The Complexity of Matching with Bonds

Abstract A bond structure in a finite, undirected, loopless graph G = ( V , E ) is a collection F of nonempty, disjoint subsets of E whose union is E . Matching with bonds (MB) is the problem of finding a maximum-weight matching in G such that either all edges in a bond are selected or none of them. Other matching problems with side constraints have been presented in the literature, we show that they are special cases of (MB). Moreover we prove that (MB) is an NP-hard problem even when the graph G is a cycle and every bond F ∈ F ∈ F has cardinality at most two.