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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References
Björklund, A., Kaski, P., Kowalik, Ł.: Constrained multilinear detection and generalized graph motifs. Algorithmica 74(2), 947–967 (2016)
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets möbius: fast subset convolution. In: Proceedings of STOC 2007, pp. 67–74 (2007)
Bhore, S., Chakraborty, S., Jana, S., Mitchell, J.S.B., Pandit, S., Roy, S.: The balanced connected subgraph problem. In: Pal, S.P., Vijayakumar, A. (eds.) CALDAM 2019. LNCS, vol. 11394, pp. 201–215. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11509-8_17
Bhore, S., Jana, S., Mitchell, J. S., Pandit, S., Roy, S.: Balanced connected subgraph problem in geometric intersection graphs. ArXiv:1909.03872 (2019)
Bodlaender, H.L., Cygan, M., Kratsch, S., Nederlof, J.: Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput. 243, 86–111 (2015)
Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)
Calders, T., Karim, A., Kamiran, F., Ali, W., Zhang, X.: Controlling attribute effect in linear regression. In: Proceedings of ICDM 2013, pp. 71–80. IEEE (2013)
Celis, L. E., Straszak, D., Vishnoi, N. K.: Ranking with fairness constraints. In: Proceedings of ICALP 2018, LIPIcs, pp. 28:1–28:15 (2018)
Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. In: Proceedings of NIPS 2017, pp. 5029–5037 (2017)
Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Matroids, matchings, and fairness. In: Proceedings of AISTATS 2019, pp. 2212–2220 (2019)
Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)
Courcelle, B., Mosbah, M.: Monadic second-order evaluations on tree-decomposable graphs. Theor. Comput. Sci. 109(1), 49–82 (1993)
Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3
Dreyfus, S.E., Wagner, R.A.: The Steiner problems. Network 1(3), 195–207 (1971)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)
Joseph, M., Kearns, M., Morgenstern, J.H., Roth, A.: Fairness in learning: classic and contextual bandits. In: Proceedings of NIPS 2016, pp. 325–333 (2016)
Kumabe, S., Maehara, T., Sin’ya, R.: Linear pseudo-polynomial factor algorithm for automaton constrained tree knapsack problem. In: Das, G.K., Mandal, P.S., Mukhopadhyaya, K., Nakano, S. (eds.) WALCOM 2019. LNCS, vol. 11355, pp. 248–260. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-10564-8_20
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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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