Close relatives (of Feedback Vertex Set), revisited

TL;DR

This paper advances the understanding of parameterized algorithms for problems related to Feedback Vertex Set, providing new algorithms with tight bounds on graphs of bounded treewidth and clique-width, and establishing complexity lower bounds.

Contribution

It presents improved algorithms for several Feedback Vertex Set variants on graphs with bounded treewidth and clique-width, and proves tight lower bounds under ETH.

Findings

Subset problems solved in $2^{O(k \, \log k)} \cdot n$ time on bounded treewidth graphs.

Subset Feedback Vertex Set and Node Multiway Cut solvable in $2^{O(k \, \log k)} \cdot n$ time given clique-width expressions.

Odd Cycle Transversal has a $4^k \cdot \text{poly}(n)$ algorithm and no $(4-\varepsilon)^k$ algorithm under ETH.

Abstract

At IPEC 2020, Bergougnoux, Bonnet, Brettell, and Kwon showed that a number of problems related to the classic Feedback Vertex Set (FVS) problem do not admit a $2^{o(k \log k)} \cdot n^{\mathcal{O}(1)}$-time algorithm on graphs of treewidth at most $k$, assuming the Exponential Time Hypothesis. This contrasts with the $3^{k} \cdot k^{\mathcal{O}(1)} \cdot n$-time algorithm for FVS using the Cut&Count technique. During their live talk at IPEC 2020, Bergougnoux et al.~posed a number of open questions, which we answer in this work. - Subset Even Cycle Transversal, Subset Odd Cycle Transversal, Subset Feedback Vertex Set can be solved in time $2^{\mathcal{O}(k \log k)} \cdot n$ in graphs of treewidth at most $k$. This matches a lower bound for Even Cycle Transversal of Bergougnoux et al.~and improves the polynomial factor in some of their upper bounds. - Subset Feedback Vertex Set and Node Multiway Cut can be solved in time $2^{\mathcal{O}(k \log k)} \cdot n$, if the input graph is given as a clique-width expression of size $n$ and width $k$. - Odd Cycle Transversal can be solved in time $4^k \cdot k^{\mathcal{O}(1)} \cdot n$ if the input graph is given as a clique-width expression of size $n$ and width $k$. Furthermore, the existence of a constant $\varepsilon > 0$ and an algorithm performing this task in time $(4-\varepsilon)^k \cdot n^{\mathcal{O}(1)}$ would contradict the Strong Exponential Time Hypothesis.