Data-Compression for Parametrized Counting Problems on Sparse Graphs

Eun Jung Kim, Maria Serna, Dimitrios M. Thilikos, Michael Wagner
2018 International Symposium on Algorithms and Computation  
We study the concept of compactor, which may be seen as a counting-analogue of kernelization in counting parameterized complexity. For a function F : Σ * → N and a parameterization κ : Σ * → 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•P, and the condensing P(x) of x has length at most s(κ(x)), for any input x ∈ Σ * . If s is a polynomial function, then the compactor is said to be of
more » ... lynomial-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 φ with one free set variable to be interpreted as a vertex subset, we want to count all A ⊆ V (G) where |A| = k and (G, A) |= φ. 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 2 n 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.
doi:10.4230/lipics.isaac.2018.20 dblp:conf/isaac/0002ST18 fatcat:xytw4ioxsbelxko3iv4tmutely