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For given a pair of nodes in a graph, the minimum non-separating path problemlooks for a minimum weight path between the two nodes such that the remaininggraph after removing the path is still connected. The balanced connectedbipartition (BCP$_2$) problem looks for a way to bipartition a graph into twoconnected subgraphs with their weights as equal as possible. In this paper wepresent an algorithm in time $O(N\log N)$ for finding a minimum weightnon-separating path between two given nodes in a grid graph of $N$ nodes withpositive weight. This result leads to a 5/4-approximation algorithm for theBCP$_2$ problem on grid graphs, which is the currently best ratio achieved inpolynomial time. We also developed an exact algorithm for the BCP$_2$ problemon grid graphs. Based on the exact algorithm and a rounding technique, we showan approximation scheme, which is a fully polynomial time approximation schemefor fixed number of rows.

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