The Unique Games Conjecture has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect completeness inherent in the Unique Games Conjecture. This work is motivated by the pursuit of a better understanding of the approximability of perfectly satisfiable instances of CSPs. We prove that an “almost Unique” version of Label Cover can be approximated within a constant factor on satisfiable instances. Our main conceptual contribution is the formulation of a (hypergraph) version of Label Cover that we call V Label Cover . Assuming a conjecture concerning the inapproximability of V Label Cover on perfectly satisfiable instances, we prove the following implications: • There is an absolute constant c 0 such that for k ≥ 3, given a satisfiable instance of Boolean k -CSP, it is hard to find an assignment satisfying more than c 0 k 2 /2 k fraction of the constraints. • Given a k -uniform hypergraph, k ≥ 2, for all ε > 0, it is hard to tell if it is q -strongly colorable or has no independent set with an ε fraction of vertices, where q =⌈ k +√ k -1/2⌉. • Given a k -uniform hypergraph, k ≥ 3, for all ε > 0, it is hard to tell if it is ( k -1)-rainbow colorable or has no independent set with an ε fraction of vertices.

中文翻译：

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寻求具有完美完整性的强不可近似性结果

Unique Games Conjecture 确定了所有约束满足问题 (CSP) 的近似性，表明自然半定规划松弛为任何 CSP 提供了最佳的最坏情况近似比。然而，由于独特游戏猜想中固有的不完善性，这幅精美的图片不适用于完全可满足的 CSP 实例。这项工作的动机是为了更好地理解完全可满足的 CSP 实例的近似性。我们证明了“几乎唯一”版本的 Label Cover 可以在可满足实例的常数因子内进行近似。我们的主要概念贡献是我们称之为标签封面的（超图）版本的制定V标签盖. 假设一个关于 V Label Cover 在完全可满足实例上的不可近似性的猜想，我们证明了以下含义： • 存在一个绝对常数C 0这样对于ķ≥ 3，给定一个可满足的 Boolean 实例ķ-CSP，很难找到令人满意的作业C 0 ķ 2/2 ķ 约束的一部分。• 给定一个ķ- 均匀超图，ķ≥ 2，对于所有 ε > 0，很难判断它是否是q- 高度可着色或没有具有 ε 部分顶点的独立集，其中q=⌈ķ+√ķ-1/2⌉。• 给定一个ķ- 均匀超图，ķ≥ 3，对于所有 ε > 0，很难判断它是否是 (ķ-1)-彩虹可着色或没有具有 ε 部分顶点的独立集。