On cone partitions for the min-cut and max-cut problems with non-negative edges

TL;DR

This paper explores the geometric structure of the min-cut and max-cut problems with non-negative edges by constructing cone partitions and analyzing their graph properties, revealing exponential degrees and diameter two.

Contribution

It introduces a novel cone partition framework for cut problems and characterizes the adjacency and complexity of the resulting graphs, highlighting differences between min-cut and max-cut.

Findings

Vertex degrees in the cone partition graphs are exponential.

The diameter of these graphs is exactly 2.

Clique numbers differ significantly between min-cut and max-cut cases.

Abstract

We consider the classical minimum and maximum cut problems: find a partition of vertices of a graph into two disjoint subsets that minimize or maximize the sum of the weights of edges with endpoints in different subsets. It is known that if the edge weights are non-negative, then the min-cut problem is polynomially solvable, while the max-cut problem is NP-hard. We construct a partition of the positive orthant into convex cones corresponding to the characteristic cut vectors, similar to a normal fan of a cut polyhedron. A graph of a cone partition is a graph whose vertices are cones, and two cones are adjacent if and only if they have a common facet. We define adjacency criteria in the graphs of cone partitions for the min-cut and max-cut problems. Based on them, we show that for both problems the vertex degrees are exponential, and the graph diameter equals 2. These results contrast with the clique numbers of graphs of cone partitions, which are linear for the minimum cut problem and exponential for the maximum cut problem.