Detecting Feedback Vertex Sets of Size $k$ in $O^{\star }(2.7^k)$ Time

Abstract

In the Feedback Vertex Set 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 test bed 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 $O^{\star }(f(k))$, where the $O^{\star }$ notation omits factors polynomial in $n$. This race seemed to be run in 2011, when a randomized algorithm $O^{\star }(3^k)$ time algorithm based on Cut\&Count was introduced. In this work, we show the contrary and give a $O^{\star }(2.7^k)$ 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 $(1-\Omega (1))k$ 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.

SODA 2020, 22 pages

References in corpus (2)

• On Problems as Hard as CNFSAT
• Randomized Algorithms for the Loop Cutset Problem