2016/11/02 by Robert Hildebrand, Robert Weismantel, Hildebrand, Robert +3
Business, Management and Accounting · Engineering · #FOS: Mathematics #Facility Location and Emergency Management #Optimization and Control (math.OC) #Optimization and Packing Problems #Vehicle Routing Optimization Methods
paper · pdf · doi:10.48550/arxiv.1611.00707
openalex publication_date 2016/11/02 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We prove that any mixed-integer linear extended formulation for the matching polytope of the complete graph on n vertices, with a polynomial number of constraints, requires Ω(√\sfracnlog n) many integer variables. By known reductions, this result extends to the traveling salesman polytope. This lower bound has various implications regarding the existence of small mixed-integer mathematical formulations of common problems in operations research. In particular, it shows that for many classic vehicle routing problems and problems involving matchings, any compact mixed-integer linear description of such a problem requires a large number of integer variables. This provides a first non-trivial lower bound on the number of integer variables needed in such settings.