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Local search for valued constraint satisfaction parameterized by treedepth

Artem Kaznatcheev

TL;DR

This paper tackles why local search can fail on sparse VCSP fitness landscapes by tying ascent behavior to the treedepth of the constraint graph. It develops a treedepth-based framework for ordered ascents, introduces leaf-smoothing to reduce problem size while maintaining structure, and proves explicit ascent-length bounds: any \( \prec_T \)-ordered ascent has length at most \( v^{\mathrm{height}(T)+1}\cdot n \), while step-steepest ascents are bounded by \( 2^{\mathrm{height}(T)+1}\cdot n \); additionally, from any initial assignment there exists an ascent of length \( 2^{d+1}\cdot n \) to a local peak, where \( d \) is the treedepth. The results imply polynomial-length ascents for graphs with logarithmic treedepth, clarifying when local search barriers arise in sparse VCSPs and linking structural sparsity to search efficiency. These insights help explain the gap between global-optimum solvability for bounded-treewidth VCSPs and local-search-based progress in sparse instances.

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.

Local search for valued constraint satisfaction parameterized by treedepth

TL;DR

This paper tackles why local search can fail on sparse VCSP fitness landscapes by tying ascent behavior to the treedepth of the constraint graph. It develops a treedepth-based framework for ordered ascents, introduces leaf-smoothing to reduce problem size while maintaining structure, and proves explicit ascent-length bounds: any -ordered ascent has length at most \( v^{\mathrm{height}(T)+1}\cdot n \), while step-steepest ascents are bounded by \( 2^{\mathrm{height}(T)+1}\cdot n \); additionally, from any initial assignment there exists an ascent of length to a local peak, where is the treedepth. The results imply polynomial-length ascents for graphs with logarithmic treedepth, clarifying when local search barriers arise in sparse VCSPs and linking structural sparsity to search efficiency. These insights help explain the gap between global-optimum solvability for bounded-treewidth VCSPs and local-search-based progress in sparse instances.

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 , I prove that from any initial assignment there always exists an ascent of length 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.
Paper Structure (3 sections, 3 theorems, 1 equation)

This paper contains 3 sections, 3 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Background
  3. Polynomial bound on ordered-ascent

Key Result

Theorem 1

In a VCSP $\mathcal{C}$ on $n$ variables with domains of size $\leq v$ and constraint graph with treedepth-decomposition $T$, any $\prec_T$-ordered-ascent has length $v^{\mathrm{height}(T) + 1}\cdot n$ or less. If, in addition to the above, $p$ is step-steepest then it has length $2^{\mathrm{height}

Theorems & Definitions (3)

  • Theorem 1
  • Lemma 2
  • Proposition 3