Online Edge Coloring Algorithms via the Nibble Method
Sayan Bhattacharya, Fabrizio Grandoni, David Wajc

TL;DR
This paper presents new online edge coloring algorithms using the Nibble Method, achieving near-optimal bounds in both random-order and adversarial models, thus resolving longstanding conjectures and improving recourse efficiency.
Contribution
It introduces the first online algorithms that match the Bar-Noy et al. conjecture in the random-order model and reduces recourse complexity in the adversarial model, adapting the Nibble Method for online use.
Findings
Achieved near-optimal edge coloring in the random-order model.
Reduced recourse complexity to poly(1/ε) in the adversarial model.
Demonstrated the effectiveness of the Nibble Method in online algorithms.
Abstract
Nearly thirty years ago, Bar-Noy, Motwani and Naor [IPL'92] conjectured that an online -edge-coloring algorithm exists for -node graphs of maximum degree . This conjecture remains open in general, though it was recently proven for bipartite graphs under \emph{one-sided vertex arrivals} by Cohen et al.~[FOCS'19]. In a similar vein, we study edge coloring under widely-studied relaxations of the online model. Our main result is in the \emph{random-order} online model. For this model, known results fall short of the Bar-Noy et al.~conjecture, either in the degree bound [Aggarwal et al.~FOCS'03], or number of colors used [Bahmani et al.~SODA'10]. We achieve the best of both worlds, thus resolving the Bar-Noy et al.~conjecture in the affirmative for this model. Our second result is in the adversarial online (and dynamic) model with…
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