Maximum $k$- vs. $\ell$-colourings of graphs
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arXiv
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| Format: | Preprint |
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2023
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| author | Nakajima, Tamio-Vesa Živný, Stanislav |
| author_facet | Nakajima, Tamio-Vesa Živný, Stanislav |
| contents | We present polynomial-time SDP-based algorithms for the following problem: For fixed $k \leq \ell$, given a real number $ε>0$ and a graph $G$ that admits a $k$-colouring with a $ρ$-fraction of the edges coloured properly, it returns an $\ell$-colouring of $G$ with an $(αρ- ε)$-fraction of the edges coloured properly in polynomial time in $G$ and $1 / ε$. Our algorithms are based on the algorithms of Frieze and Jerrum [Algorithmica'97] and of Karger, Motwani and Sudan [JACM'98].
When $k$ is fixed and $\ell$ grows large, our algorithm achieves an approximation ratio of $α= 1 - o(1 / \ell)$. When $k, \ell$ are both large, our algorithm achieves an approximation ratio of $α= 1 - 1 / \ell + 2 \ln \ell / k \ell - o(\ln \ell / k \ell) - O(1 / k^2)$; if we fix $d = \ell - k$ and allow $k, \ell$ to grow large, this is $α= 1 - 1 / \ell + 2 \ln \ell / k \ell - o(\ln \ell / k \ell)$.
By extending the results of Khot, Kindler, Mossel and O'Donnell [SICOMP'07] to the promise setting, we show that for large $k$ and $\ell$, assuming Khot's Unique Games Conjecture (\UGC), it is \NP-hard to achieve an approximation ratio $α$ greater than $1 - 1 / \ell + 2 \ln \ell / k \ell + o(\ln \ell / k \ell)$, provided that $\ell$ is bounded by a function that is $o(\exp(\sqrt[3]{k}))$. For the case where $d = \ell - k$ is fixed, this bound matches the performance of our algorithm up to $o(\ln \ell / k \ell)$. Furthermore, by extending the results of Guruswami and Sinop [ToC'13] to the promise setting, we prove that it is \NP-hard to achieve an approximation ratio greater than $1 - 1 / \ell + 8 \ln \ell / k \ell + o(\ln \ell / k \ell)$, provided again that $\ell$ is bounded as before (but this time without assuming the \UGC). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_00440 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Maximum $k$- vs. $\ell$-colourings of graphs Nakajima, Tamio-Vesa Živný, Stanislav Data Structures and Algorithms Computational Complexity Discrete Mathematics We present polynomial-time SDP-based algorithms for the following problem: For fixed $k \leq \ell$, given a real number $ε>0$ and a graph $G$ that admits a $k$-colouring with a $ρ$-fraction of the edges coloured properly, it returns an $\ell$-colouring of $G$ with an $(αρ- ε)$-fraction of the edges coloured properly in polynomial time in $G$ and $1 / ε$. Our algorithms are based on the algorithms of Frieze and Jerrum [Algorithmica'97] and of Karger, Motwani and Sudan [JACM'98]. When $k$ is fixed and $\ell$ grows large, our algorithm achieves an approximation ratio of $α= 1 - o(1 / \ell)$. When $k, \ell$ are both large, our algorithm achieves an approximation ratio of $α= 1 - 1 / \ell + 2 \ln \ell / k \ell - o(\ln \ell / k \ell) - O(1 / k^2)$; if we fix $d = \ell - k$ and allow $k, \ell$ to grow large, this is $α= 1 - 1 / \ell + 2 \ln \ell / k \ell - o(\ln \ell / k \ell)$. By extending the results of Khot, Kindler, Mossel and O'Donnell [SICOMP'07] to the promise setting, we show that for large $k$ and $\ell$, assuming Khot's Unique Games Conjecture (\UGC), it is \NP-hard to achieve an approximation ratio $α$ greater than $1 - 1 / \ell + 2 \ln \ell / k \ell + o(\ln \ell / k \ell)$, provided that $\ell$ is bounded by a function that is $o(\exp(\sqrt[3]{k}))$. For the case where $d = \ell - k$ is fixed, this bound matches the performance of our algorithm up to $o(\ln \ell / k \ell)$. Furthermore, by extending the results of Guruswami and Sinop [ToC'13] to the promise setting, we prove that it is \NP-hard to achieve an approximation ratio greater than $1 - 1 / \ell + 8 \ln \ell / k \ell + o(\ln \ell / k \ell)$, provided again that $\ell$ is bounded as before (but this time without assuming the \UGC). |
| title | Maximum $k$- vs. $\ell$-colourings of graphs |
| topic | Data Structures and Algorithms Computational Complexity Discrete Mathematics |
| url | https://arxiv.org/abs/2311.00440 |