束矩阵元素为单变量多项式且子行列式在特定集\(S\)中的情况，对识别和优化两个关键问题，通过特定集合\(S\)给出肯定回答，所涉矩阵有独立研究价值。
AI 中文摘要
我们引入了一个框架来解决与参数约束矩阵相关的离散优化问题。具体而言，约束矩阵的元素是单变量多项式，且这些矩阵的所有子行列式都是给定规定集\(S\)中的多项式。在此背景下出现两个关键问题。一是识别问题：这种形式的矩阵能否在多项式时间内被识别？二是优化问题：给定一个约束矩阵为此形式的整数规划，能否在多项式时间内求解？对于由九个线性形式组成的特定集合\(S\)，我们对这两个问题都给出了肯定回答。我们所考虑的矩阵本身就具有独立的研究价值；它们是某些双模态矩阵的矩阵投影，这些双模态矩阵允许两个不同的幺模投影。
英文摘要
We introduce a framework for tackling questions in discrete optimization associated with parametric constraint matrices. More precisely, the constraint matrices have entries that are polynomials in one variable and all subdeterminants of these matrices are polynomials in a given prescribed set $S$. Two key problems arise in this context. The first is the recognition problem: can a matrix of this form be recognized in polynomial time? The second is the optimization problem: given an integer program whose constraint matrix is of this form, can it be solved in polynomial time? We answer both questions affirmatively for a particular set $S$ consisting of nine linear forms. The matrices we consider are of themselves independent interest; they arise as matrix projections of certain bimodular matrices that admit two distinct unimodular projections.
CommentsThis paper is an extension of a conference proceedings version by the same authors that was presented at IPCO 2024
