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), 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) ≤Δ for some constant Δ∈ℤ_≥ 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.