Solving connectivity problems parameterized by treewidth in single exponential time

TL;DR

This paper introduces a new technique called Cut&Count that significantly improves algorithms for connectivity problems parameterized by treewidth, achieving single exponential time solutions for many problems previously limited to slower methods.

Contribution

The paper presents the Cut&Count technique, enabling single exponential time algorithms for a broad class of connectivity problems parameterized by treewidth, answering longstanding open questions.

Findings

Cut&Count yields c^tw V^O(1) algorithms for connectivity problems.

It solves problems like Hamiltonian Path and Feedback Vertex Set efficiently.

Some problems remain hard, confirming limits of the technique.

Abstract

For the vast majority of local graph problems standard dynamic programming techniques give c^tw V^O(1) algorithms, where tw is the treewidth of the input graph. On the other hand, for problems with a global requirement (usually connectivity) the best-known algorithms were naive dynamic programming schemes running in tw^O(tw) V^O(1) time. We breach this gap by introducing a technique we dubbed Cut&Count that allows to produce c^tw V^O(1) Monte Carlo algorithms for most connectivity-type problems, including Hamiltonian Path, Feedback Vertex Set and Connected Dominating Set, consequently answering the question raised by Lokshtanov, Marx and Saurabh [SODA'11] in a surprising way. We also show that (under reasonable complexity assumptions) the gap cannot be breached for some problems for which Cut&Count does not work, like CYCLE PACKING. The constant c we obtain is in all cases small (at most 4 for undirected problems and at most 6 for directed ones), and in several cases we are able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail. Our results have numerous consequences in various fields, like FPT algorithms, exact and approximate algorithms on planar and H-minor-free graphs and algorithms on graphs of bounded degree. In all these fields we are able to improve the best-known results for some problems.