Parity-SAT is the problem of determining whether a given CNF formula has an odd number of satisfying assignments. As a canonical $\oplus$P-complete problem, it represents a fundamental variant of the exact model counting problem (#SAT). Under the Strong Exponential Time Hypothesis (SETH), Parity-SAT admits no $O^*((2-\varepsilon)^n)$-time or $O^*((2-\varepsilon)^m)$-time algorithm for any constant $\varepsilon>0$, where $n$ and $m$ denote the numbers of variables and clauses, respectively. Thus, breaking the $2^n$ or $2^m$ barrier appears impossible in full generality. In this work, we revisit this barrier through structural restrictions and a refined exploitation of parity. We study Parity-$d$-occ-SAT, where each variable appears in at most $d$ clauses, and obtain three main results. First, we design a randomized $O^*(2^{m(1-1/O(d))})$-time algorithm, thereby breaking the $2^m$ barrier for every fixed $d$. Second, for the special case $d=2$, we develop a significantly sharper branching algorithm running in $O^*(1.1193^n)$ time or $O^*(1.3248^m)$ time. Third, leveraging the structural insights underlying the $d=2$ case, we obtain an $O^*(1.1052^L)$-time algorithm for general Parity-SAT, where $L$ denotes the formula length. All algorithms use only polynomial space. Notably, our running-time bounds are better than the best known bounds for the corresponding exact counting counterparts, highlighting a genuine algorithmic advantage of parity over counting. Conceptually, our results demonstrate that parity admits finer structural reductions and more efficient branching than exact model counting, and that bounded occurrence can be systematically leveraged to circumvent classical exponential barriers.

翻译：偶奇性-SAT问题是判断给定CNF公式是否具有奇数个满足赋值的问题。作为一个典型的$\oplus$P完全问题，它代表了精确模型计数问题(#SAT)的基本变体。在强指数时间假说(SETH)下，对于任意常数$\varepsilon>0$，偶奇性-SAT都不存在$O^*((2-\varepsilon)^n)$时间或$O^*((2-\varepsilon)^m)$时间的算法，其中$n$和$m$分别表示变量和子句的数量。因此，在完全一般性下突破$2^n$或$2^m$障碍似乎不可能。在本文中，我们通过结构限制和对偶奇性的精细利用重新审视了这一障碍。我们研究了偶奇性-$d$次出现-SAT问题，其中每个变量最多出现在$d$个子句中，并获得了三个主要结果。首先，我们设计了一个随机化的$O^*(2^{m(1-1/O(d))})$时间算法，从而对每个固定$d$突破了$2^m$障碍。其次，对于$d=2$的特殊情况，我们开发了一个显著更锐化的分支算法，运行时间为$O^*(1.1193^n)$或$O^*(1.3248^m)$。第三，利用$d=2$情况下的结构洞见，我们获得了适用于一般偶奇性-SAT问题的$O^*(1.1052^L)$时间算法，其中$L$表示公式长度。所有算法仅使用多项式空间。值得注意的是，我们的运行时间上界优于相应精确计数问题已知的最佳上界，凸显了偶奇性相对于计数的真正算法优势。从概念上讲，我们的结果表明，与精确模型计数相比，偶奇性允许更精细的结构归约和更高效的分支策略，并且可以系统地利用有界出现来规避经典的指数障碍。