On $b$-Matching and Fully-Dynamic Maximum $k$-Edge Coloring
Antoine El-Hayek, Kathrin Hanauer, Monika Henzinger
TL;DR
This work advances the study of fully dynamic maximum $k$-edge coloring (MkEC) by uncovering a close relationship with dynamic $oldsymbol{b}$-matching and delivering three adaptive strategies that achieve near-polylog update times. Central contributions include a new integrality-gap result for the $oldsymbol{b}$-matching polytope, $ rac{3eta}{3eta-1}$ with $eta=eta_{ ext{min}}$, and a sparsification-based rounding framework that enables efficient dynamic MkEC against adaptive adversaries. Specifically, MatchO attains a $(2+ ightε) rac{k+1}{k}$-approximation against oblivious adversaries in polylog update time, while MatchA applies sparsification and rounding to achieve a $(7+ ightε) rac{3k+3}{3k-1}$-approximation (and $(7+ε)$-approximation in bipartite graphs) against adaptive adversaries; a simple greedy method yields a $2.16$-approximation with $O(oldsymbol{Δ}+k)$ updates. The results are modular: improvements in dynamic $oldsymbol{b}$-matching automatically translate into better MkEC performance, and the bipartite cases provide stronger constants, underscoring practical relevance for dynamic network configurations.
Abstract
Given a graph $G$ that is modified by a sequence of edge insertions and deletions, we study the Maximum $k$-Edge Coloring problem Having access to $k$ colors, how can we color as many edges of $G$ as possible such that no two adjacent edges share the same color? While this problem is different from simply maintaining a $b$-matching with $b=k$, the two problems are closely related: a maximum $k$-matching always contains a $\frac{k+1}k$-approximate maximum $k$-edge coloring. However, maximum $b$-matching can be solved efficiently in the static setting, whereas the Maximum $k$-Edge Coloring problem is NP-hard and even APX-hard for $k \ge 2$. We present new results on both problems: For $b$-matching, we show a new integrality gap result and for the case where $b$ is a constant, we adapt Wajc's matching sparsification scheme~[STOC20]. Using these as basis, we give three new algorithms for the dynamic Maximum $k$-Edge Coloring problem: Our MatchO algorithm builds on the dynamic $(2+ε)$-approximation algorithm of Bhattacharya, Gupta, and Mohan~[ESA17] for $b$-matching and achieves a $(2+ε)\frac{k+1} k$-approximation in $O(poly(\log n, ε^{-1}))$ update time against an oblivious adversary. Our MatchA algorithm builds on the dynamic $8$-approximation algorithm by Bhattacharya, Henzinger, and Italiano~[SODA15] for fractional $b$-matching and achieves a $(8+ε)\frac{3k+3}{3k-1}$-approximation in $O(poly(\log n, ε^{-1}))$ update time against an adaptive adversary. Moreover, our reductions use the dynamic $b$-matching algorithm as a black box, so any future improvement in the approximation ratio for dynamic $b$-matching will automatically translate into a better approximation ratio for our algorithms. Finally, we present a greedy algorithm that runs in $O(Δ+k)$ update time, while guaranteeing a $2.16$~approximation factor.