1 research outputs found
Detecting and Counting Small Patterns in Planar Graphs in Subexponential Parameterized Time
We present an algorithm that takes as input an -vertex planar graph
and a -vertex pattern graph , and computes the number of (induced) copies
of in in time. If is a matching,
independent set, or connected bounded maximum degree graph, the runtime reduces
to .
While our algorithm counts all copies of , it also improves the fastest
algorithms that only detect copies of . Before our work, no time algorithms for detecting unrestricted patterns were
known, and by a result of Bodlaender et al. [ICALP 2016] a 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 in time
probabilistically. For counting problems, it was a repeatedly asked open
question whether 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 in time proportional
to its number of non-isomorphic separations of order . 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].Comment: 25 pages, 1 figure, under submissio