Vizing's Theorem in Near-Linear Time
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908590113030144 |
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| author | Assadi, Sepehr Behnezhad, Soheil Bhattacharya, Sayan Costa, Martín Solomon, Shay Zhang, Tianyi |
| author_facet | Assadi, Sepehr Behnezhad, Soheil Bhattacharya, Sayan Costa, Martín Solomon, Shay Zhang, Tianyi |
| contents | Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $Δ$ can be edge colored using at most $Δ+ 1$ different colors [Vizing, 1964]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in $O(mn)$ time. This was subsequently improved to $\tilde O(m\sqrt{n})$ time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985].
Very recently, independently and concurrently, using randomization, this runtime bound was further improved to $\tilde{O}(n^2)$ by [Assadi, 2024] and $\tilde O(mn^{1/3})$ by [Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024] (and subsequently to $\tilde O(mn^{1/4})$ time by [Bhattacharya, Costa, Solomon and Zhang, 2024]).
In this paper, we present a randomized algorithm that computes a $(Δ+1)$-edge coloring in near-linear time -- in fact, only $O(m\logΔ)$ time -- with high probability, giving a near-optimal algorithm for this fundamental problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_05240 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Vizing's Theorem in Near-Linear Time Assadi, Sepehr Behnezhad, Soheil Bhattacharya, Sayan Costa, Martín Solomon, Shay Zhang, Tianyi Data Structures and Algorithms Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $Δ$ can be edge colored using at most $Δ+ 1$ different colors [Vizing, 1964]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in $O(mn)$ time. This was subsequently improved to $\tilde O(m\sqrt{n})$ time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. Very recently, independently and concurrently, using randomization, this runtime bound was further improved to $\tilde{O}(n^2)$ by [Assadi, 2024] and $\tilde O(mn^{1/3})$ by [Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024] (and subsequently to $\tilde O(mn^{1/4})$ time by [Bhattacharya, Costa, Solomon and Zhang, 2024]). In this paper, we present a randomized algorithm that computes a $(Δ+1)$-edge coloring in near-linear time -- in fact, only $O(m\logΔ)$ time -- with high probability, giving a near-optimal algorithm for this fundamental problem. |
| title | Vizing's Theorem in Near-Linear Time |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2410.05240 |