Sparse Integer Programming Is Fixed-Parameter Tractable

IF 1.4 3区数学 Q2 MATHEMATICS, APPLIED

Mathematics of Operations Research Pub Date : 2024-08-19 DOI:10.1287/moor.2023.0162

Friedrich Eisenbrand, Christoph Hunkenschröder, Kim-Manuel Klein, Martin Koutecký, Asaf Levin, Shmuel Onn

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引用次数: 0

Abstract

We study the general integer programming problem where the number of variables n is a variable part of the input. We consider two natural parameters of the constraint matrix A: its numeric measure a and its sparsity measure d. We present an algorithm for solving integer programming in time [Formula: see text], where g is some computable function of the parameters a and d, and L is the binary encoding length of the input. In particular, integer programming is fixed-parameter tractable parameterized by a and d, and is solvable in polynomial time for every fixed a and d. Our results also extend to nonlinear separable convex objective functions.Funding: F. Eisenbrand, C. Hunkenschröder, and K.-M. Klein were supported by the Swiss National Science Foundation (SNSF) within the project “Convexity, geometry of numbers, and the complexity of integer programming” [Grant 163071]. A. Levin and S. Onn are partially supported by the Israel Science Foundation [Grant 308/18]. A. Levin is also partially supported by the Israel Science Foundation [Grant 1467/22]. S. Onn is also partially supported by the Dresner Chair at the Technion. M. Koutecký is partially supported by Charles University project UNCE 24/SCI/008, and by the project 22-22997S of the Grantová Agentura České Republiky (GA ČR).

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稀疏整数编程是固定参数可实现的

我们研究的是一般整数编程问题，其中变量数 n 是输入变量的一部分。我们考虑了约束矩阵 A 的两个自然参数：其数值度量 a 和其稀疏度量 d。我们提出了一种在时间内求解整数编程的算法[公式：见正文]，其中 g 是参数 a 和 d 的某个可计算函数，L 是输入的二进制编码长度。我们的结果还扩展到非线性可分离凸目标函数：F. Eisenbrand、C. Hunkenschröder和K.-M. Klein得到了瑞士国家科学院的资助。克莱因得到了瑞士国家科学基金会 (SNSF) 在 "凸性、数字几何和整数编程的复杂性 "项目 [163071] 中的支持。A. Levin 和 S. Onn 得到了以色列科学基金会 [308/18 号拨款] 的部分资助。A. Levin 还得到以色列科学基金会 [1467/22 号拨款] 的部分资助。S. Onn 也得到了以色列理工学院 Dresner 讲座的部分资助。M. Koutecký 由查尔斯大学项目 UNCE 24/SCI/008 和 Grantová Agentura České Republiky (GA ČR) 项目 22-22997S 提供部分资助。

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来源期刊

Mathematics of Operations Research 管理科学-应用数学

CiteScore

3.40

自引率

5.90%

发文量

178

审稿时长

15.0 months

期刊介绍： Mathematics of Operations Research is an international journal of the Institute for Operations Research and the Management Sciences (INFORMS). The journal invites articles concerned with the mathematical and computational foundations in the areas of continuous, discrete, and stochastic optimization; mathematical programming; dynamic programming; stochastic processes; stochastic models; simulation methodology; control and adaptation; networks; game theory; and decision theory. Also sought are contributions to learning theory and machine learning that have special relevance to decision making, operations research, and management science. The emphasis is on originality, quality, and importance; correctness alone is not sufficient. Significant developments in operations research and management science not having substantial mathematical interest should be directed to other journals such as Management Science or Operations Research.