Abstract
We study the complexity of a generic hitting problem H -Subgraph Hitting , where given a fixed pattern graph H and an input graph G, we seek for the minimum size of a set X ⊆ V(G) that hits all subgraphs of G isomorphic to H. In the colorful variant of the problem, each vertex of G is precolored with some color from V(H) and we require to hit only H-subgraphs with matching colors. Standard techniques (e.g., Courcelle’s theorem) show that, for every fixed H and the problem is fixed-parameter tractable parameterized by the treewidth of G; however, it is not clear how exactly the running time should depend on treewidth. For the colorful variant, we demonstrate matching upper and lower bounds showing that the dependence of the running time on treewidth of G is tightly governed by μ(H), the maximum size of a minimal vertex separator in H. That is, we show for every fixed H that, on a graph of treewidth t, the colorful problem can be solved in time \(2^{\mathcal{O}(t^{\mu (H)})}\cdot|V(G)|\), but cannot be solved in time \(2^{o(t^{\mu (H)})}\cdot |V(G)|^{O(1)}\), assuming the Exponential Time Hypothesis (ETH). Furthermore, we give some preliminary results showing that, in the absence of colors, the parameterized complexity landscape of H -Subgraph Hitting is much richer.
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 267959, and n. 280152, as well as OTKA grant NK10564 and Polish National Science Centre grant DEC-2012/05/D/ST6/03214.
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References
Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12(2), 308–340 (1991)
Betzler, N., Bredereck, R., Niedermeier, R., Uhlmann, J.: On bounded-degree vertex deletion parameterized by treewidth. Discrete Appl. Math. 160(1-2), 53–60 (2012)
Bodlaender, H.L., Cygan, M., Kratsch, S., Nederlof, J.: Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 196–207. Springer, Heidelberg (2013)
Bodlaender, H.L., Drange, P.G., Dregi, M.S., Fomin, F.V., Lokshtanov, D., Pilipczuk, M.: An \(\mathcal{O}(c^k n)\) 5-approximation algorithm for treewidth. In: FOCS, pp. 499–508 (2013)
Courcelle, B.: The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Inf. Comput. 85, 12–75 (1990)
Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J.M.M., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time. In: FOCS, pp. 150–159 (2011)
Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through colors and iDs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 378–389. Springer, Heidelberg (2009)
Fomin, F.V., Lokshtanov, D., Saurabh, S.: Efficient computation of representative sets with applications in parameterized and exact algorithms. In: SODA, pp. 142–151 (2014)
Jansen, B.M.P., Lokshtanov, D., Saurabh, S.: A near-optimal planarization algorithm. In: SODA, pp. 1802–1811 (2014)
Lokshtanov, D., Marx, D., Saurabh, S.: Known algorithms on graphs on bounded treewidth are probably optimal. In: SODA, pp. 777–789 (2011)
Lokshtanov, D., Marx, D., Saurabh, S.: Lower bounds based on the exponential time hypothesis. Bulletin of the EATCS 105, 41–72 (2011)
Lokshtanov, D., Marx, D., Saurabh, S.: Slightly superexponential parameterized problems. In: SODA, pp. 760–776 (2011)
Pilipczuk, M.: Problems parameterized by treewidth tractable in single exponential time: A logical approach. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 520–531. Springer, Heidelberg (2011)
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Cygan, M., Marx, D., Pilipczuk, M., Pilipczuk, M. (2014). Hitting Forbidden Subgraphs in Graphs of Bounded Treewidth. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds) Mathematical Foundations of Computer Science 2014. MFCS 2014. Lecture Notes in Computer Science, vol 8635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44465-8_17
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