We study the problem of optimizing an arbitrary weight function w'z over the metric polytope of a graph G=(V,E), a well-known relaxation of the cut polytope. We define the signed graph (G, E^-), where E^- consists of the edges of G having negative weight. We characterize the sign patterns of the weight vector w such that all optimal vertices of the metric polytope are integral, and we show that they correspond to signed graphs with no odd-K_5 minor. Our result has significant implications for unconstrained zero-one quadratic programming problems. We relate the strength of the cycle relaxation of the boolean quadric polytope to the sign pattern of the objective function. In this case, the sign patterns such that all optimal vertices of the cycle relaxation are integral correspond to signed graphs with no odd-K_4 minor.

## Citation

University of Wisconsin - Madison, 2021.