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.
This work is partially supported by JSPS KAKENHI Grant Number JP17H01788 and JST CREST JPMJCR1401.
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Kobayashi, Y., Kojima, K., Matsubara, N., Sone, T., Yamamoto, A. (2019). Algorithms and Hardness Results for the Maximum Balanced Connected Subgraph Problem. In: Li, Y., Cardei, M., Huang, Y. (eds) Combinatorial Optimization and Applications. COCOA 2019. Lecture Notes in Computer Science(), vol 11949. Springer, Cham. https://doi.org/10.1007/978-3-030-36412-0_24
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