发表机构
Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证实了4对1游戏猜想，证明区分具有完美完备性的4对1游戏的最优值为1和不超过ε是NP困难的，并由此推导出图着色与超图独立集问题的NP困难性结果。
AI 中文摘要
我们证明，对于所有ε>0，存在正整数k，使得给定字母大小至多为k的4对1游戏Ψ，区分val(Ψ)=1与val(Ψ)≤ε的问题是NP困难的。这证实了Khot在CCC 2002中提出的4对1游戏猜想。此前，Dinur、Khot、Kindler、Minzer和Safra取得的最佳结果确立了几乎完美完备性版本（但应用于更严格的2对1游戏问题）。利用文献中的结果，我们得到以下推论：(1) 对于所有k∈ℕ，给定一个3-可着色图G，找到其正确的k着色是NP困难的；(2) 对于所有δ>0，给定一个2-可着色3-均匀超图G，在其中找到包含至少δ比例顶点的独立集是NP困难的。我们的证明是一个三步构造，基于Dinur等人的两步框架：在外层PCP步骤中，我们使用二次方程获得完美完备性；然后构造一个新的中间PCP，执行低秩测试同时保留关键覆盖性质；最后，基于标准Grassmann编码的张量及其由Golowich在FOCS 2023提出的低秩变体，构造一个新的内层PCP。
英文摘要
We prove that for all $\varepsilon>0$, there exists a positive integer $k$ such that given a $4$-to-$1$ game $Ψ$ with alphabet size at most $k$, it is $\mathbf{NP}$-hard to distinguish between the case that $\mathrm{val}(Ψ)=1$ and the case that $\mathrm{val}(Ψ)\leq \varepsilon$. This confirms the $4$-to-$1$ Games Conjecture from [Khot, \textit{CCC 2002}]. Previously, the best known result, due to [Dinur, Khot, Kindler, Minzer, Safra], established the almost-perfect completeness version (but applied to the stricter problem of $2$-to-$1$ games). Using results from the literature, we get the following implications: (1) for all $k\in \mathbb{N}$, given a $3$-colorable graph $G$, it is $\mathbf{NP}$-hard to find a proper $k$-coloring; (2) for all $δ>0$, given a $2$-colorable $3$-uniform hypergraph $G$, it is $\mathbf{NP}$-hard to find in it an independent set containing at least $δ$ fraction of the vertices. Our proof is a three-step construction that builds on the two-step framework of [Dinur, Khot, Kindler, Minzer, Safra]. In the outer-PCP step, we use quadratic equations to gain perfect completeness. We then construct a new middle PCP that performs low-rank tests while preserving a key covering property. Finally, we construct a new inner PCP based on a tensor of the standard Grassmann encoding with its low-rank variant due to [Golowich, FOCS 2023].
