稀疏 ILP 与超图多重装填/多重覆盖问题的更快算法
Faster Algorithms for Sparse ILP and Hypergraph Multi-Packing/Multi-Cover Problems
AI总结：
本文研究稀疏矩阵 ILP 的可行性、求解与计数问题，并提出图及超图多重装填/多重覆盖问题的更快指数算法框架。
摘要：
在本文中，我们考虑以下一般问题：假设 P 是由具有稀疏矩阵 A 的系统 Ax≤b 或 Ax=b、x≥0 定义的多面体，检查可行性、计算可行解的数量、寻找最优解，以及计算 P∩Z^n 中最优解的数量。我们为这些问题开发了算法；在元素有界的稀疏实例上，这些算法优于最先进的 ILP 和计数算法。我们利用已知方法和新方法，为图与超图上的边/顶点多重装填/多重覆盖问题开发了新的指数算法。该框架包含许多不同问题，例如 Stable Multi-set、Vertex Multi-cover、Dominating Multi-set、Set Multi-cover、Multi-set Multi-cover 和 Hypergraph Multi-matching 问题，它们是标准 Stable Set、Vertex Cover、Dominating Set、Set Cover 和 Maximal Matching 问题的自然推广。
英文摘要：
In our paper, we consider the following general problems: check feasibility, count the number of feasible solutions, find an optimal solution, and count the number of optimal solutions in $P \cap Z^n$, assuming that $P$ is a polyhedron, defined by systems $A x \leq b$ or $Ax = b,\, x \geq 0$ with a sparse matrix $A$. We develop algorithms for these problems that outperform state of the art ILP and counting algorithms on sparse instances with bounded elements. We use known and new methods to develop new exponential algorithms for Edge/Vertex Multi-Packing/Multi-Cover Problems on graphs and hypergraphs. This framework consists of many different problems, such as the Stable Multi-set, Vertex Multi-cover, Dominating Multi-set, Set Multi-cover, Multi-set Multi-cover, and Hypergraph Multi-matching problems, which are natural generalizations of the standard Stable Set, Vertex Cover, Dominating Set, Set Cover, and Maximal Matching problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
