Algorithms for the minimum non-separating path and the balanced connected bipartition problems on grid graphs (With errata)

For given a pair of nodes in a graph, the minimum non-separating path problem looks for a minimum weight path between the two nodes such that the remaining graph after removing the path is still connected. The balanced connected bipartition (BCP2) problem looks for a way to bipartition a graph into two connected subgraphs with their weights as equal as possible. In this paper we present an algorithm in time O(N logN) for finding a minimum weight non-separating path between two given nodes in a grid graph of N nodes with positive weight. This result leads to a 5/4-approximation algorithm for the BCP2 problem on grid graphs, which is the currently best ratio achieved in polynomial time. We also developed an exact algorithm for the BCP2 problem on grid graphs. Based on the exact algorithm and a rounding technique, we show an approximation scheme, which is a fully polynomial time approximation scheme for fixed number of rows.