Abstract
For some years it was believed that for “connectivity” problems such as Hamiltonian Cycle, algorithms running in time \(2^{O({\mathbf {tw}})}\cdot n^{O(1)}\) –called single-exponential– existed only on planar and other sparse graph classes, where \({\mathbf {tw}}\) stands for the treewidth of the \(n\)-vertex input graph. This was recently disproved by Cygan et al. [FOCS 2011], Bodlaender et al. [ICALP 2013], and Fomin et al. [SODA 2014], who provided single-exponential algorithms on general graphs for most connectivity problems that were known to be solvable in single-exponential time on sparse graphs. In this article we further investigate the role of planarity in connectivity problems parameterized by treewidth, and convey that several problems can indeed be distinguished according to their behavior on planar graphs. Known results from the literature imply that there exist problems, like Cycle Packing, that cannot be solved in time \(2^{o({\mathbf {tw}}\log {\mathbf {tw}})} \cdot n^{O(1)}\) on general graphs but that can be solved in time \(2^{O({\mathbf {tw}})} \cdot n^{O(1)}\) when restricted to planar graphs. Our main contribution is to show that there exist natural problems that can be solved in time \(2^{O({\mathbf {tw}}\log {\mathbf {tw}})} \cdot n^{O(1)}\) on general graphs but that cannot be solved in time \(2^{o({\mathbf {tw}}\log {\mathbf {tw}})} \cdot n^{O(1)}\) even when restricted to planar graphs. Furthermore, we prove that Planar Cycle Packing and Planar Disjoint Paths cannot be solved in time \(2^{o({\mathbf {tw}})} \cdot n^{O(1)}\). The mentioned negative results hold unless the ETH fails. We feel that our results constitute a first step in a subject that can be further exploited.
Research supported by the Languedoc-Roussillon Project “Chercheur d’avenir” KERNEL and by the grant EGOS ANR-12-JS02-002-01.
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Notes
- 1.
That is, certificates consisting of a constant number of bits per vertex that can be checked by a cardinality check and by iteratively looking at the neighborhoods of the input graph.
- 2.
The ETH states that there exists a positive real number \(s\) such that 3-CNF-Sat with \(n\) variables and \(m\) clauses cannot be solved in time \(2^{sn}\cdot (n+m)^{O(1)}\). See [15] for more details.
- 3.
In [15] that Planar Vertex Cover problem is mentioned, which is equivalent to solving Planar Independent Set, as the complement of a vertex cover is an independent set.
References
Baste, J., Sau, I.: The role of planarity in connectivity problems parameterized by treewidth. CoRR, abs/1312.2889 (2013)
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)
Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inform. Comput. 85(1), 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: Proceeding of the 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 150–159. IEEE Computer Society (2011)
Deineko, V.G., Steiner, G., Xue, Z.: Robotic-cell scheduling: special polynomially solvable cases of the traveling salesman problem on permuted monge matrices. J. Comb. Optim. 9, 381–399 (2005)
Diestel, R.: Graph Theory, 3rd edn. Springer, New York (2005)
Dorn, F., Fomin, F.V., Thilikos, D.M.: Fast subexponential algorithm for non-local problems on graphs of bounded genus. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 172–183. Springer, Heidelberg (2006)
Dorn, F., Fomin, F.V., Thilikos, D.M.: Catalan structures and dynamic programming in \(H\)-minor-free graphs. J. Syst. Sci. 78(5), 1606–1622 (2012)
Dorn, F., Penninkx, E., Bodlaender, H.L., Fomin, F.V.: Efficient exact algorithms on planar graphs: exploiting sphere cut decompositions. Algorithmica 58(3), 790–810 (2010)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Theoretical Computer Science. Springer, Berlin (2006)
Fomin, F.V., Lokshtanov, D., Saurabh, S.: Efficient Computation of Representative Sets with Applications in Parameterized and Exact Algorithms. In: Proceeding of SODA’14. CoRR, abs/1304.4626 (2013)
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)
Kloks, T., Lee, C.M., Liu, J.: New algorithms for \(k\)-face cover, \(k\)-feedback vertex set, and \(k\)-disjoint cycles on plane and planar graphs. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 282–295. Springer, Heidelberg (2002)
Lokshtanov, D., Marx, D., Saurabh, S.: Known algorithms on graphs of bounded treewidth are probably optimal. In: Proceeding of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 777–789 (2011)
Lokshtanov, D., Marx, D., Saurabh, S.: Lower bounds based on the exponential time hypothesis. Bull. EATCS 105, 41–72 (2011)
Lokshtanov, D., Marx, D., Saurabh, S.: Slightly superexponential parameterized problems. In: Proceeding of the 22nd Annual ACM-SIAM Symposium on Discrete algorithms (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)
Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986)
Rué, J., Sau, I., Thilikos, D.M.: Dynamic programming for graphs on surfaces. In: Short Version in the Proceeding of ICALP’10 in ACM Transactions on Algorithms (TALG). CoRR, abs/1104.2486 (2011)
Bhattacharya, B., Kameda, T.: A linear time algorithm for computing minmax regret 1-median on a tree. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 1–12. Springer, Heidelberg (2012)
Scheffler, P.: A practical linear time algorithm for disjoint paths in graphs with bounded tree-width. Fachbereich 3 Mathematik, Technical Report 396/1994, FU Berlin (1994)
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We would like to thank the anonymous referees for helpful remarks that improved the presentation of the manuscript.
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Baste, J., Sau, I. (2014). The Role of Planarity in Connectivity Problems Parameterized by Treewidth. In: Cygan, M., Heggernes, P. (eds) Parameterized and Exact Computation. IPEC 2014. Lecture Notes in Computer Science(), vol 8894. Springer, Cham. https://doi.org/10.1007/978-3-319-13524-3_6
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