The Balanced Connected Subgraph Problem
The problem of computing induced subgraphs that satisfy some specified restrictions arises in various applications of graph algorithms and has been well studied. In this paper, we consider the following Open image in new window (shortly, Open image in new window ) problem. The input is a graph \(G=(V,E)\), with each vertex in the set V having an assigned color, “ Open image in new window ” or “ Open image in new window ”. We seek a maximum-cardinality subset \(V'\subseteq V\) of vertices that is Open image in new window (having exactly \(|V'|/2\) red nodes and \(|V'|/2\) blue nodes), such that the subgraph induced by the vertex set \(V'\) in G is connected. We show that the BCS problem is NP-hard, even for bipartite graphs G (with red/blue color assignment not necessarily being a proper 2-coloring). Further, we consider this problem for various classes of the input graph G, including, e.g., planar graphs, chordal graphs, trees, split graphs, bipartite graphs with a proper red/blue 2-coloring, and graphs with diameter 2. For each of these classes either we prove NP-hardness or design a polynomial time algorithm.
KeywordsBalanced connected subgraph Trees Split graphs Chordal graphs Planar graphs Bipartite graphs NP-hard Color-balanced
We thank Florian Sikora for pointing out the connection with the Graph Motif problem.
- 1.Aichholzer, O., et al.: Balanced islands in two colored point sets in the plane. arXiv preprint arXiv:1510.01819 (2015)
- 2.Balachandran, N., Mathew, R., Mishra, T.K., Pal, S.P.: System of unbiased representatives for a collection of bicolorings. arXiv preprint arXiv:1704.07716 (2017)
- 5.Biniaz, A., Maheshwari, A., Smid, M.H.: Bottleneck bichromatic plane matching of points. In: CCCG (2014)Google Scholar
- 12.El-Kebir, M., Klau, G.W.: Solving the maximum-weight connected subgraph problem to optimality. CoRR abs/1409.5308 (2014)Google Scholar
- 17.Kaneko, A., Kano, M.: Discrete geometry on red and blue points in the plane—a survey—. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry, vol. 25, pp. 551–570. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-642-55566-4_25CrossRefzbMATHGoogle Scholar
- 18.Kaneko, A., Kano, M., Watanabe, M.: Balancing colored points on a line by exchanging intervals. J. Inf. Process. 25, 551–553 (2017)Google Scholar