Local search for valued constraint satisfaction parameterized by treedepth

TL;DR

This paper investigates the structure of local search in valued constraint satisfaction problems (VCSPs), showing how the treedepth of the constraint graph influences ascent lengths and the difficulty of reaching local peaks.

Contribution

It establishes bounds on ascent lengths based on treedepth and highlights the complexity barriers in sparse VCSPs for local search algorithms.

Findings

Short ascents exist in VCSPs with logarithmic treedepth.

Superpolynomial ascents can occur in VCSPs with loglog treedepth.

All ascents are superpolynomial from certain initial assignments in polylog treedepth VCSPs.

Abstract

Sometimes local search algorithms cannot efficiently find even local peaks. To understand why, I look at the structure of ascents in fitness landscapes from valued constraint satisfaction problems (VCSPs). Given a VCSP with a constraint graph of treedepth $d$, I prove that from any initial assignment there always exists an ascent of length $2^{d + 1} \cdot n$ to a local peak. This means that short ascents always exist in fitness landscapes from constraint graphs of logarithmic treedepth, and thus also for all VCSPs of bounded treewidth. But this does not mean that local search algorithms will always find and follow such short ascents in sparse VCSPs. I show that with loglog treedepth, superpolynomial ascents exist; and for polylog treedepth, there are initial assignments from which all ascents are superpolynomial. Together, these results suggest that the study of sparse VCSPs can help us better understand the barriers to efficient local search.