Degeneracy-Guided List Compression for Greedy Graph Coloring

2026-07-18

AI Summary

This work addresses the problem of efficiently compressing color lists for greedy graph coloring under a known graph structure, aiming to achieve high success rates with minimal storage overhead. The authors propose a degeneracy-based list compression framework that leverages a color-agnostic vertex ordering and assigns independent, uniformly random color lists based on backward neighborhoods. A key innovation is the introduction of removal-profile-based analysis to precisely characterize local color budgets, enabling structure-aware asymmetric palette sparsification. The method achieves linear list volume on higher-order forests and core–periphery graph families, substantially outperforming conventional logarithmic-squared sampling bounds. Empirical evaluation demonstrates that P-SAPST Lite reduces average list size by 47.6% compared to APST across 40,320 trials while maintaining a greedy coloring success rate of 99.8%, and achieves a load ratio as low as 0.193 on power-law graphs such as Enron.

Abstract

We study degeneracy guided list compression for greedy graph coloring when graph structure is available before colors are sampled. Our exposure calibrated ordering framework assigns each vertex an independent uniform list according to its backward neighborhood in a color independent order. Its certified instantiation, Profiled Structure Aware Asymmetric Palette Sparsification, or P-SAPST, reverses a minimum degree removal sequence and obtains every backward exposure from the removal profile. For each fixed profile, we characterize the exact local budget required by independent uniform lists under history robust greedy recovery. The profile yields linear list volume on high degree forests and on a core fringe family where reciprocal rank allocation requires Theta(n log^2 n) sampled colors. Exact conflict expectation, concentration, and a dense exposure barrier complete the theoretical description. The evaluation contains 40,320 runs over SAPBench and two SNAP networks. At the theorem scale, P-SAPST reduces mean list size by 47.6 percent relative to calibrated APST while attaining 99.8 percent observed greedy success. P-SAPST Lite replaces peeling with a degree order and provides a lower latency order choice within the same framework. On stress graphs with 250,000 vertices and up to 1,251,868 edges, Lite obtains a payload ratio of 0.865, while calibrated APST obtains 7.886. On email Enron, the corresponding ratios are 0.193 and 5.814. Compression is strongest on hub dominated and power law graphs and disappears near the dense exposure barrier. The method complements edge oblivious streaming APST by addressing an offline regime in which structural plans can be reused.

Problem

• graph coloring
• list compression
• degeneracy
• greedy algorithm
• color sampling

Innovation

• Degeneracy-guided list compression
• Greedy graph coloring
• Exposure calibrated ordering
• Palette sparsification
• Structural profile