The Exact Bipartite Matching Polytope Has Exponential Extension Complexity

Xinrui Jia; Ola Svensson; Weiqiang Yuan

Home/Archive/arXiv

arXiv · 2211.09106

The Exact Bipartite Matching Polytope Has Exponential Extension Complexity

Xinrui Jia·Ola Svensson·Weiqiang Yuan

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.CC cs.DM

Keep this discovery

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

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗

arXiv2026

Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor

We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard--Milani\v{c}--\v{S}torgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.

cs.CC↗

arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗