Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal

TL;DR

This paper establishes tight lower bounds on the running times of algorithms for various problems on graphs with bounded treewidth, assuming the Strong Exponential Time Hypothesis, indicating these algorithms are likely optimal.

Contribution

It proves that many known algorithms for problems on bounded treewidth graphs are essentially optimal under the Strong Exponential Time Hypothesis.

Findings

Lower bounds match the best known algorithms' running times.

Assumes SAT cannot be solved faster than (2-)^{n}m^{O(1)}.

Shows no significantly faster algorithms exist for these problems under the hypothesis.

Abstract

We obtain a number of lower bounds on the running time of algorithms solving problems on graphs of bounded treewidth. We prove the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi. In particular, assuming that SAT cannot be solved in (2-\epsilon)^{n}m^{O(1)} time, we show that for any e > 0; {\sc Independent Set} cannot be solved in (2-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Dominating Set} cannot be solved in (3-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Max Cut} cannot be solved in (2-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Odd Cycle Transversal} cannot be solved in (3-e)^{tw(G)}|V(G)|^{O(1)} time, For any $q \geq 3$, $q$-{\sc Coloring} cannot be solved in (q-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Partition Into Triangles} cannot be solved in (2-e)^{tw(G)}|V(G)|^{O(1)} time. Our lower bounds match the running times for the best known algorithms for the problems, up to the e in the base.