We define a novel notion of ``non-backtracking'' matrix associated to any symmetric matrix, and we prove a ``Ihara-Bass'' type formula for it. We use this theory to prove new results on polynomial-time strong refutations of random constraint satisfaction problems with $k$ variables per constraints (k-CSPs). For a random k-CSP instance constructed out of a constraint that is satisfied by a $p$ fraction of assignments, if the instance contains $n$ variables and $n^{k/2} / \epsilon^2$ constraints, we can efficiently compute a certificate that the optimum satisfies at most a $p+O_k(\epsilon)$ fraction of constraints. Previously, this was known for even $k$, but for odd $k$ one needed $n^{k/2} (\log n)^{O(1)} / \epsilon^2$ random constraints to achieve the same conclusion. Although the improvement is only polylogarithmic, it overcomes a significant barrier to these types of results. Strong refutation results based on current approaches construct a certificate that a certain matrix associated to the k-CSP instance is quasirandom. Such certificate can come from a Feige-Ofek type argument, from an application of Grothendieck's inequality, or from a spectral bound obtained with a trace argument. The first two approaches require a union bound that cannot work when the number of constraints is $o(n^{\lceil k/2 \rceil})$ and the third one cannot work when the number of constraints is $o(n^{k/2} \sqrt{\log n})$. We further apply our techniques to obtain a new PTAS finding assignments for $k$-CSP instances with $n^{k/2} / \epsilon^2$ constraints in the semi-random settings where the constraints are random, but the sign patterns are adversarial.

翻译：我们定义了一种与任意对称矩阵相关的新型“非回溯”矩阵，并证明了其伊哈拉-巴斯型公式。利用这一理论，我们得到了关于随机约束满足问题（每约束含k个变量，即k-CSP）的多项式时间强反驳的新结果。对于由满足概率为p的赋值约束构建的随机k-CSP实例，若实例包含n个变量和n^{k/2}/ε^2个约束，则我们能高效计算一个证书，证明最优解满足的约束比例至多为p+O_k(ε)。此前，这一结果仅对偶数k成立；对于奇数k，需要n^{k/2} (log n)^{O(1)}/ε^2个随机约束才能达到相同结论。尽管改进仅是多对数级别的，但它突破了此类结果的关键障碍。基于现有方法的强反驳结果需构建一个与k-CSP实例相关的特定矩阵为准随机的证书。此类证书可通过Feige-Ofek型论证、Grothendieck不等式的应用或迹方法获得的谱界得到。前两种方法依赖于联合界，当约束数量为o(n^{⌈k/2⌉})时失效；第三种方法在约束数量为o(n^{k/2}√(log n))时失效。我们进一步应用该技术，在约束随机但符号模式对抗的半随机设置下，针对含n^{k/2}/ε^2个约束的k-CSP实例，得到了一种新的多项式时间近似方案（PTAS）以寻找赋值。