An O(n^4) time algorithm to compute the bisection width of solid grid graphs

The bisection problem asks for a partition of the n vertices of a graph into two sets of size at most dn/2e, so that the number of edges connecting the two sets is minimised. A grid graph is a finite connected subgraph of the infinite two-dimensional grid. It is called solid if it has no holes. Papadimitriou and Sideri [7] gave an O(n) time algorithm to solve the bisection problem on solid grid graphs. We propose a novel approach that exploits structural properties of optimal cuts within a dynamic program. We show that our new technique leads to an O(n)