Abstract
In this paper, we prove several inapproximability results on the P 3-convexity and the geodetic convexity on graphs. We prove that determining the P 3-hull number and the geodetic hull number are APX-hard problems. We prove that the Carathéodory number, the Radon number and the convexity number of both convexities are O(n 1 − ε)-inapproximable in polynomial time for every ε > 0, unless P=NP. We also prove that the interval numbers of both convexities are W[2]-hard and O(logn)-inapproximable in polynomial time, unless P=NP. Moreover, these results hold for bipartite graphs in the P 3-convexity.
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References
Araújo, J., Campos, V., Giroire, F., Sampaio, L., Soares, R.: On the hull number of some graph classes. Elec. Notes Disc. Math. 38, 49–55 (2011)
Araújo, R.T., Sampaio, R.M., Szwarcfiter, J.L.: The convexity of induced paths of order three. Electronic Notes in Discrete Mathematics (2013) (to appear)
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Combinatorial Optimization Problems and Their Approximability Properties. Springe, Berlin (1999)
Ausiello, G., D’Atri, A., Protasi, M.: Structure preserving reductions among convex optimization problems. J. Comput. System Sci. 21, 136–153 (1980)
Barbosa, R.M., Coelho, E.M.M., Dourado, M.C., Rautenbach, D., Szwarcfiter, J.L.: On the Carathéodory number for the convexity of paths of order three. SIAM J. Discrete Math. 26, 929–939 (2012)
Dourado, M.C., Rautenbach, D., dos Santos, V.F., Schäfer, P.M., Szwarcfiter, J.L., Toman, A.: Algorithmic and structural aspects of the P 3-Radon number. Ann. Oper. Res. 206, 75–91 (2013)
Cáceres, J., Hernando, C., Mora, M., Pelayo, I.M., Puertas, M.L., Seara, C.: On geodetic sets formed by boundary vertices. Discrete Math. 306, 188–198 (2006)
Centeno, C., Dourado, M., Penso, L., Rautenbach, D., Szwarcfiter, J.L.: Irreversible conversion of graphs. Theoretical Computer Science 412, 3693–3700 (2011)
Centeno, C.C., Dantas, S., Dourado, M.C., Rautenbach, D., Szwarcfiter, J.L.: Convex Partitions of Graphs induced by Paths of Order Three. Discrete Mathematics and Theoretical Computer Science 12, 175–184 (2010)
Centeno, C.C., Dourado, Szwarcfiter, J.L.: On the convexity of Paths of length two in undirected graphs. Elect. Notes in Disc. Math. 32, 11–18 (2009)
Changat, M., Mathew, J.: On triangle path convexity in graphs. Discrete Mathematics 206, 91–95 (1999)
Changat, M., Klavz̃ar, S., Mulder, H.M.: The all-paths transit function of a graph. Czech. Math. J. 51 (126), 439��448 (2001)
Deagan, F.F., Nicolai, F., Bransdstädt, A.: Convexity and HHD-graphs. SIAM J. Discrete Mathematics 12, 119–135 (1999)
Dourado, M.C., Protti, F., Rautenbach, D., Szwarcfiter, J.L.: On the convexity number of graphs. Graphs and Combinatorics 28, 333–345 (2012)
Dourado, M., Protti, F., Rautenbach, D., Szwarcfiter, J.L.: Some remarks on the geodetic number of a graph. Discrete Mathematics 310, 832–837 (2012)
Dourado, M.C., Protti, F., Rautenbach, D., Szwarcfiter, J.L.: On the hull number of triangle-free graphs. SIAM J. Discrete Math. 23, 2163–2172 (2010)
Dourado, M.C., Protti, F., Szwarcfiter, J.L.: Complexity results related to monophonic convexity. Discrete Appl. Math. 158, 1269–1274 (2010)
Dourado, M.C., Rautenbach, D., dos Santos, V.F., Schäfer, P.M., Szwarcfiter, J.L.: On the Carathéodory number of interval and graph convexities (to appear)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)
Duchet, P.: Convex sets in graphs II: Minimal path convexity. J. Combin. Theory, Ser. B 44, 307–316 (1988)
Erdős, P., Fried, E., Hajnal, A., Milner, E.C.: Some remarks on simple tournaments. Algebra Univers. 2, 238–245 (1972)
Everett, M.G., Seidman, S.B.: The hull number of a graph. Discrete Math. 57, 217–223 (1985)
Farber, M., Jamison, R.E.: Convexity in graphs and hypergraphs. SIAM J. Algebraic Discrete Methods 7, 433–444 (1986)
Farber, M., Jamison, R.E.: On local convexity in graphs. Discrete Math. 66, 231–247 (1987)
Parker, D.B., Westhoff, R.F., Wolf, M.J.: On two-path convexity in multipartite tournaments. European J. Combin. 29, 641–651 (2008)
Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proc. of the 29th Annual ACM Symposium on Theory of Computing, pp. 475–484 (1987)
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: Proc. 38th ACM Symp. Theory of Computing (STOC 2006), pp. 681–690 (2006)
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Coelho, E.M.M., Dourado, M.C., Sampaio, R.M. (2014). Inapproximability Results for Graph Convexity Parameters. In: Kaklamanis, C., Pruhs, K. (eds) Approximation and Online Algorithms. WAOA 2013. Lecture Notes in Computer Science, vol 8447. Springer, Cham. https://doi.org/10.1007/978-3-319-08001-7_9
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DOI: https://doi.org/10.1007/978-3-319-08001-7_9
Publisher Name: Springer, Cham
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