Steepest ascent can be exponential in bounded treewidth problems

TL;DR

This paper demonstrates that steepest ascent local search can require exponential time to reach a local optimum even in problems with small variable domains and bounded treewidth, highlighting limitations of local search in certain structured problems.

Contribution

The authors show that bounded treewidth does not guarantee polynomial-time convergence of steepest ascent local search, improving previous results that required unbounded treewidth.

Findings

Exponential steepest ascent can occur with treewidth 7

Bounded treewidth does not ensure polynomial local search convergence

Constructed landscapes require exponential steps to reach local optima

Abstract

We investigate the complexity of local search based on steepest ascent. We show that even when all variables have domains of size two and the underlying constraint graph of variable interactions has bounded treewidth (in our construction, treewidth 7), there are fitness landscapes for which an exponential number of steps may be required to reach a local optimum. This is an improvement on prior recursive constructions of long steepest ascents, which we prove to need constraint graphs of unbounded treewidth.