Algorithms and Hardness Results for the Maximum Balanced Connected Subgraph Problem

TL;DR

This paper advances the understanding of the Maximum Balanced Connected Subgraph problem by improving algorithms for specific graph classes, analyzing its weighted variant's complexity, and providing an exponential-time solution for general graphs.

Contribution

It presents faster algorithms for trees and interval graphs, analyzes the weighted version's NP-hardness, and offers an exponential algorithm for general graphs.

Findings

BCS solvable in O(n^2) for trees

Weighted BCS is weakly NP-hard on stars

Exponential algorithm runs in 2^{n/2}n^{O(1)} time

Abstract

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.