In 2016, Chandrasekaran, Végh, and Vempala (Mathematics of Operations Research, 41(1):23–48) published a method to solve the minimum-cost perfect matching problem on an arbitrary graph by solving a strictly polynomial number of linear programs. However, their method requires a strong uniqueness condition, which they imposed by using perturbations of the form . On large graphs (roughly ), these perturbations lead to cost values that exceed the precision of floating-point formats used by typical linear programming solvers for numerical calculations. We demonstrate, by a sequence of counterexamples, that perturbations are required for the algorithm to work, motivating our formulation of a general method that arrives at the same solution to the problem as Chandrasekaran et al. but overcomes the limitations described above by solving multiple linear programs without using perturbations. The key ingredient of our method is an adaptation of an algorithm for lexicographic linear goal programming due to Ignizio (Journal of the Operational Research Society, 36(6):507–515, 1985). We then give an explicit algorithm that exploits our method, and show that this new algorithm still runs in strongly polynomial time.
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@article{OJMO_2020__1__A2_0, author = {Chen, Amber Q. and Cheung, Kevin K. H. and Kielstra, P. Michael and Winn, Avery D.}, title = {Revisiting a {Cutting-Plane} {Method} for {Perfect} {Matchings}}, journal = {Open Journal of Mathematical Optimization}, eid = {2}, pages = {1--15}, publisher = {Universit\'e de Montpellier}, volume = {1}, year = {2020}, doi = {10.5802/ojmo.2}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/ojmo.2/} }
TY - JOUR AU - Chen, Amber Q. AU - Cheung, Kevin K. H. AU - Kielstra, P. Michael AU - Winn, Avery D. TI - Revisiting a Cutting-Plane Method for Perfect Matchings JO - Open Journal of Mathematical Optimization PY - 2020 SP - 1 EP - 15 VL - 1 PB - Université de Montpellier UR - http://archive.numdam.org/articles/10.5802/ojmo.2/ DO - 10.5802/ojmo.2 LA - en ID - OJMO_2020__1__A2_0 ER -
%0 Journal Article %A Chen, Amber Q. %A Cheung, Kevin K. H. %A Kielstra, P. Michael %A Winn, Avery D. %T Revisiting a Cutting-Plane Method for Perfect Matchings %J Open Journal of Mathematical Optimization %D 2020 %P 1-15 %V 1 %I Université de Montpellier %U http://archive.numdam.org/articles/10.5802/ojmo.2/ %R 10.5802/ojmo.2 %G en %F OJMO_2020__1__A2_0
Chen, Amber Q.; Cheung, Kevin K. H.; Kielstra, P. Michael; Winn, Avery D. Revisiting a Cutting-Plane Method for Perfect Matchings. Open Journal of Mathematical Optimization, Tome 1 (2020), article no. 2, 15 p. doi : 10.5802/ojmo.2. http://archive.numdam.org/articles/10.5802/ojmo.2/
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