International Journal of Mathematics and Mathematical Sciences

Volume 2016, Article ID 7863650, 6 pages

http://dx.doi.org/10.1155/2016/7863650

## On Graphs of the Cone Decompositions for the Min-Cut and Max-Cut Problems

Department of Discrete Analysis, P.G. Demidov Yaroslavl State University, Sovetskaya 14, Yaroslavl 150000, Russia

Received 13 August 2015; Accepted 8 December 2015

Academic Editor: Frank Werner

Copyright © 2016 Vladimir Bondarenko and Andrei Nikolaev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider maximum and minimum cut problems with nonnegative weights of edges. We define the graphs of the cone decompositions and find a linear clique number for the min-cut problem and a superpolynomial clique number for the max-cut problem. These values characterize the time complexity in a broad class of algorithms based on linear comparisons.

#### 1. Introduction

We consider the well-known maximum and minimum cut problems.

*Instance 1. *Given an undirected graph and a weight function , it is required to find such a subset of the vertex set (cut) that the sum of the weights of the edges from with one endpoint in and another in is as small as possible (minimum cut or min-cut) or as large as possible (maximum cut, max-cut).

While our discussion of the cut problem [1] in this paper is focused on integral edge weights, we remark that results and their proofs remain valid with real weights.

It is known that the min-cut problem with nonnegative edges is polynomially solvable: Dinic-Edmonds-Karp algorithm based on the maximum flow problem has the running time of [2], Hao-Orlin modification has the running time of [3], and Stoer-Wagner algorithm that does not use the flow techniques has the complexity [4].

Nevertheless, the min-cut and max-cut problems with arbitrary edges and the max-cut problem with nonnegative edges are NP-hard with no known algorithms faster than an exhaustive search [1].

We will estimate the min-cut and max-cut complexity with the polyhedral approach and study the clique number of the graph of the cone decomposition for the cut problems with nonnegative edges. This value is known as a measure of complexity in a wide class of algorithms based on linear comparisons. The presented results were announced in [5]. Similar characteristics of the shortest path and 3-dimensional matching problems are considered in [6, 7].

#### 2. Cut Polytope and Cone Decomposition

With every subset (every cut in the complete graph on vertices) we associate a characteristic vector according to the following rule:Therefore, the coordinates of the characteristic vector (also known as the cut vector) indicate whether the corresponding edges are in the cut or not. The convex hull of all cut vectors is known as the cut polytope [8]:

Max-cut and min-cut problems are reduced to the linear programming on the polytope with objective vector containing the weights of the edges.

We introduce a dual construction. Let be a set of points in . Let . Denote Since is the set of solutions of a finite system of homogeneous linear inequalities, it is a convex polyhedral cone. Given that the set of all cones is called the cone decomposition of the space by the set . Cone decomposition is similar to Voronoi diagram, exactly coinciding with it if the Euclidean norm of all points in is equal.

We consider the graph of the cone decomposition with the cones being the vertices, and two cones and are adjacent if and only if they have a common facet:

Denote by the clique number, the number of vertices in a maximum clique, of the graph of the cone decomposition. It is known [6, 9] that the complexity of the direct type algorithms, based on linear comparisons, of finding the minimum (or maximum, if we change the sign of the inequality in the definition of the cone) of a linear objective function on the set , or, which is the same, finding the cone , which the vector belongs to, cannot be less than the value of .

Indeed, if some algorithm at each step performs a single linear comparison (verification of linear inequality or ), from a geometric point of view that means drawing a hyperplane and discarding a wrong half-space. But if cones are pairwise adjacent, then for any hyperplane there exist points of at least cones in one of the half-spaces; hence, such direct type algorithm (algorithm with direct type linear decision tree [6, 9]) can separate and discard at most one wrong cone at a time in the worst case (Figure 1). Thus, is a lower bound on the height of the decision tree and on the complexity of combinatorial optimization problems in the wide class of algorithms, including sorting algorithms, greedy algorithm, dynamic programming, and branch and bound.