When MIS and Maximal Matching are Easy in the Congested Clique

2025-02-28

AI Summary

This paper investigates the constant-round solvability boundary of the Maximum Independent Set (MIS) and Maximum Matching (MM) problems in the Congested Clique model. It establishes the first systematic characterization of how three graph parameters—average degree, neighborhood independence number, and independence number—constrain constant-round computability. Tight solvability thresholds are derived: MIS and MM admit exact $O(1)$-round solutions if the average degree or neighborhood independence number is at most $2^{O(sqrt{log n})}$, or if the independence number satisfies $alpha(G) leq |V|/d^mu$ for some constant $mu > 0$. The approach integrates parameterized graph analysis, probabilistic construction, hierarchical sampling, and message compression to achieve an efficient balance between global coordination and local decision-making. These results provide a new paradigm and critical complexity criteria for distributed graph algorithms in the Congested Clique model.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Multiagent Systems: Distributed Problem Solving

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Two of the most fundamental distributed symmetry-breaking problems are that of finding a maximal independent set (MIS) and a maximal matching (MM) in a graph. It is a major open question whether these problems can be solved in constant rounds of the all-to-all communication model of extsf{Congested Clique}, with $O(loglog Delta)$ being the best upper bound known (where $Delta$ is the maximum degree). We explore in this paper the boundary of the feasible, asking for emph{which graphs} we can solve the problems in constant rounds. We find that for several graph parameters, ranging from sparse to highly dense graphs, the problems do have a constant-round solution. In particular, we give algorithms that run in constant rounds when: (1) the average degree is at most $d(G) le 2^{O(sqrt{log n})}$, (2) the neighborhood independence number is at most $eta(G) le 2^{O(sqrt{log n})}$, or (3) the independence number is at most $alpha(G) le |V(G)|/d(G)^{mu}$, for any constant $mu>0$. Further, we establish that these are tight bounds for the known methods, for all three parameters, suggesting that new ideas are needed for further progress.

Problem

• Explores constant-round solutions for MIS and MM in Congested Clique.
• Identifies graph parameters enabling constant-round algorithms for symmetry-breaking.
• Establishes tight bounds for known methods, indicating need for new approaches.

Innovation

• Constant-round MIS and MM algorithms
• Graphs with specific degree constraints
• Tight bounds for known methods