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]. 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 Khot's Unique Games Conjecture (\UGC), 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 $o(\ln \ell / k \ell)$. 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).

翻译：我们提出了基于半正定规划（SDP）的多项式时间算法，用于解决以下问题：对于固定的$k \leq \ell$，给定实数$\epsilon>0$以及一个允许$k$-着色且具有$\rho$比例边被正确着色的图$G$，该算法在关于$G$和$1 / \epsilon$的多项式时间内，返回$G$的一个$\ell$-着色，其中具有$(\alpha \rho - \epsilon)$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。当$k$固定且$\ell$趋于无穷大时，我们的算法达到近似比$\alpha = 1 - o(1 / \ell)$。当$k$和$\ell$都很大时，我们的算法达到近似比$\alpha = 1 - 1 / \ell + 2 \ln \ell / k \ell - o(\ln \ell / k \ell) - O(1 / k^2)$；如果我们固定$d = \ell - k$并让$k$和$\ell$趋于无穷大，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] 的结果推广到承诺设定，我们证明对于大的$k$和$\ell$，在假设Khot的唯一游戏猜想（\UGC）的前提下，若$\ell$被一个$o(\exp(\sqrt[3]{k}))$的函数所限制，则实现大于$1 - 1 / \ell + 2 \ln \ell / k \ell + o(\ln \ell / k \ell)$的近似比$\alpha$是\NP-难的。对于$d = \ell - k$固定的情况，该下界与我们的算法性能在$o(\ln \ell / k \ell)$范围内匹配。此外，通过将Guruswami和Sinop [ToC'13] 的结果推广到承诺设定，我们证明（同样在$\ell$被如前所述限制的条件下，但这次无需假设\UGC）实现大于$1 - 1 / \ell + 8 \ln \ell / k \ell + o(\ln \ell / k \ell)$的近似比是\NP-难的。