Total Matching and Subdeterminants
Luca Ferrarini, Samuel Fiorini, Stefan Kober, Yelena Yuditsky
TL;DR
This paper investigates the total matching problem through the lens of the maximum subdeterminant $Δ(G)$ of the constraint matrix $M(G)$. It proves that the total matching is solvable in strongly polynomial time when $Δ(G) ≤ Δ$ for a fixed constant, via an $O(2^{O(Δ \log Δ)}(|V|+|E|))$-time algorithm, and provides an FPT procedure to compute $Δ(G)$ or certify $Δ(G) > Δ$; it also analyzes $Δ(G)$ for forests. The approach combines structural graph results, including component-wise factorization and cycle/near-pencil submatrix analysis, with a reduction to a small core plus path components that enables dynamic programming on the reduced instance. The work clarifies the role of determinant bounds in discrete optimization, offering both practical algorithms and open questions about computing $Δ(G)$ and its relation to other invariants such as odd cycle packing.
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 $Δ(G)$ 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 $Δ(G) \leq Δ$ for some constant $Δ\in \mathbb{Z}_{\ge 1}$. We also show that the problem of computing $Δ(G)$ admits an FPT algorithm. We also establish further results on $Δ(G)$ when $G$ is a forest.