We present polynomial-time SDP-based algorithms for the following problem: For fixed $k \leq \ell$, given a real number $\epsilon>0$ and a graph $G$ that admits a $k$-colouring with a $\rho$-fraction of the edges coloured properly, it returns an $\ell$-colouring of $G$ with an $(\alpha \rho - \epsilon)$-fraction of the edges coloured properly in polynomial time in $G$ and $1 / \epsilon$. Our algorithms are based on the algorithms of Frieze and Jerrum [Algorithmica'97] and of Karger, Motwani and Sudan [JACM'98]. For $k = 2, \ell = 3$, our algorithm achieves an approximation ratio $\alpha = 1$, which is the best possible. When $k$ is fixed and $\ell$ grows large, our algorithm achieves an approximation ratio of $\alpha = 1 - o(1 / \ell)$. When $k, \ell$ are both large, our algorithm achieves an approximation ratio of $\alpha = 1 - 1 / \ell + 2 \ln \ell / k \ell - o(\ln \ell / k \ell) - O(1 / k^2)$; if we fix $d = \ell - k$ and allow $k, \ell$ to grow large, this is $\alpha = 1 - 1 / \ell + 2 \ln \ell / k \ell - o(\ln \ell / k \ell)$. 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 the Unique Games Conjecture, it is \NP-hard to achieve an approximation ratio $\alpha$ 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 $o(\ln \ell / k \ell)$.

翻译：我们提出了基于多项式时间半定规划（SDP）的算法，用于解决以下问题：对于固定的 $k \leq \ell$，给定一个实数 $\epsilon>0$ 和一个图 $G$，该图允许一种 $k$-着色，其中 $\rho$-比例的边被正确着色。该算法在 $G$ 和 $1 / \epsilon$ 的多项式时间内，返回 $G$ 的一个 $\ell$-着色，其中 $(\alpha \rho - \epsilon)$-比例的边被正确着色。我们的算法基于 Frieze 和 Jerrum [Algorithmica'97] 以及 Karger、Motwani 和 Sudan [JACM'98] 的算法。对于 $k = 2, \ell = 3$，我们的算法实现了近似比 $\alpha = 1$，这是最佳可能值。当 $k$ 固定而 $\ell$ 增大时，我们的算法实现了近似比 $\alpha = 1 - o(1 / \ell)$。当 $k, \ell$ 都很大时，我们的算法实现了近似比 $\alpha = 1 - 1 / \ell + 2 \ln \ell / k \ell - o(\ln \ell / k \ell) - O(1 / k^2)$；如果我们固定 $d = \ell - k$ 并让 $k, \ell$ 增大，则近似比为 $\alpha = 1 - 1 / \ell + 2 \ln \ell / k \ell - o(\ln \ell / k \ell)$。通过将 Khot、Kindler、Mossel 和 O'Donnell [SICOMP'07] 的结果扩展到承诺设置，我们证明对于大的 $k$ 和 $\ell$，假设唯一游戏猜想成立，则实现近似比 $\alpha$ 大于 $1 - 1 / \ell + 2 \ln \ell / k \ell + o(\ln \ell / k \ell)$ 是 \NP-难的，前提是 $\ell$ 被一个函数 $o(\exp(\sqrt[3]{k}))$ 所界定。对于 $d = \ell - k$ 固定的情况，该界限与我们的算法性能在 $o(\ln \ell / k \ell)$ 级别相匹配。