Exponential Steepest Ascent from Valued Constraint Graphs of Pathwidth Four

TL;DR

This paper investigates the complexity of local search methods in valued constraint satisfaction problems, demonstrating how to simulate ordered ascents with steepest ascents and constructing VCSPs with exponential ascent lengths, improving previous bounds.

Contribution

It introduces a padding technique to simulate ordered ascents by steepest ascents and constructs VCSPs with shorter arity and pathwidth that exhibit exponential ascent lengths.

Findings

Constructed VCSP with exponential steepest ascents

Improved bounds on arity and pathwidth for long ascents

Demonstrated simulation of ordered ascents by steepest ascents

Abstract

We examine the complexity of maximising fitness via local search on valued constraint satisfaction problems (VCSPs). We consider two kinds of local ascents: (1) steepest ascents, where each step changes the domain that produces a maximal increase in fitness; and (2) $\prec$-ordered ascents, where -- of the domains with available fitness increasing changes -- each step changes the $\prec$-minimal domain. We provide a general padding argument to simulate any ordered ascent by a steepest ascent. We construct a VCSP that is a path of binary constraints between alternating 2-state and 3-state domains with exponentially long ordered ascents. We apply our padding argument to this VCSP to obtain a Boolean VCSP that has a constraint (hyper)graph of arity 5 and pathwidth 4 with exponential steepest ascents. This is an improvement on the previous best known construction for long steepest ascents, which had arity 8 and pathwidth 7.