Fast algorithms for Vizing's theorem on bounded degree graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915426028486656 |
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| author | Bernshteyn, Anton Dhawan, Abhishek |
| author_facet | Bernshteyn, Anton Dhawan, Abhishek |
| contents | Vizing's theorem states that every graph $G$ of maximum degree $Δ$ can be properly edge-colored using $Δ+ 1$ colors. The fastest currently known $(Δ+1)$-edge-coloring algorithm for general graphs is due to Sinnamon and runs in time $O(m\sqrt{n})$, where $n :=|V(G)|$ and $m :=|E(G)|$. We investigate the case when $Δ$ is constant, i.e., $Δ= O(1)$. In this regime, the runtime of Sinnamon's algorithm is $O(n^{3/2})$, which can be improved to $O(n \log n)$, as shown by Gabow, Nishizeki, Kariv, Leven, and Terada. Here we give an algorithm whose running time is only $O(n)$, which is obviously best possible. Prior to this work, no linear-time $(Δ+1)$-edge-coloring algorithm was known for any $Δ\geq 4$. Using some of the same ideas, we also develop new algorithms for $(Δ+1)$-edge-coloring in the $\mathsf{LOCAL}$ model of distributed computation. Namely, when $Δ$ is constant, we design a deterministic $\mathsf{LOCAL}$ algorithm with running time $\tilde{O}(\log^5 n)$ and a randomized $\mathsf{LOCAL}$ algorithm with running time $O(\log ^2 n)$. Although our focus is on the constant $Δ$ regime, our results remain interesting for $Δ$ up to $\log^{o(1)} n$, since the dependence of their running time on $Δ$ is polynomial. The key new ingredient in our algorithms is a novel application of the entropy compression method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_05408 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fast algorithms for Vizing's theorem on bounded degree graphs Bernshteyn, Anton Dhawan, Abhishek Data Structures and Algorithms Distributed, Parallel, and Cluster Computing Discrete Mathematics Combinatorics Vizing's theorem states that every graph $G$ of maximum degree $Δ$ can be properly edge-colored using $Δ+ 1$ colors. The fastest currently known $(Δ+1)$-edge-coloring algorithm for general graphs is due to Sinnamon and runs in time $O(m\sqrt{n})$, where $n :=|V(G)|$ and $m :=|E(G)|$. We investigate the case when $Δ$ is constant, i.e., $Δ= O(1)$. In this regime, the runtime of Sinnamon's algorithm is $O(n^{3/2})$, which can be improved to $O(n \log n)$, as shown by Gabow, Nishizeki, Kariv, Leven, and Terada. Here we give an algorithm whose running time is only $O(n)$, which is obviously best possible. Prior to this work, no linear-time $(Δ+1)$-edge-coloring algorithm was known for any $Δ\geq 4$. Using some of the same ideas, we also develop new algorithms for $(Δ+1)$-edge-coloring in the $\mathsf{LOCAL}$ model of distributed computation. Namely, when $Δ$ is constant, we design a deterministic $\mathsf{LOCAL}$ algorithm with running time $\tilde{O}(\log^5 n)$ and a randomized $\mathsf{LOCAL}$ algorithm with running time $O(\log ^2 n)$. Although our focus is on the constant $Δ$ regime, our results remain interesting for $Δ$ up to $\log^{o(1)} n$, since the dependence of their running time on $Δ$ is polynomial. The key new ingredient in our algorithms is a novel application of the entropy compression method. |
| title | Fast algorithms for Vizing's theorem on bounded degree graphs |
| topic | Data Structures and Algorithms Distributed, Parallel, and Cluster Computing Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2303.05408 |