Neighborhood persistency of the linear optimization relaxation of integer linear optimization

• 成果类型: Article; Early Access
• 署名作者: Kimura, Kei; Nakayama, Kotaro
• 署名单位: Kyushu University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-024-02174-0
• 发表日期: 2024
• 关键词: 2-sat

摘要

For an integer linear optimization (ILO) problem, persistency of its linear optimization (LO) relaxation is a property that for every optimal solution of the relaxation that assigns integer values to some variables, there exists an optimal solution of the ILO problem in which these variables retain the same values. Although persistency has been used to develop heuristic, approximation, and fixed-parameter algorithms for special cases of ILO, its applicability remains unknown in the literature. In this paper, we reveal a maximal subclass of ILO such that its LO relaxation has persistency. Specifically, we show that the LO relaxation of ILO on unit-two-variable-per-inequality (UTVPI) systems has persistency and is (in a certain sense) maximal among such ILO. Our result generalizes the persistency results by Nemhauser and Trotter, Hochbaum et al., and Fiorini et al. Even more, we propose a stronger property called neighborhood persistency and show that the LO relaxation of ILO on UTVPI systems in general has this property. Using this stronger result, we obtain a fixed-parameter algorithm (where the parameter is the optimal value) and another proof of 2-approximability for ILO on UTVPI systems where objective functions and variables are non-negative.