An O(|E|)-linear Model for the MaxCut Problem

A polytope P is a model for a combinatorial problem on finite graphs G whose variables are indexed by the edge set E of G if the points of P with (0,1)-coordinates are precisely the characteristic vectors of the subset of edges inducing the feasible configurations for the problem. In the case of the (simple) MaxCut Problem, which is the one that concern us here, the feasible subsets of edges are the ones inducing the bipartite subgraphs of G. In this paper we introduce a new polytope P12 ⊂ R|E| given by at most 11|E| inequalities, which is a model for the MaxCut Problem on G. Moreover, the left side of each inequality is the sum of at most 4 edge variables with coefficients ±1 and right side 0,1, or 2. We restrict our analysis to the case of G = Kz, the complete graph in z vertices, where z is an even positive integer z ≥ 4. This case is sufficient to study because the simple MaxCut problem for general graphs G can be reduced to the complete graph Kz by considering the obective function of the associated integer programming as the characteristic vector of the edges in G ⊆ Kz. This is a polynomial algorithmic transformation. 1 Notation and Preliminaries The MaxCut Problem [3] is one of the first NP-complete problems. This problem can be stated as follows. Given a graph G does it has a bipartite subgraph with n edges? It is a very special problem which has been acting as a paradigm for great theoretical developments. See, for instance [4], where an algorithm with a rather peculiar worse case performance (greater than 87%) can be established as a fraction of type (solution found/optimum solution). This result constitutes a landmark in the theory of approximation algorithms. Our approach is a theoretical investigation on polytopes associated to complete graphs. The main result is that there is a set of at most 11|E| short inequalities (each involving no more than 4 edge variables with coefficients ±1) so that the polytope in R|E| formed by these inequalities has its all integer coordinate points in 1-1 correpondence with the characteristic vectors of the complete bipartite subgraphs of Kz, z even. Thick graphs into closed surfaces. A surface is closed if it is compact and has no boundary. A closed surface is characterized by its Euler characterisitic and the information whether or not is orientable. We use the following combinatorial counterpart for a graph G cellularly ∗2010 Mathematics Subject Classification: 68Q25 (primary), 68R10, 05C85 (secondary).