Abstract
We introduce the Red-Blue Separation problem on graphs, where we are given a graph \(G = (V, E)\) whose vertices are colored either red or blue, and we want to select a (small) subset \(\mathcal{S} \subseteq V\), called red-blue separating set, such that for every red-blue pair of vertices, there is a vertex \(s \in \mathcal S\) whose closed neighborhood contains exactly one of the two vertices of the pair. We study the computational complexity of Red-Blue Separation, in which one asks whether a given red-blue colored graph has a red-blue separating set of size at most a given integer. We prove that the problem is NP-complete even for restricted graph classes. We also show that it is always approximable in polynomial time within a factor of \(2\ln n\), where n is the input graph’s order. In contrast, for triangle-free graphs and for graphs of bounded maximum degree, we show that Red-Blue Separation is solvable in polynomial time when the size of the smaller color class is bounded by a constant. However, on general graphs, we show that the problem is W[2]-hard even when parameterized by the solution size plus the size of the smaller color class. We also consider the problem Max Red-Blue Separation where the coloring is not part of the input. Here, given an input graph G, we want to determine the smallest integer k such that, for every possible red-blue-coloring of G, there is a red-blue separating set of size at most k. We derive tight bounds on the cardinality of an optimal solution of Max Red-Blue Separation, showing that it can range from logarithmic in the graph order, up to the order minus one. We also give bounds with respect to related parameters. For trees however we prove an upper bound of two-thirds the order. We then show that Max Red-Blue Separation is NP-hard, even for graphs of bounded maximum degree, but can be approximated in polynomial time within a factor of \(O(\ln ^ 2 n)\).
This study has been carried out in the frame of the “Investments for the future” Programme IdEx Bordeaux - SysNum (ANR-10-IDEX-03-02).
F. Foucaud—Research financed by the French government IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25) and by the ANR project GRALMECO (ANR-21-CE48-0004-01).
T. Lehtilä—Research supported by the Finnish Cultural Foundation and by the Academy of Finland grant 338797.
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Notes
- 1.
One class consists of vertices colored red and the other class consists of vertices colored blue.
- 2.
A graph \(G = (V, E)\) is called a split graph when the vertices in V can be partitioned into an independent set and a clique.
References
Bollobás, B., Scott, A.D.: On separating systems. Eur. J. Comb. 28, 1068–1071 (2007)
Bondy, J.A.: Induced subsets. J. Comb. Theory Ser. B 12(2), 201–202 (1972)
Bonnet, É., Giannopoulos, P., Lampis, M.: On the parameterized complexity of red-blue points separation. J. Comput. Geom. 10(1), 181–206 (2019)
Bousquet, N., Lagoutte, A., Li, Z., Parreau, A., Thomassé, S.: Identifying codes in hereditary classes of graphs and VC-dimension. SIAM J. Discret. Math. 29(4), 2047–2064 (2015)
Cǎlinescu, G., Dumitrescu, A., Karloff, H.J., Wan, P.: Separating points by axis-parallel lines. Int. J. Comput. Geom. Appl. 15(6), 575–590 (2005)
Charbit, E., Charon, I., Cohen, G., Hudry, O., Lobstein, A.: Discriminating codes in bipartite graphs: bounds, extremal cardinalities, complexity. Adv. Math. Commun. 2(4), 403–420 (2008)
Chlebus, B.S., Nguyen, S.H.: On finding optimal discretizations for two attributes. In: Polkowski, L., Skowron, A. (eds.) RSCTC 1998. LNCS (LNAI), vol. 1424, pp. 537–544. Springer, Heidelberg (1998). https://doi.org/10.1007/3-540-69115-4_74
Crowston, R., Gutin, G., Jones, M., Muciaccia, G., Yeo, A.: Parameterizations of test cover with bounded test sizes. Algorithmica 74(1), 367–384 (2016)
Dey, S., Foucaud, F., Nandy, S.C., Sen, A.: Discriminating codes in geometric setups. In: Proceedings of the 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics, vol. 181, pp. 24:1–24:16 (2020)
De Bontridder, K.M.J., et al.: Approximation algorithms for the test cover problem. Math. Program. Ser. B 98, 477–491 (2003)
Dinur, I., Steurer, D.: Analytical approach to parallel repetition. In: ACM Symposium on Theory of Computing, vol. 46, pp. 624–633 (2014)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
Erdös, P.: Some combinatorial, geometric and set theoretic problems in measure theory. In: Kölzow, D., Maharam-Stone, D. (eds.) Measure Theory Oberwolfach 1983. LNM, vol. 1089, pp. 321–327. Springer, Heidelberg (1984). https://doi.org/10.1007/BFb0072626
Foucaud, F., Guerrini, E., Kovše, M., Naserasr, R., Parreau, A., Valicov, P.: Extremal graphs for the identifying code problem. Eur. J. Comb. 32(4), 628–638 (2011)
Gledel, V., Parreau, A.: Identification of points using disks. Discret. Math. 342, 256–269 (2019)
Har-Peled, S., Jones, M.: On separating points by lines. Discret. Comput. Geom. 63, 705–730 (2020)
Henning, M.A., Yeo, A.: Distinguishing-transversal in hypergraphs and identifying open codes in cubic graphs. Graphs Comb. 30(4), 909–932 (2014)
Karpovsky, M.G., Chakrabarty, K., Levitin, L.B.: On a new class of codes for identifying vertices in graphs. IEEE Trans. Inf. Theory 44, 599–611 (1998)
Kratsch, S., Masařík, T., Muzi, I., Pilipczuk, M., Sorge, M.: Optimal discretization is fixed-parameter tractable. In: Proceedings of the 32nd ACM-SIAM Symposium on Discrete Algorithms (SODA 2021), pp. 1702–1719 (2021)
Kujala, J., Elomaa, T.: Improved algorithms for univariate discretization of continuous features. In: Kok, J.N., Koronacki, J., Lopez de Mantaras, R., Matwin, S., Mladenič, D., Skowron, A. (eds.) PKDD 2007. LNCS (LNAI), vol. 4702, pp. 188–199. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74976-9_20
Lobstein, A.: Watching systems, identifying, locating-dominating and discriminating codes in graphs: a bibliography. https://www.lri.fr/~lobstein/debutBIBidetlocdom.pdf
Misra, N., Mittal, H., Sethia, A.: Red-blue point separation for points on a circle. In: Proceedings of the 32nd Canadian Conference on Computational Geometry (CCCG 2020), pp. 266–272 (2020)
Moret, B.M.E., Shapiro, H.D.: On minimizing a set of tests. SIAM J. Sci. Stat. Comput. 6(4), 983–1003 (1985)
Rényi, A.: On random generating elements of a finite Boolean algebra. Acta Scientiarum Mathematicarum Szeged 22, 75–81 (1961)
Tovey, C.A.: A simplified NP-complete satisfiability problem. Discret. Appl. Math. 8(1), 85–89 (1984)
Ungrangsi, R., Trachtenberg, A., Starobinski, D.: An implementation of indoor location detection systems based on identifying codes. In: Aagesen, F.A., Anutariya, C., Wuwongse, V. (eds.) INTELLCOMM 2004. LNCS, vol. 3283, pp. 175–189. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30179-0_16
Vapnik, V.N., Chervonenkis, A.Y.: On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl. 16(2), 264–280 (1971)
Zvervich, I.E., Zverovich, V.E.: An induced subgraph characterization of domination perfect graphs. J. Graph Theory 20(3), 375–395 (1995)
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Dev, S.R., Dey, S., Foucaud, F., Klasing, R., Lehtilä, T. (2022). The Red-Blue Separation Problem on Graphs. In: Bazgan, C., Fernau, H. (eds) Combinatorial Algorithms. IWOCA 2022. Lecture Notes in Computer Science, vol 13270. Springer, Cham. https://doi.org/10.1007/978-3-031-06678-8_21
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