A 1.43-Competitive Online Graph Edge Coloring Algorithm In The Random Order Arrival Model

Bahmani, Bahman; Mehta, Aranyak; Motwani, Rajeev

doi:10.1137/1.9781611973075.4

Cited by 10 publications

(17 citation statements)

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“…We note that the above stipulation that each edge have bounded multiplicity is necessary in order to obtain (1 + o(1)) competitiveness for known ∆. 4 Observation F.1. No algorithm is (1 + o(1)) competitive on multigraphs of arbitrary multiplicity.…”

Section: F Extension To Multigraphsmentioning

confidence: 99%

“…Online Edge Coloring. Several previous papers studied edge coloring in online settings [1,4,5,18,20,21,45,46]. Mikkelsen [45,46] studied the online edge coloring problem, but with advice about the future.…”

Section: Related Workmentioning

confidence: 99%

“…The only positive results are under random-order edge arrival in "dense" bipartite multigraphs. Specifically, under such stochastic arrivals, Aggarwal et al [1] showed how to obtain a ∆(1 + o(1)) edge coloring of n-vertex multigraphs if ∆ = ω(n 2 ) and Bahmani et al [4] showed how to obtain a 1.26∆ edge coloring under the milder assumption that ∆ = ω(log n). The lack of progress for adversarial arrival order is likely explained by the following theorem of Bar-Noy et al [5].…”

Section: Introductionmentioning

confidence: 99%

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Tight Bounds for Online Edge Coloring

Cohen

Peng

Wajc³

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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Vizing's celebrated theorem asserts that any graph of maximum degree ∆ admits an edge coloring using at most ∆ + 1 colors. In contrast, Bar-Noy, Naor and Motwani showed over a quarter century 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 low-degree 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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Section: F Extension To Multigraphsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Tight Bounds for Online Edge Coloring

Cohen

Peng

Wajc³

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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“…Aggarwal, Motwani, Shah, and Zhu [AMSZ03] were the first to show that for very dense multi-graphs with ∆ = ω(n 2 ) one can get a near-optimal (1 + o(1))∆-edge coloring algorithm. For simple graphs, Bahmani, Mehta, and Motwani [BMM12] gave a 1.26∆-edge coloring algorithm when ∆ = ω(log n). Very recently, Bhattacharya, Grandoni, and Wajc [BGW21] showed that one can get the best of both these results by presenting a (1 + o(1))∆-edge coloring algorithm for simple graphs with ∆ = ω(log n) using an adaptation of the Nibble method.…”

Section: A Brief History Of the Problemmentioning

confidence: 99%

Online Edge Coloring via Tree Recurrences and Correlation Decay

Kulkarni¹,

Liu²,

Sah³

et al. 2021

Preprint

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We give an online algorithm that with high probability computes a e e−1 + o(1) ∆ edge coloring on a graph G with maximum degree ∆ = ω(log n) under online edge arrivals against oblivious adversaries, making first progress on the conjecture of Bar-Noy, Motwani, and Naor in this general setting. Our algorithm is based on reducing to a matching problem on locally treelike graphs, and then applying a tree recurrences based approach for arguing correlation decay.

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“…By contrast, the assumption that the input is ordered randomly improves the competitive ratios in many optimization problems. This includes packing problems [7,9,18], scheduling problems [19], and graph problems [20,21]. Therefore, developing new techniques for secretary problems may, more generally, yield relevant insights for this input model as well.…”

Section: Introductionmentioning

confidence: 99%

New Results for the $k$-Secretary Problem

Albers¹,

Ladewig²

2020

Preprint

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Suppose that n items arrive online in random order and the goal is to select k of them such that the expected sum of the selected items is maximized. The decision for any item is irrevocable and must be made on arrival without knowing future items. This problem is known as the k-secretary problem, which includes the classical secretary problem with the special case k = 1. It is well-known that the latter problem can be solved by a simple algorithm of competitive ratio 1/e which is optimal for n → ∞. Existing algorithms beating the threshold of 1/e either rely on involved selection policies already for k = 2, or assume that k is large.In this paper we present results for the k-secretary problem, considering the interesting and relevant case that k is small. We focus on simple selection algorithms, accompanied by combinatorial analyses. As a main contribution we propose a natural deterministic algorithm designed to have competitive ratios strictly greater than 1/e for small k ≥ 2. This algorithm is hardly more complex than the elegant strategy for the classical secretary problem, optimal for k = 1, and works for all k ≥ 1. We derive its competitive ratios for k ≤ 100, ranging from 0.41 for k = 2 to 0.75 for k = 100.Moreover, we consider an algorithm proposed earlier in the literature, for which no rigorous analysis is known. We show that its competitive ratio is 0.4168 for k = 2, implying that the previous analysis was not tight. Our analysis reveals a surprising combinatorial property of this algorithm, which might be helpful to find a tight analysis for all k.

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A 1.43-Competitive Online Graph Edge Coloring Algorithm In The Random Order Arrival Model

Cited by 10 publications

References 0 publications

Tight Bounds for Online Edge Coloring

Tight Bounds for Online Edge Coloring

Online Edge Coloring via Tree Recurrences and Correlation Decay

New Results for the $k$-Secretary Problem

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