The Query Complexity of Local Search in Rounds on General Graphs

2026-01-19

AI Summary

This study investigates the query complexity of locating a local minimum—defined as a vertex \( v \) such that \( f(v) \leq f(u) \) for all neighbors \( u \)—on general graphs using at most \( t \) rounds of queries. By leveraging the graph’s separation number \( s \) and maximum degree \( \Delta \), the authors present both deterministic and randomized algorithms and establish the first matching upper and lower bounds for this problem: a deterministic upper bound of \( O(t n^{1/t} (s\Delta)^{1-1/t}) \) and a randomized lower bound of \( \Omega(t n^{1/t} - t) \). Furthermore, they introduce a warm-start-based parallel steepest descent strategy that significantly improves efficiency on graphs with high separation numbers. This work elucidates the critical role of structural graph parameters in local search complexity and extends theoretical understanding beyond grid graphs to general graphs.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We analyze the query complexity of finding a local minimum in $t$ rounds on general graphs. More precisely, given a graph $G = (V,E)$ and oracle access to an unknown function $f : V \to \mathbb{R}$, the goal is to find a local minimum--a vertex $v$ such that $f(v) \leq f(u)$ for all $(u,v) \in E$--using at most $t$ rounds of interaction with the oracle. The query complexity is well understood on grids, but much less is known beyond. This abstract problem captures many optimization tasks, such as finding a local minimum of a loss function during neural network training. For each graph with $n$ vertices, we prove a deterministic upper bound of $O(t n^{1/t} (s\Delta)^{1-1/t})$, where $s$ is the separation number and $\Delta$ is the maximum degree of the graph. We complement this result with a randomized lower bound of $\Omega(t n^{1/t}-t)$ that holds for any connected graph. We also find that parallel steepest descent with a warm start provides improved bounds for graphs with high separation number and bounded degree. To obtain our results, we utilized an advanced version of Gemini at various stages of our research. We discuss our experience in a methodology section.

Problem

• local search
• query complexity
• rounds
• general graphs
• local minimum

Innovation

• query complexity
• local search
• rounds
• separation number
• parallel steepest descent