All ascents exponential from valued constraint graphs of pathwidth three

TL;DR

This paper investigates the limits of strict local search on pseudo-Boolean optimization problems by modeling fitness landscapes as valued constraint graphs and using pathwidth as a key sparsity parameter. It first reviews prior Max-Cut constructions, showing that their high pathwidth (at least four) aligns with known exponential-ascents results. The authors then introduce a novel pathwidth-three construction, the controlled doubling gadget, and prove that all ascents from a designated start are exponential, thereby forcing any strict local search to take exponentially many steps on relatively sparse graphs. This work sharpens our understanding of how structural graph parameters constrain local-search dynamics and identifies open questions about the potential existence of pathwidth-two instances with all-ascents-long behavior, highlighting a boundary between local and non-local optimization techniques.

Abstract

Many combinatorial optimization problems can be formulated as finding as assignment that maximized 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. From this, 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.

Figures (4)

• Figure 1: Gadget $\mathcal{C}^\text{MT}_k$ of Monien and Tscheuschner construction MT2010 ($\mathcal{C}^\text{MT}$) with constraint weights omitted. Dotted edges and vertices illustrate connections to neighboring gadgets. Node labels refer to grid position in layout of Figure 5 of MT2010: letters for rows, numbers for columns.
• Figure 2: Gadget $\mathcal{C}^\text{MS}_k$ of Michel and Scott construction MS2024: the weights of unary constraints are next to their variables and the weights of binary constraints are on the edges that specify their scope. The dotted edges and vertices illustrate the connection to the neighboring gadgets.
• Figure 3: A gadget $\mathcal{C}^{CD}_k$ of the controlled doubling construction with $m_k=2^{k-1}(8n+16)-16$ and $s_k = 8(n-k)$. The constraints of the $k$th of $n$ gadgets are shown: the weights of unary constraints are next to their variables, the weights of binary constraints are on the edges that specify their scope, the weights of ternary constraints are in the center of the shaded area that specify their scope. The dotted edges and vertices illustrate the connection to the neighboring gadgets.
• Figure 4: The first gadget ($\mathcal{C}^{CD}_1$) of the controlled doubling construction with $m_1=s_0 = 8n$. The constraints of the first gadget are shown: the weights of unary constraints are next to their variables, the weights of binary constraints are on the edges that specify their scope, the weights of ternary constraints are in the center of the shaded area that specify their scope. The dotted edge and vertices illustrate the connection to the second gadget.

Theorems & Definitions (9)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Theorem 4
• proof
• Conjecture 5