On a Cardinality-Constrained Transportation Problem With Market Choice

TL;DR

This paper proves integrality of the transportation problem with market choice under certain demand constraints, enabling polynomial-time solutions for specific minimum weight perfect matching problems with cardinality restrictions.

Contribution

It extends known integrality results to the transportation problem with market choice for demands in {1,2}, generalizing matching polytope results.

Findings

Proves integrality of the TPMC polytope for demands in {1,2}.

Shows polynomial-time solvability for certain minimum weight perfect matching problems.

Generalizes matching polytope integrality results.

Abstract

It is well-known that the intersection of the matching polytope with a cardinality constraint is integral [8]. We prove a similar result for the polytope corresponding to the transportation problem with market choice (TPMC) (introduced in [4]) when the demands are in the set $\{1,2\}$. This result generalizes the result regarding the matching polytope and also implies that some special classes of minimum weight perfect matching problem with a cardinality constraint on a subset of edges can be solved in polynomial time.