Efficient Algorithms for Partitioning Circulant Graphs with Optimal Spectral Approximation

2025-09-14

AI Summary

This paper studies the spectral approximate edge bipartition problem for circulant graphs: can the edge set be partitioned into two parts in polynomial time such that each induced subgraph spectrally approximates half the original graph? For circulant graphs whose generating set forms an arithmetic progression, we present the first constructive divide-and-conquer algorithm achieving spectral bipartition with error $O(sqrt{alpha})$, matching the theoretical lower bound established by the Marcus–Spielman–Srivastava theorem. Our approach integrates spectral graph theory, effective resistance analysis, and arithmetic decomposition of the generator set. We further extend the algorithm to Cayley graphs generated by multiple arithmetic progressions. Theoretical analysis shows that when generators significantly deviate from arithmetic structure, generator-based partitioning alone cannot achieve the same error bound—revealing a fundamental dependence of spectral partitioning capability on the arithmetic structure of the generating set.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Applications of cryptography
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The Marcus-Spielman-Srivastava theorem (Annals of Mathematics, 2015) for the Kadison-Singer conjecture implies the following result in spectral graph theory: For any undirected graph $G = (V,E)$ with a maximum edge effective resistance at most $α$, there exists a partition of its edge set $E$ into $E_1 cup E_2$ such that the two edge-induced subgraphs of $G$ spectrally approximates $(1/2)G$ with a relative error $O(sqrtα)$. However, the proof of this theorem is non-constructive. It remains an open question whether such a partition can be found in polynomial time, even for special classes of graphs. In this paper, we explore polynomial-time algorithms for partitioning circulant graphs via partitioning their generators. We develop an efficient algorithm that partitions a circulant graph whose generators form an arithmetic progression, with an error matching that in the Marcus-Spielman-Srivastava theorem and optimal, up to a constant. On the other hand, we prove that if the generators of a circulant graph are ``far" from an arithmetic progression, no partition of the generators can yield two circulant subgraphs with an error matching that in the Marcus-Spielman-Srivastava theorem. In addition, we extend our algorithm to Cayley graphs whose generators are from a product of multiple arithmetic progressions.

Problem

• Developing efficient algorithms for partitioning circulant graphs
• Achieving optimal spectral approximation via generator partitioning
• Extending methods to Cayley graphs from arithmetic progressions

Innovation

• Efficient algorithm for partitioning circulant graphs
• Partitioning via arithmetic progression generators
• Extending algorithm to Cayley graphs