Abstract
For an integer \(d \ge 2\), a distance-d independent set of an unweighted graph \(G = (V, E)\) is a subset \(S \subseteq V\) of vertices such that for any pair of vertices \(u, v \in S\), the number of edges in any path between u and v is at least d in G. Given an unweighted graph G, the goal of Maximum Distance-d Independent Set problem (MaxD d IS) is to find a maximum-cardinality distance-d independent set of G. In this paper we focus on MaxD3IS on cubic (3-regular) graphs. For every fixed integer \(d\ge 3\), MaxD d IS is NP-hard even for planar bipartite graphs of maximum degree three. Furthermore, when \(d =3\), it is known that there exists no \(\sigma \)-approximation algorithm for MaxD3IS oncubic graphs for constant \(\sigma < 1.00105\). On the other hand, the previously best approximation ratio known for MaxD3IS on cubic graphs is 2. In this paper, we improve the approximation ratio into 1.875 for MaxD3IS on cubic graphs.
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This work is partially supported by JSPS KAKENHI Grant Numbers JP15J05484, JP15H00849, JP16K00004, and JP26330017.
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Agnarsson, G., Damaschke, P., Halldórsson, M.H.: Powers of geometric intersection graphs and dispersion algorithms. Discret. Appl. Math. 132, 3–16 (2004)
Berman, P., Fujito, T.: On approximation properties of the independent set problem for low degree graphs. Theory Comput. Syst. 32(2), 115–132 (1999)
Brandstädt, A., Giakoumakis, V.: Maximum weight independent sets in hole- and co-chair-free graphs. Inf. Process. Lett. 112, 67–71 (2012)
Chlebík, M., Chlebíková, J.: Complexity of approximating bounded variants of optimization problems. Theoret. Comput. Sci. 354, 320–338 (2006)
Eto, H., Guo, F., Miyano, E.: Distance-\(d\) independent set problems for bipartite and chordal graphs. J. Comb. Optim. 27(1), 88–99 (2014)
Eto, H., Ito, T., Liu, Z., Miyano, E.: Approximability of the distance independent set problem on regular graphs and planar graphs. In: Chan, T.-H.H., Li, M., Wang, L. (eds.) COCOA 2016. LNCS, vol. 10043, pp. 270–284. Springer, Heidelberg (2016)
Gavril, F.: Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of chordal graph. SIAM J. Comput. 1, 180–187 (1972)
Gavril, F.: Algorithms on circular-arc graphs. Networks 4, 357–369 (1974)
Golumbic, M.C.: The complexity of comparability graph recognition and coloring. Computing 18, 199–208 (1977)
Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)
Lozin, V.V., Milanič, M.: A polynomial algorithm to find an independent set of maximum weight in a fork-free graph. J. Discret. Algorithm 6, 595–604 (2008)
Minty, G.J.: On maximal independent sets of vertices in claw-free graphs. J. Comb. Theory Ser. B 28, 284–304 (1980)
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3(1), 103–128 (2007)
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Eto, H., Ito, T., Liu, Z., Miyano, E. (2017). Approximation Algorithm for the Distance-3 Independent Set Problem on Cubic Graphs. In: Poon, SH., Rahman, M., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2017. Lecture Notes in Computer Science(), vol 10167. Springer, Cham. https://doi.org/10.1007/978-3-319-53925-6_18
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DOI: https://doi.org/10.1007/978-3-319-53925-6_18
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