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arXiv · 2312.17630

Total Matching and Subdeterminants

Luca Ferrarini·Samuel Fiorini·Stefan Kober·Yelena Yuditsky
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Abstract

In the total matching problem, one is given a graph $G$ with weights on the vertices and edges. The goal is to find a maximum weight set of vertices and edges that is the non-incident union of a stable set and a matching. We consider the natural formulation of the problem as an integer program (IP), with variables corresponding to vertices and edges. Let $M = M(G)$ denote the constraint matrix of this IP. We define $\Delta(G)$ as the maximum absolute value of the determinant of a square submatrix of $M$. We show that the total matching problem can be solved in strongly polynomial time provided $\Delta(G) \leq \Delta$ for some constant $\Delta \in \mathbb{Z}_{\ge 1}$. We also show that the problem of computing $\Delta(G)$ admits an FPT algorithm. We also establish further results on $\Delta(G)$ when $G$ is a forest.

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BibTeXRIS

Luca Ferrarini, Samuel Fiorini, Stefan Kober, Yelena Yuditsky. 2023-12-29. Total Matching and Subdeterminants. https://arxiv.org/abs/2312.17630

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