A Bound on the Pathwidth of Sparse Graphs with Applications to Exact Algorithms

Abstract

We present a bound of $m/5.769+O(\log n)$ on the pathwidth of graphs with $m$ edges. Respective path decompositions can be computed in polynomial time. Using a well-known framework for algorithms that rely on tree decompositions, this directly leads to runtime bounds of $O^*(2^{m/5.769})$ for Max-2SAT and Max-Cut. Both algorithms require exponential space due to dynamic programming. If we agree to accept a slightly larger bound of $m/5.217+3$, we even obtain path decompositions with a rather simple structure: all bags share a large set of common nodes. Using branching based algorithms, this allows us to solve the same problems in polynomial space and time $O^*(2^{m/5.217})$.