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

TL;DR

This paper introduces efficient algorithms for the minimum non-separating path and the balanced connected bipartition problems on grid graphs, including an optimal algorithm, a near-optimal approximation, and an exact solution for specific cases.

Contribution

It presents an $O(N ext{log}N)$ algorithm for the minimum non-separating path and a 5/4-approximation for BCP$_2$, along with an exact algorithm and a PTAS for fixed rows.

Findings

Optimal $O(N ext{log}N)$ algorithm for minimum non-separating path.

Best polynomial-time approximation ratio of 5/4 for BCP$_2$ on grid graphs.

Development of an exact algorithm and a PTAS for BCP$_2$ on grid graphs.

Abstract

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 (BCP$_2$) 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 $O(N\log N)$ 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 BCP$_2$ problem on grid graphs, which is the currently best ratio achieved in polynomial time. We also developed an exact algorithm for the BCP$_2$ 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.