Affiliation:
1. Carnegie Mellon University, Pittsburg, Kansas
2. Utrecht University, Utrecht, The Netherlands
Abstract
In the Feedback Vertex Set (FVS) problem, one is given an undirected graph
G
and an integer
k
, and one needs to determine whether there exists a set of
k
vertices that intersects all cycles of
G
(a so-called feedback vertex set). Feedback Vertex Set is one of the most central problems in parameterized complexity: It served as an excellent testbed for many important algorithmic techniques in the field such as Iterative Compression [Guo et al. (JCSS’06)], Randomized Branching [Becker et al. (J. Artif. Intell. Res’00)] and Cut&Count [Cygan et al. (FOCS’11)]. In particular, there has been a long race for the smallest dependence
f(k)
in run times of the type
O
⋆
(f(k))
, where the
O
⋆
notation omits factors polynomial in
n
. This race seemed to have reached a conclusion in 2011, when a randomized
O
⋆
(3
k
) time algorithm based on Cut&Count was introduced.
In this work, we show the contrary and give a
O
⋆
(2.7
k
)
time randomized algorithm. Our algorithm combines all mentioned techniques with substantial new ideas: First, we show that, given a feedback vertex set of size
k
of bounded average degree, a tree decomposition of width
(1-Ω (1))k
can be found in polynomial time. Second, we give a randomized branching strategy inspired by the one from [Becker et al. (J. Artif. Intell. Res’00)] to reduce to the aforementioned bounded average degree setting. Third, we obtain significant run time improvements by employing fast matrix multiplication.
Funder
Netherlands Organization for Scientific Research
European Research Council
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
2 articles.
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