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 \(\varDelta (G)\) 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 \(\varDelta (G) \le \varDelta \) for some constant \(\varDelta \in \mathbb {Z}_{\ge 1}\). We also show that the problem of computing \(\varDelta (G)\) admits an FPT algorithm. We also establish further results on \(\varDelta (G)\) when G is a forest.
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Notes
- 1.
We use \(\delta (v)\) to denote the set of edges incident to vertex v.
- 2.
Notice that \(B'\) is not always a true incidence matrix since there might be bichromatic edges e such that only one endpoint of e is bichromatic.
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Ferrarini, L., Fiorini, S., Kober, S., Yuditsky, Y. (2024). Total Matching and Subdeterminants. In: Basu, A., Mahjoub, A.R., Salazar González, J.J. (eds) Combinatorial Optimization. ISCO 2024. Lecture Notes in Computer Science, vol 14594. Springer, Cham. https://doi.org/10.1007/978-3-031-60924-4_15
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