Vizing’s celebrated theorem asserts that any graph of maximum degree ∆ admits an edge coloring using at most ∆ + 1 colors. In contrast, Bar-Noy, Motwani and Naor showed over a quarter century ago that the trivial greedy algorithm, which uses 2∆−1 colors, is optimal among online algorithms. Their lower bound has a caveat, however: it only applies to lowdegree graphs, with ∆ = O(log n), and they conjectured the existence of online algorithms using ∆(1 + o(1)) colors for ∆ = ω(log n). Progress towards resolving this conjecture was only made under stochastic arrivals (Aggarwal et al., FOCS’03 and Bahmani et al., SODA’10). We resolve the above conjecture for adversarial vertex arrivals in bipartite graphs, for which we present a (1+o(1))∆-edge-coloring algorithm for ∆ = ω(log n) known a priori. Surprisingly, if ∆ is not known ahead of time, we show that no (e/(e−1)−Ω(1))∆-edge-coloring algorithm exists.We then provide an optimal, (e/(e−1) +o(1))∆-edge-coloring algorithm for unknown ∆ = ω(log n). Key to our results, and of possible independent interest, is a novel fractional relaxation for edge coloring, for which we present optimal fractional online algorithms and a near-lossless online rounding scheme, yielding our optimal randomized algorithms.
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Efficiently list‐edge coloring multigraphs asymptotically optimally
We give polynomial time algorithms for the seminal results of Kahn, who showed that the Goldberg–Seymour and list‐coloring conjectures for (list‐)edge coloring multigraphs hold asymptotically. Kahn's arguments are based on the probabilistic method and are non‐constructive. Our key insight is that we can combine sophisticated techniques due to Achlioptas, Iliopoulos, and Kolmogorov for the analysis of local search algorithms with correlation decay properties of the probability spaces on matchings used by Kahn in order to construct efficient edge‐coloring algorithms.
more » « less- Award ID(s):
- 1815328
- PAR ID:
- 10444388
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Random Structures & Algorithms
- Volume:
- 61
- Issue:
- 4
- ISSN:
- 1042-9832
- Page Range / eLocation ID:
- p. 724-753
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract DP‐coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvořák and Postle [J. Combin. Theory Ser. B 129 (2018), pp. 38–54]. In this paper we introduce and study the fractional DP‐chromatic number . We characterize all connected graphs such that : they are precisely the graphs with no odd cycles and at most one even cycle. By a theorem of Alon, Tuza, and Voigt [Discrete Math. 165–166 (1997), pp. 31–38], the fractional list‐chromatic number of any graph equals its fractional chromatic number . This equality does not extend to fractional DP‐colorings. Moreover, we show that the difference can be arbitrarily large, and, furthermore, for every graph of maximum average degree . On the other hand, we show that this asymptotic lower bound is tight for a large class of graphs that includes all bipartite graphs as well as many graphs of high girth and high chromatic number.
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