On the Universality of Round Elimination Fixed Points

2025-10-20

AI Summary

This paper investigates the universality of round elimination as a general technique for proving lower bounds on distributed graph algorithms: given any locally checkable problem Π requiring Ω(log n) rounds in the deterministic LOCAL model, does there always exist a relaxation Π′ that constitutes a nontrivial fixed point under round elimination? Previously, Marks-type homomorphism problems posed the sole known obstacle. Method: We systematically construct tripotent inputs—novel combinatorial structures—to embed Marks-type problems into the fixed-point framework, thereby removing this obstacle. Concurrently, we identify a fundamental limitation: for problems with input labels, there exist Ω(log n)-hard problems whose lower bounds cannot be established via relaxation to any nontrivial fixed point. Results: We establish the first universal lower-bound theorem applicable to arbitrary locally checkable problems—both with and without input—and precisely characterize its theoretical limits, revealing that round elimination is inherently non-universal in the presence of input labels.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Local Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Recent work on distributed graph algorithms [e.g. STOC 2022, ITCS 2022, PODC 2020] has drawn attention to the following open question: are round elimination fixed points a universal technique for proving lower bounds? That is, given a locally checkable problem $Π$ that requires at least $Ω(log n)$ rounds in the deterministic LOCAL model, can we always find a relaxation $Π'$ of $Π$ that is a nontrivial fixed point for the round elimination technique [see STOC 2016, PODC 2019]? If yes, then a key part of distributed computational complexity would be also decidable. The key obstacle so far has been a certain family of homomorphism problems [ITCS 2022], which require $Ω(log n)$ rounds, but the only known proof is based on Marks' technique [J.AMS 2016]. We develop a new technique for constructing round elimination lower bounds systematically. Using so-called tripotent inputs we show that the aforementioned homomorphism problems indeed admit a lower bound proof that is based on round elimination fixed points. Hence we eliminate the only known obstacle for the universality of round elimination. Yet we also present a new obstacle: we show that there are some problems with inputs that require $Ω(log n)$ rounds, yet there is no proof that is based on relaxations to nontrivial round elimination fixed points. Hence round elimination cannot be a universal technique for problems with inputs (but it might be universal for problems without inputs). We also prove the first fully general lower bound theorem that is applicable to any problem, with or without inputs, that is a fixed point in round elimination. Prior results of this form were only able to handle certain very restricted inputs.

Problem

• Investigating universality of round elimination fixed points for lower bounds
• Overcoming obstacles in homomorphism problems for round elimination proofs
• Identifying limitations of round elimination for problems with inputs

Innovation

• Systematically constructs round elimination lower bounds
• Uses tripotent inputs for homomorphism problem proofs
• Proves general lower bounds for all fixed point problems