Vizing’s theorem guarantees that every graph with maximum degree $\Delta$ admits an edge coloring using $\Delta+1$ colors. In online settings-where edges arrive one at a time and must be colored immediately-a simple greedy algorithm uses at most $2 \Delta-1$ colors. Over thirty years ago, Bar-Noy, Motwani, and Naor [IPL’92] proved that this guarantee is optimal among deterministic algorithms when $\Delta=O(\log n)$, and among randomized algorithms when $\Delta=O(\sqrt{\log n})$. While deterministic improvements seemed out of reach, they conjectured that for graphs with $\Delta=\omega(\log n)$, randomized algorithms can achieve $(1+o(1)) \Delta$ edge coloring. This conjecture was recently resolved in the affirmative: a $(1+o(1)) \Delta$ coloring is achievable online using randomization for all graphs with $\Delta=\omega(\log n)$ [BSVW STOC’24]. Our results go further, uncovering two findings not predicted by the original conjecture. First, we give a deterministic online algorithm achieving $(1+o(1)) \Delta$-colorings for all $\Delta=\omega(\log n)$. Second, we give a randomized algorithm achieving $(1+o(1)) \Delta$ colorings already when $\Delta=\omega(\sqrt{\log n})$. Our results establish sharp thresholds for when greedy can be surpassed, and nearoptimal guarantees can be achieved - matching the impossibility results of [BNMN IPL’92], both deterministically and randomly.

We study online bipartite edge coloring, with nodes on one side of the graph revealed sequentially.

To meet stringent performance requirements, communication networks are becoming increasingly programmable and flexible, supporting fast and frequent adjustments. However, reconfiguring networks in a dependable and transiently consistent manner is known to be algorithmically challenging. This paper revisits the fundamental problem of how to update the routes in a network in a (transiently) loop-free manner, considering both the Strong Loop-Freedom (SLF) and the Relaxed Loop-Freedom (RLF) property.We present two fast algorithms to solve the SLF and RLF problem variants exactly, to optimality. Our algorithms are based on a parameterized integer linear program which would be intractable to solve directly by a classic solver. Our main technical contribution is a lazy cycle breaking strategy which, by adding constraints lazily, improves performance dramatically, and outperforms the state-of-the-art exact algorithms by an order of magnitude on realistic medium-sized networks. We further explore approximate algorithms and show that while a relaxation approach is relatively slow, with a local search approach short update schedules can be found, outperforming the state-of-the-art heuristics.On the theoretical front, we also provide an approximation lower bound for the update time of the state-of-the-art algorithm in the literature. As a contribution to the research community, we made all our code and implementations publicly available.

The classic theorem of Vizing (Diskret. Analiz.’64) asserts that any graph of maximum degree Δ can be edge colored (offline) using no more than Δ+1 colors (with Δ being a trivial lower bound). In the online setting, Bar-Noy, Motwani and Naor (IPL’92) conjectured that a (1+o(1))Δ-edge-coloring can be computed online in n-vertex graphs of maximum degree Δ=ω(logn). Numerous algorithms made progress on this question, using a higher number of colors or assuming restricted arrival models, such as random-order edge arrivals or vertex arrivals (e.g., AGKM FOCS’03, BMM SODA’10, CPW FOCS’19, BGW SODA’21, KLSST STOC’22). In this work, we resolve this longstanding conjecture in the affirmative in the most general setting of adversarial edge arrivals. We further generalize this result to obtain online counterparts of the list edge coloring result of Kahn (J. Comb. Theory. A’96) and of the recent “local” edge coloring result of Christiansen (STOC’23).