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# Subexponential Parameterized Algorithms for Graphs of Polynomial Growth

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Dániel Marx and Marcin Pilipczuk. Subexponential Parameterized Algorithms for Graphs of Polynomial Growth. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ESA.2017.59

## Abstract

We show that for a number of parameterized problems for which only 2^{O(k)} n^{O(1)} time algorithms are known on general graphs, subexponential parameterized algorithms with running time 2^{O(k^{1-1/(1+d)} log^2 k)} n^{O(1)} are possible for graphs of polynomial growth with growth rate (degree) d, that is, if we assume that every ball of radius r contains only O(r^d) vertices. The algorithms use the technique of low-treewidth pattern covering, introduced by Fomin et al. [FOCS 2016] for planar graphs; here we show how this strategy can be made to work for graphs of polynomial growth. Formally, we prove that, given a graph G of polynomial growth with growth rate d and an integer k, one can in randomized polynomial time find a subset A of V(G) such that on one hand the treewidth of G[A] is O(k^{1-1/(1+d)} log k), and on the other hand for every set X of vertices of size at most k, the probability that X is a subset of A is 2^{-O(k^{1-1/(1+d)} log^2 k)}. Together with standard dynamic programming techniques on graphs of bounded treewidth, this statement gives subexponential parameterized algorithms for a number of subgraph search problems, such as Long Path or Steiner Tree, in graphs of polynomial growth. We complement the algorithm with an almost tight lower bound for Long Path: unless the Exponential Time Hypothesis fails, no parameterized algorithm with running time 2^{k^{1-1/d-epsilon}}n^{O(1)} is possible for any positive epsilon and any integer d >= 3.
##### Keywords
• polynomial growth
• subexponential algorithm
• low treewidth pattern covering

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