2026-08-14T03:59:56Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:9968 2024-03-06T10:45:13Z ddc:004 open_access

Data-Compression for Parametrized Counting Problems on Sparse Graphs Kim, Eun Jung Serna, Maria Thilikos, Dimitrios M. Parameterized counting compactor protrusion decomposition We study the concept of compactor, which may be seen as a counting-analogue of kernelization in counting parameterized complexity. For a function F:Sigma^* -> N and a parameterization kappa: Sigma^* -> N, a compactor (P,M) consists of a polynomial-time computable function P, called condenser, and a computable function M, called extractor, such that F=M o P, and the condensing P(x) of x has length at most s(kappa(x)), for any input x in Sigma^*. If s is a polynomial function, then the compactor is said to be of polynomial-size. Although the study on counting-analogue of kernelization is not unprecedented, it has received little attention so far. We study a family of vertex-certified counting problems on graphs that are MSOL-expressible; that is, for an MSOL-formula phi with one free set variable to be interpreted as a vertex subset, we want to count all A subseteq V(G) where |A|=k and (G,A) models phi. In this paper, we prove that every vertex-certified counting problems on graphs that is MSOL-expressible and treewidth modulable, when parameterized by k, admits a polynomial-size compactor on H-topological-minor-free graphs with condensing time O(k^2n^2) and decoding time 2^{O(k)}. This implies the existence of an FPT-algorithm of running time O(n^2 k^2)+2^{O(k)}. All aforementioned complexities are under the Uniform Cost Measure (UCM) model where numbers can be stored in constant space and arithmetic operations can be done in constant time. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Eun Jung Kim and Maria Serna and Dimitrios M. Thilikos 2018 Is Part Of LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ISAAC.2018.20 urn:nbn:de:0030-drops-99688 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.20 eng https://creativecommons.org/licenses/by/3.0/legalcode