Online Edge Coloring Algorithms via the Nibble Method

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 $(1+o(1))\Delta$-edge-coloring algorithm exists for $n$-node graphs of maximum degree $\Delta=\omega(\log n)$. 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 \emph{recourse}. A recent algorithm of Duan et al.~[SODA'19] yields a $(1+\epsilon)\Delta$-edge-coloring with poly$(\log n/\epsilon)$ recourse. We achieve the same with poly$(1/\epsilon)$ recourse, thus removing all dependence on $n$. Underlying our results is one common offline algorithm, which we show how to implement in these two online models. Our algorithm, based on the R\"odl Nibble Method, is an adaptation of the distributed algorithm of Dubhashi et al.~[TCS'98]. The Nibble Method has proven successful for distributed edge coloring. We display its usefulness in the context of online algorithms.