Generalized cut and metric polytopes of graphs and simplicial complexes

The metric polytope METP ( K n ) of the complete graph on n nodes is defined by the triangle inequalities x ( i , j ) ≤ x ( i , k ) + x ( k , j ) and x ( i , j ) + x ( j , k ) + x ( k , i ) ≤ 2 for all triples i , j , k of { 1 , ⋯ , n } . The cut polytope CUTP ( K n ) is the convex hull of the { 0 , 1 } vectors of METP ( K n ) . For a graph G on n vertices the metric polytope METP ( G ) and cut polytope CUTP ( G ) are the projections of METP ( K n ) and CUTP ( K n ) on the edge set of G . The facets of the cut polytopes are of special importance in optimization and are studied here in some detail for many simple graphs. Then we define variants QMETP ( G ) for quasi-metrics, i.e. not necessarily symmetric distances and we give an explicit description by inequalities. Finally we generalize distances to m -dimensional area between m + 1 points and this defines an hemimetric. In that setting the generalization of the notion of graph is the notion of m -dimensional simplicial complex K for which we define a cone of hemimetric HMET ( K ) .