Graph Coloring Below Guarantees via Co-Triangle Packing

2025-09-15

AI Summary

This paper studies “below-guarantee” parameterized variants of Graph Coloring: given an $n$-vertex graph, decide whether it admits a proper $(n-k)$-coloring. It further introduces the refined parameter $(omega + overline{mu} - k)$-Coloring, where $omega$ is the clique number and $overline{mu}$ the maximum matching size in the complement graph. The authors propose a novel co-triangle packing framework, integrating greedy construction with win-win analysis to reduce the problem to subinstances of bounded motif size. Technically, the approach combines randomized algorithms, parameterized complexity theory, and structural analysis of the complement graph. Key contributions are: (1) a randomized $O^*(2^{3k/2})$ algorithm for $(n-k)$-Coloring, improving upon the prior $O^*(4^k)$; (2) an $O^*(2^{6k})$ FPT algorithm for $(omega + overline{mu} - k)$-Coloring, establishing its fixed-parameter tractability; and (3) a proof that neither $(omega - k)$- nor $(overline{mu} - k)$-Coloring is FPT, refuting two natural parameterizations.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Tracking, profiling, and countermeasures against them

Abstract

In the $ell$-Coloring Problem, we are given a graph on $n$ nodes, and tasked with determining if its vertices can be properly colored using $ell$ colors. In this paper we study below-guarantee graph coloring, which tests whether an $n$-vertex graph can be properly colored using $g-k$ colors, where $g$ is a trivial upper bound such as $n$. We introduce an algorithmic framework that builds on a packing of co-triangles $overline{K_3}$ (independent sets of three vertices): the algorithm greedily finds co-triangles and employs a win-win analysis. If many are found, we immediately return YES; otherwise these co-triangles form a small co-triangle modulator, whose deletion makes the graph co-triangle-free. Extending the work of [Gutin et al., SIDMA 2021], who solved $ell$-Coloring (for any $ell$) in randomized $O^*(2^{k})$ time when given a $overline{K_2}$-free modulator of size $k$, we show that this problem can likewise be solved in randomized $O^*(2^{k})$ time when given a $overline{K_3}$-free modulator of size~$k$. This result in turn yields a randomized $O^{*}(2^{3k/2})$ algorithm for $(n-k)$-Coloring (also known as Dual Coloring), improving the previous $O^{*}(4^{k})$ bound. We then introduce a smaller parameterization, $(ω+overlineμ-k)$-Coloring, where $ω$ is the clique number and $overlineμ$ is the size of a maximum matching in the complement graph; since $ω+overlineμle n$ for any graph, this problem is strictly harder. Using the same co-triangle-packing argument, we obtain a randomized $O^{*}(2^{6k})$ algorithm, establishing its fixed-parameter tractability for a smaller parameter. Complementing this finding, we show that no fixed-parameter tractable algorithm exists for $(ω-k)$-Coloring or $(overlineμ-k)$-Coloring under standard complexity assumptions.

Problem

• Solving graph coloring below trivial guarantees using co-triangle packing
• Developing FPT algorithms for (n-k)-Coloring via co-triangle-free modulators
• Establishing FPT for (ω+μ̄-k)-Coloring while proving hardness for components

Innovation

• Co-triangle packing for win-win analysis
• Randomized O*(2^k) algorithm with modulator
• Fixed-parameter tractability for smaller parameters