Generalized cut and metric polytopes of graphs and simplicial complexes

Abstract

Given a graph G one can define the cut polytope CUTP(G) and the metric polytope METP(G) of this graph and those polytopes encode in a nice way the metric on the graph. According to Seymour's theorem, CUTP(G) = METP(G) if and only if K 5 is not a minor of G.We consider possibly extensions of this framework: (1) We compute the CUTP(G) and METP(G) for many graphs.(2) We define the oriented cut polytope WOMCUTP(G) and oriented multicut polytope OMCUTP(G) as well as their oriented metric version QMETP(G) and WQMETP(G). (3) We define an n-dimensional generalization of metric on simplicial complexes.

Cycle algebras and polytopes of matroids

“…Proof. The desired result follows from (10) and since clearly I Cyc(M ) = 0 if and only if height I Cyc(M ) = 0, since it is a prime ideal. Proof.…”

“…For simplicity, let d(M) := the number of coparallel classes of M. As it was mentioned in Section 3, it is known that dim P Cyc (M) = d(M). This together with [4,Proposition 4.22] imply that the Krull dimension of the cycle algebra of M is given by dim K[Cyc(M)] = d(M) + 1, and hence (10) height…”

“…for any A ⊆ V and e ∈ E. The cut polytope of G has been intensively studied by many authors; see, e.g., [6,9,10,12,28,27,33,34].…”

“…A prominent example with many applications is the cut polytope Cut (G) of a graph G which has taken the attention of researchers from many fields. Cut polytopes are well studied, see for example [3,9,10,14]. Indeed, cut polytopes are closely related to some well-known problems, like the MAXCUT problem (see for example [11,12,13]) and the four color theorem in graph theory (see [24]).…”

Complete list of tight Bell inequalities for two parties with four binary settings

“…For four settings on each side, the number of facet inequalities grows to 175, where 169 of these inequalities genuinely use the four settings. The list of 175 inequalities has been found in [3][4][5], however the realization that the list is complete is due to [6]. Therefore, one could find an almost complete list of the inequalities distributed in the literature.…”