Close relatives of Feedback Vertex Set without single-exponential algorithms parameterized by treewidth

TL;DR

This paper establishes tight lower bounds for several graph problems parameterized by treewidth, showing that they likely cannot be solved in single-exponential time, unlike Feedback Vertex Set.

Contribution

It proves that certain subset problems do not admit single-exponential algorithms in treewidth, matching upper bounds with tight lower bounds under ETH.

Findings

Subset Feedback Vertex Set and related problems have no $2^{o(tw \, log \, tw)}n^{O(1)}$ algorithms under ETH.

Algorithms exist with $2^{O(tw \, log \, tw)}n^{O(1)}$ complexity, matching the lower bounds.

Lower bounds also hold for pathwidth and the weighted case, with a contrast for Even Cycle Transversal.

Abstract

The Cut & Count technique and the rank-based approach have lead to single-exponential FPT algorithms parameterized by treewidth, that is, running in time $2^{O(tw)}n^{O(1)}$, for Feedback Vertex Set and connected versions of the classical graph problems (such as Vertex Cover and Dominating Set). We show that Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Restricted Edge-Subset Feedback Edge Set, Node Multiway Cut, and Multiway Cut are unlikely to have such running times. More precisely, we match algorithms running in time $2^{O(tw \log tw)}n^{O(1)}$ with tight lower bounds under the Exponential-Time Hypothesis (ETH), ruling out $2^{o(tw \log tw)}n^{O(1)}$, where $n$ is the number of vertices and $tw$ is the treewidth of the input graph. Our algorithms extend to the weighted case, while our lower bounds also hold for the larger parameter pathwidth and do not require weights. We also show that, in contrast to Odd Cycle Transversal, there is no $2^{o(tw \log tw)}n^{O(1)}$-time algorithm for Even Cycle Transversal under the ETH.