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：我们针对以下问题提出了基于半定规划（SDP）的多项式时间算法：给定固定整数 $k \leq \ell$，输入一个实数 $\epsilon>0$ 和一个允许 $k$-着色的图 $G$（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，使得被正确着色的边比例达到 $(\alpha \rho - \epsilon)$。我们的算法基于 Frieze 和 Jerrum [Algorithmica'97] 以及 Karger、Motwani 和 Sudan [JACM'98] 的算法。当 $k=2, \ell=3$ 时，我们的算法实现了近似比 $\alpha=1$，这是最优结果。当 $k$ 固定而 $\ell$ 增长时，算法达到近似比 $\alpha=1 - o(1/\ell)$。当 $k, \ell$ 均较大时，算法达到近似比 $\alpha=1 - 1/\ell + 2\ln\ell/k\ell - o(\ln\ell/k\ell) - O(1/k^2)$；若固定 $d=\ell-k$ 并令 $k,\ell$ 增长，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] 的结果推广到承诺问题设定，我们证明：对于较大的 $k$ 和 $\ell$，假设 Khot 的唯一游戏猜想（UGC）成立，当 $\ell$ 被某个 $o(\exp(\sqrt[3]{k}))$ 函数有界时，实现近似比 $\alpha$ 大于 $1 - 1/\ell + 2\ln\ell/k\ell + o(\ln\ell/k\ell)$ 是 \NP-困难的。对于 $d=\ell-k$ 固定的情况，该下界与我们的算法性能差距在 $o(\ln\ell/k\ell)$ 内。此外，通过将 Guruswami 和 Sinop [ToC'13] 的结果扩展到承诺问题设定，我们证明：在相同 $\ell$ 有界条件下（此时不依赖 UGC），实现近似比大于 $1 - 1/\ell + 8\ln\ell/k\ell + o(\ln\ell/k\ell)$ 是 \NP-困难的。