arXiv 1801.094882018-01-29cs.DS

哪一个NP难SAT和CSP问题允许指数改进算法？

Which NP-Hard SAT and CSP Problems Admit Exponentially Improved Algorithms?

Victor Lagerkvist, Magnus Wahlström

摘要：

本文研究了在变量否定下闭的潜在无限语言$Γ$（符号对称语言）的SAT（$Γ$）问题的复杂性。通过代数连接，这减少到$Γ$的受限部分多态性的研究，我们称之为\emph{pSDI-operations}（部分，自对偶和幂等）。首先，我们学习语言课程本身。我们对限制性最小的pSDI-操作的结构进行了分类，对应于最强大的语言$Γ$，并发现这些操作可以分为\emph{levels}，对应于粗略的难度概念;在每一个层次中都有一个最强的操作（部分$k$ -NU操作，保留$(k-1)$ -SAT）和最弱操作（$k$ -通用操作$u_k$，保留可通过有界次数多项式定义的问题）。我们发现，每一个符号对称的$Γ$不保留$u_k$实现所有$k$ -条款，因此，如果$Γ$不保留$u_k$的任何$k$，那么SAT（$Γ$）是平凡的SETH-硬，不能解决速度比$O^*(2^n)$，除非SETH失败。其次，我们研究了此类语言的SAT（$Γ$）的上界和下界。我们发现，在层次结构中的几个类对应的问题，可以解决的速度比$2^n$使用以前已知的算法策略，如子集和式满足在中间和快速矩阵乘法。此外，如果向日葵猜想对于具有k个集合的向日葵成立，则部分k-NU语言具有通过局部搜索的改进算法。补充这一点，我们表明，对于每一类有一个具体的下限$c$，这样SAT（$Γ$）不能解决的速度比$O^*(c^n)$的类中的所有问题，除非SETH失败。这给出了第一个已知的情况下，SAT问题，同时具有非平凡的上限和下限下SETH。

英文摘要：

We study the complexity of SAT($Γ$) problems for potentially infinite languages $Γ$ closed under variable negation (sign-symmetric languages). Via an algebraic connection, this reduces to the study of restricted partial polymorphisms of $Γ$ we refer to as \emph{pSDI-operations} (for partial, self-dual and idempotent). First, we study the language classes themselves. We classify the structure of the least restrictive pSDI-operations, corresponding to the most powerful languages $Γ$, and find that these operations can be divided into \emph{levels}, corresponding to a rough notion of difficulty; and that within each level there is a strongest operation (the partial $k$-NU operation, preserving $(k-1)$-SAT) and a weakest operation (the $k$-universal operation $u_k$, preserving problems definable via bounded-degree polynomials). We show that every sign-symmetric $Γ$ not preserved by $u_k$ implements all $k$-clauses; thus if $Γ$ is not preserved by $u_k$ for any $k$, then SAT($Γ$) is trivially SETH-hard and cannot be solved faster than $O^*(2^n)$ unless SETH fails. Second, we study upper and lower bounds for SAT($Γ$) for such languages. We show that several classes in the hierarchy correspond to problems which can be solved faster than $2^n$ using previously known algorithmic strategies such as Subset Sum-style meet-in-the-middle and fast matrix multiplication. Furthermore, if the sunflower conjecture holds for sunflowers with k sets, then the partial k-NU language has an improved algorithm via local search. Complementing this, we show that for every class there is a concrete lower bound $c$ such that SAT($Γ$) cannot be solved faster than $O^*(c^n)$ for all problems in the class unless SETH fails. This gives the first known case of a SAT-problem which simultaneously has non-trivial upper and lower bounds under SETH.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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