On minimum bisection and related partition problems in graphs with bounded tree width

Abstract

Minimum Bisection denotes the NP-hard problem to partition the vertex set of a graph into two sets of equal sizes while minimizing the number of edges between these two sets. We consider this problem in bounded degree graphs with a given tree decomposition (T, X) and prove an upper bound for their minimum bisection width in terms of the structure and width of (T, X). When (T, X) is provided as input, a bisection satisfying our bound can be computed in time proportional to the encoding length of (T, X). Furthermore, our result can be generalized to k-section, which is known to be APX-hard even when restricted to trees with bounded degree.