Abstract
This work further explores the applications of co-nondeterminism for showing kernelization lower bounds. The only known example excludes polynomial kernelizations for the Ramsey problem of finding an independent set or a clique of at least k vertices in a given graph (Kratsch 2012, SODA). We study the more general problem of finding induced subgraphs on k vertices fulfilling some hereditary property Π, called Π-Induced Subgraph. The problem is NP-hard for all non-trivial choices of Π by a classic result of Lewis and Yannakakis (JCSS 1980). The parameterized complexity of this problem was classified by Khot and Raman (TCS 2002) depending on the choice of Π. The interesting cases for kernelization are for Π containing all independent sets and all cliques, since the problem is trivial or W[1]-hard otherwise.
Our results are twofold. Regarding Π-Induced Subgraph, we show that for a large choice of natural graph properties Π, including chordal, perfect, cluster, and cograph, there is no polynomial kernel with respect to k. This is established by two theorems: one using a co-nondeterministic variant of cross-composition and one by a polynomial parameter transformation from Ramsey.
Additionally, we show how to use improvement versions of NP-hard problems as source problems for lower bounds, without requiring their NP-hardness. E.g., for Π-Induced Subgraph our compositions may assume existing solutions of size k − 1. We believe this to be useful for further lower bound proofs, since improvement versions simplify the construction of a disjunction (OR) of instances required in compositions. This adds a second way of using co-nondeterminism for lower bounds.
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References
Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. Journal of Computer and System Sciences 75(8), 423–434 (2009)
Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Cross-composition: A new technique for kernelization lower bounds. In: STACS, pp. 165–176 (2011)
Dell, H., Marx, D.: Kernelization of packing problems. In: SODA, pp. 68–81 (2012)
Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: STOC, pp. 251–260 (2010)
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)
Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. Journal of Computer and System Sciences 77, 91–106 (2011)
Hermelin, D., Wu, X.: Weak compositions and their applications to polynomial lower bounds for kernelization. In: SODA, pp. 104–113 (2012)
Kratsch, S.: Co-nondeterminism in compositions: a kernelization lower bound for a ramsey-type problem. In: SODA, pp. 114–122 (2012)
Harnik, D., Naor, M.: On the compressibility of NP instances and cryptographic applications. SIAM Journal on Computing 39(5), 1667–1713 (2010)
Khot, S., Raman, V.: Parameterized complexity of finding subgraphs with hereditary properties. Theoretical Computer Science 289, 997–1008 (2002)
Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. Journal of Computer and System Sciences 20(2), 219–230 (1980)
Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compositio Mathematica 2, 463–470 (1935)
Erdős, P.: Some remarks on the theory of graphs. Bulletin of the American Mathematical Society 53, 292–294 (1947)
Erdős, P., Hajnal, A.: Ramsey-type theorems. Discrete Applied Mathematics 25(1-2), 37–52 (1989)
Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Information Processing Letters 58, 171–176 (1996)
Marx, D.: Chordal deletion is fixed-parameter tractable. Algorithmica 57(4), 747–768 (2010)
Marx, D., Schlotter, I.: Obtaining a planar graph by vertex deletion. Algorithmica 62(3-4), 807–822 (2012)
van Bevern, R., Komusiewicz, C., Moser, H., Niedermeier, R.: Measuring Indifference: Unit Interval Vertex Deletion. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 232–243. Springer, Heidelberg (2010)
Villanger, Y.: Proper Interval Vertex Deletion. In: Raman, V., Saurabh, S. (eds.) IPEC 2010. LNCS, vol. 6478, pp. 228–238. Springer, Heidelberg (2010)
Diestel, R.: Graph Theory. Springer (2005)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, 2nd edn. Elsevier Science (2004)
Alon, N., Pach, J., Solymosi, J.: Ramsey-type theorems with forbidden subgraphs. Combinatorica 21(2), 155–170 (2001)
Lovasz, L.: Perfect graphs. In: Beineke, L.W., Wilson, R.J. (eds.) Selected Topics in Graph Theory, vol. 2, pp. 55–67. Academic Press, London (1983)
Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theoretical Computer Science 412(35), 4570–4578 (2011)
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Kratsch, S., Pilipczuk, M., Rai, A., Raman, V. (2012). Kernel Lower Bounds Using Co-nondeterminism: Finding Induced Hereditary Subgraphs. In: Fomin, F.V., Kaski, P. (eds) Algorithm Theory – SWAT 2012. SWAT 2012. Lecture Notes in Computer Science, vol 7357. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31155-0_32
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