Abstract
Given a simple undirected graph G = (V, E) and an integer k < |V|, the Sparsest k -Subgraph problem asks for a set of k vertices which induces the minimum number of edges. As a generalization of the classical independent set problem, Sparsest k -Subgraph is \(\mathcal{NP}\)-hard and even not approximable unless \(\mathcal{P} = \mathcal{NP}\) in general graphs. Thus, we investigate Sparsest k -Subgraph in graph classes where independent set is polynomial-time solvable, such as subclasses of perfect graphs. Our two main results are the \(\mathcal{NP}\)-hardness of Sparsest k -Subgraph on chordal graphs, and a greedy 2-approximation algorithm. Finally, we also show how to derive a PTAS for Sparsest k -Subgraph on proper interval graphs.
This work has been funded by grant ANR 2010 BLAN 021902.
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References
Apollonio, N., Sebő, A.: Minconvex factors of prescribed size in graphs. SIAM Journal of Discrete Mathematics 23(3), 1297–1310 (2009)
Apollonio, N., Simeone, B.: The maximum vertex coverage problem on bipartite graphs. Preprint (2013)
Bhaskara, A., Charikar, M., Chlamtac, E., Feige, U., Vijayaraghavan, A.: Detecting high log-densities: An \(\mathcal{O}(n^{1/4})\) approximation for densest k-subgraph. In: Proceedings of the 42nd ACM symposium on Theory of Computing, pp. 201–210. ACM (2010)
Bourgeois, N., Giannakos, A., Lucarelli, G., Milis, I., Paschos, V., Pottié, O.: The max quasi-independent set problem. Journal of Combinatorial Optimization 23(1), 94–117 (2012)
Broersma, H., Golovach, P.A., Patel, V.: Tight complexity bounds for FPT subgraph problems parameterized by clique-width. In: Marx, D., Rossmanith, P. (eds.) IPEC 2011. LNCS, vol. 7112, pp. 207–218. Springer, Heidelberg (2012)
Bshouty, N., Burroughs, L.: Massaging a linear programming solution to give a 2-approximation for a generalization of the vertex cover problem. In: Meinel, C., Morvan, M., Krob, D. (eds.) STACS 1998. LNCS, vol. 1373, pp. 298–308. Springer, Heidelberg (1998)
Cai, L.: Parameterized complexity of cardinality constrained optimization problems. Computer Journal 51(1), 102–121 (2008)
Corneil, D.G., Perl, Y.: Clustering and domination in perfect graphs. Discrete Applied Mathematics 9(1), 27–39 (1984)
Fulkerson, D., Gross, O.: Incidence matrices and interval graphs. Pacific J. Math. 15, 835–855 (1965)
Goldschmidt, O., Hochbaum, D.S.: k-edge subgraph problems. Discrete Applied Mathematics 74(2), 159–169 (1997)
Guo, J., Niedermeier, R., Wernicke, S.: Parameterized complexity of vertex cover variants. Theory of Computing Systems 41(3), 501–520 (2007)
Joret, G., Vetta, A.: Reducing the rank of a matroid. CoRR, abs/1211.4853 (2012)
Kneis, J., Langer, A., Rossmanith, P.: Improved upper bounds for partial vertex cover. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 240–251. Springer, Heidelberg (2008)
Liazi, M., Milis, I., Zissimopoulos, V.: A constant approximation algorithm for the densest k-subgraph problem on chordal graphs. Information Processing Letters 108(1), 29–32 (2008)
Nonner, T.: PTAS for densest k-subgraph in interval graphs. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 631–641. Springer, Heidelberg (2011)
Watrigant, R., Bougeret, M., Giroudeau, R.: The k-sparsest subgraph problem. Technical Report RR-12019, LIRMM (2012)
Watrigant, R., Bougeret, M., Giroudeau, R.: Approximating the sparsest k-subgraph in chordal graphs. Technical Report hal-00868188, LIRMM (2013)
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Watrigant, R., Bougeret, M., Giroudeau, R. (2014). Approximating the Sparsest k-Subgraph in Chordal Graphs. 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_7
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DOI: https://doi.org/10.1007/978-3-319-08001-7_7
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