🤖 AI Summary
This paper studies the maximum $k$-edge coloring problem on dynamic graphs: maintaining a proper edge coloring using at most $k$ colors to maximize the number of colored edges under edge insertions and deletions. It establishes, for the first time, a black-box reduction framework between dynamic $b$-matching and dynamic $k$-edge coloring, revealing their approximate equivalence. The authors propose three novel algorithms: (1) an efficient dynamic algorithm based on fractional $b$-matching and sparsification, achieving $(2+varepsilon)(k+1)/k$ and $(8+varepsilon)(3k+3)/(3k-1)$ approximation ratios with $O(mathrm{poly}(log n, varepsilon^{-1}))$ update time; (2) a greedy algorithm with $O(Delta + k)$ per-update time, attaining a $2.16$-approximation; (3) the first dynamic $k$-edge coloring algorithm robust against adaptive adversaries. Additionally, the paper derives a new upper bound on the integrality gap of $b$-matching.
📝 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 $(2+epsilon)$-approximation algorithm of Bhattacharya, Gupta, and Mohan~[ESA17] for $b$-matching and achieves a $(2+epsilon)frac{k+1} k$-approximation in $O(poly(log n, epsilon^{-1}))$ 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 $(8+epsilon)frac{3k+3}{3k-1}$-approximation in $O(poly(log n, epsilon^{-1}))$ 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 $O(Delta+k)$ update time, while guaranteeing a $2.16$~approximation factor.
