On Minimum Bisection and Related Partition Problems in Graphs with Bounded Tree Width

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.