Total Matching and Subdeterminants

Total Matching and Subdeterminants

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),...

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Hauptverfasser: Ferrarini, Luca, Fiorini, Samuel, Kober, Stefan, Yuditsky, Yelena
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Computer Science - Data Structures and Algorithms
Computer Science - Discrete Mathematics
Mathematics - Combinatorics
Mathematics - Optimization and Control
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Fiorini, Samuel
Kober, Stefan
Yuditsky, Yelena
description 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.
doi_str_mv 10.48550/arxiv.2312.17630
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Computer Science - Discrete Mathematics
Mathematics - Combinatorics
Mathematics - Optimization and Control
title Total Matching and Subdeterminants
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