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 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：我们提出基于多项式时间半正定规划（SDP）的算法解决以下问题：对于固定整数 $k \leq \ell$，给定实数 $\epsilon>0$ 和图 $G$（该图存在一个 $k$ 染色方案，其中 $\rho$ 比例的边被正确着色），算法能在 $G$ 和 $1/\epsilon$ 的多项式时间内返回 $G$ 的一个 $\ell$ 染色方案，使得正确着色的边比例达到 $(\alpha \rho - \epsilon)$。我们的算法基于 Frieze 与 Jerrum [Algorithmica'97] 以及 Karger、Motwani 与 Sudan [JACM'98] 的算法。当 $k=2,\ell=3$ 时，该算法达到最佳逼近比 $\alpha=1$。当 $k$ 固定而 $\ell$ 增长时，算法实现逼近比 $\alpha=1-o(1/\ell)$。当 $k$ 与 $\ell$ 均较大时，逼近比为 $\alpha=1-1/\ell+2\ln\ell/k\ell-o(\ln\ell/k\ell)-O(1/k^2)$；若固定 $d=\ell-k$ 并令 $k,\ell$ 增长，则 $\alpha=1-1/\ell+2\ln\ell/k\ell-o(\ln\ell/k\ell)$。通过将 Khot、Kindler、Mossel 与 O'Donnell [SICOMP'07] 的结果推广至承诺问题场景，我们证明：在 $\ell$ 受限于 $o(\exp(\sqrt[3]{k}))$ 的函数条件下，若假设 Khot 的唯一性游戏猜想（UGC），对于大型 $k$ 与 $\ell$，任何大于 $1-1/\ell+2\ln\ell/k\ell+o(\ln\ell/k\ell)$ 的逼近比 $\alpha$ 都难以通过 NP 算法实现。当 $d=\ell-k$ 固定时，该下界与算法性能的差距仅达 $o(\ln\ell/k\ell)$。此外，通过将 Guruswami 与 Sinop [ToC'13] 的结果扩展至承诺问题场景，我们证明（在相同 $\ell$ 有界条件下，但无需假设 UGC）：逼近比大于 $1-1/\ell+8\ln\ell/k\ell+o(\ln\ell/k\ell)$ 是 NP 难题。