All ascents exponential from valued constraint graphs of pathwidth three

TL;DR

This paper demonstrates that strict local search algorithms can require exponential time even on simple valued constraint graphs with pathwidth three, highlighting limitations in their efficiency.

Contribution

It introduces a new construction of VCSPs with pathwidth three where all ascents are exponentially long, improving previous bounds.

Findings

All ascents are exponential in length on the constructed VCSPs.

The construction applies to valued constraint graphs of pathwidth three.

Strict local search algorithms can be forced to take exponential steps even on simple graphs.

Abstract

Many combinatorial optimization problems can be formulated as finding an assignment that maximizes some pseudo-Boolean function (that we call the fitness function). Strict local search starts with some assignment and follows some update rule to proceed to an adjacent assignment of strictly higher fitness. This means that strict local search algorithms follow ascents in the fitness landscape of the pseudo-Boolean function. The complexity of the pseudo-Boolean function (and the fitness landscapes that it represents) can be parameterized by properties of the valued constraint satisfaction problem (VCSP) that encodes the pseudo-Boolean function. We focus on properties of the constraint graphs of the VCSP, with the intuition that spare graphs are less complex than dense ones. Specifically, we argue that pathwidth is the natural sparsity parameter for understanding limits on the power of strict local search. We show that prior constructions of sparse VCSPs where all ascents are exponentially long had pathwidth greater than or equal to four. We improve this this with our controlled doubling construction: a valued constraint satisfaction problem of pathwidth three where all ascents are exponentially long from a designated initial assignment. We conclude that all strict local search algorithms can be forced to take an exponential number of steps even on simple valued constraint graphs of pathwidth three.