Detecting and Counting Small Patterns in Planar Graphs in Subexponential Parameterized Time
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We present an algorithm that takes as input an n-vertex planar graph G and a k-vertex pattern graph P, and computes the number of (induced) copies of P in G in 2^O(k/ k)n^O(1) time. If P is a matching, independent set, or connected bounded maximum degree graph, the runtime reduces to 2^Õ(√(k))n^O(1). While our algorithm counts all copies of P, it also improves the fastest algorithms that only detect copies of P. Before our work, no 2^O(k/ k)n^O(1) time algorithms for detecting unrestricted patterns P were known, and by a result of Bodlaender et al. [ICALP 2016] a 2^o(k/ k)n^O(1) time algorithm would violate the Exponential Time Hypothesis (ETH). Furthermore, it was only known how to detect copies of a fixed connected bounded maximum degree pattern P in 2^Õ(√(k))n^O(1) time probabilistically. For counting problems, it was a repeatedly asked open question whether 2^o(k)n^O(1) time algorithms exist that count even special patterns such as independent sets, matchings and paths in planar graphs. The above results resolve this question in a strong sense by giving algorithms for counting versions of problems with running times equal to the ETH lower bounds for their decision versions. Generally speaking, our algorithm counts copies of P in time proportional to its number of non-isomorphic separations of order Õ(√(k)). The algorithm introduces a new recursive approach to construct families of balanced cycle separators in planar graphs that have limited overlap inspired by methods from Fomin et al. [FOCS 2016], a new `efficient' inclusion-exclusion based argument and uses methods from Bodlaender et al. [ICALP 2016].
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