Algorithms and Hardness Results for the Maximum Balanced Connected Subgraph Problem.

The Balanced Connected Subgraph problem (BCS) was recently introduced by Bhore et al. (CALDAM 2019). In this problem, we are given a graph G whose vertices are colored by red or blue. The goal is to find a maximum connected subgraph of G having the same number of blue vertices and red vertices. They showed that this problem is NP-hard even on planar graphs, bipartite graphs, and chordal graphs. They also gave some positive results: BCS can be solved in \(O(n^3)\) time for trees and \(O(n + m)\) time for split graphs and properly colored bipartite graphs, where n is the number of vertices and m is the number of edges. In this paper, we show that BCS can be solved in \(O(n^2)\) time for trees and \(O(n^3)\) time for interval graphs. The former result can be extended to bounded treewidth graphs. We also consider a weighted version of BCS (WBCS). We prove that this variant is weakly NP-hard even on star graphs and strongly NP-hard even on split graphs and properly colored bipartite graphs, whereas the unweighted counterpart is tractable on those graph classes. Finally, we consider an exact exponential-time algorithm for general graphs. We show that BCS can be solved in \(2^{n/2}n^{O(1)}\) time. This algorithm is based on a variant of Dreyfus-Wagner algorithm for the Steiner tree problem.