The Exact Bipartite Matching Polytope Has Exponential Extension Complexity

Xinrui Jia; Ola Svensson; Weiqiang Yuan

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arXiv · 2211.09106

The Exact Bipartite Matching Polytope Has Exponential Extension Complexity

Xinrui Jia·Ola Svensson·Weiqiang Yuan

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Abstract

Given a graph with edges colored red or blue and an integer $k$, the exact perfect matching problem asks if there exists a perfect matching with exactly $k$ red edges. There exists a randomized polylogarithmic-time parallel algorithm to solve this problem, dating back to the eighties, but no deterministic polynomial-time algorithm is known, even for bipartite graphs. In this paper we show that there is no sub-exponential sized linear program that can describe the convex hull of exact matchings in bipartite graphs. In fact, we prove something stronger, that there is no sub-exponential sized linear program to describe the convex hull of perfect matchings with an odd number of red edges.

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BibTeX RIS

Xinrui Jia, Ola Svensson, Weiqiang Yuan. 2022-11-16. The Exact Bipartite Matching Polytope Has Exponential Extension Complexity. https://arxiv.org/abs/2211.09106

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