Easy and difficult objective functions for max cut

This note investigates the boundary between polynomially-solvable Max Cut and NP Hard Max Cut instances when they are classified only on the basis of the sign pattern of the objective function coefficients, i.e., of the orthant containing the objective function vector. It turns out that the matching number of the subgraph induced by the positive edges is the key parameter that allows us to differentiate between polynomially-solvable and hard instances of the problem. We give some applications of the polynomially solvable cases. A cut in a weighted undirected graph G = (V,E,w) is defined by a node set S ⊆ V as the set of all edges with exactly one endpoint in S, and it is denoted by δ(S). The weight of a cut is the sum of the weights of its edges. Given a weighted graph G, the Max Cut ∗The research of this author was partially supported by an NSERC Research Grant.