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Neighborhood persistency of the linear optimization relaxation of integer linear optimization

MATHEMATICAL PROGRAMMING 2024年 1-19页

作者： Kimura, Kei Nakayama, Kotaro Kyushu Univ Fac Informat Sci & Elect Engn Fukuoka Japan Oplan Inc Tokyo Japan

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.

关键词： Integer linear optimization Linear optimization unit-two-variable-per-inequality system persistency 90 C 10

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Neighborhood persistency of the Linear Optimization Relaxation of Integer Linear Optimization 7th

Neighborhood Persistency of the Linear Optimization Relaxati...

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7th International Symposium on Combinatorial Optimization (ISCO)

作者： Kimura, Kei Nakayama, Kotaro Kyushu Univ Fac Informat Sci & Elect Engn Fukuoka Japan Oplan Inc Tokyo Japan

ISBN: (纸本)9783031185298;9783031185304

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 propose a stronger property called neighborhood persistency and show that the LO relaxation of ILO on unit-two-variable-per-inequality (UTVPI) systems is a maximal class of ILO such that its LO relaxation has (neighborhood) persistency. Our result on neighborhood persistency generalizes the previous results of Nemhauser and Trotter, Hochbaum et al., and Fiorini et al., and implies fixed-parameter tractability and two-approximability for ILO on UTVPI systems where the objective function and the variables are non-negative.

关键词： Integer linear optimization Linear optimization unit-two-variable-per-inequality system persistency

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