arXiv 2108.059142021-08-12cs.DS

子空间中的CNF可满足性及相关问题

CNF Satisfiability in a Subspace and Related Problems

Vikraman Arvind, Venkatesan Guruswami

摘要：

我们引入的问题，找到一个满意的分配CNF公式，必须进一步属于一个规定的输入子空间。问题的等价公式包括找到子空间并集外的一点（子空间避免并集（USA）问题），以及找到$\F_2$上多项式系统的公共零点，每个多项式都是仿射形式的乘积。我们专注于k-CNF公式的情况下（的k-SAT-SAT问题）。显然，它并不比k-SAT容易，而且可能更难。事实上，通过简单的约化，我们证明了k=2的NP-硬度和W[1]-硬度由子空间的余维参数化。我们还证明了优化版本Max-2-SUB-SAT是NP-难的近似比平凡的3/4比，即使在可满足的情况下。在算法方面，我们研究了快速指数算法，它提供了非平凡的节省暴力算法。我们给出了一个运行时间为1.5^r的2-SUB-SAT分支算法，其中$r$是子空间维数，和一个O^*（1.4312）^n时间算法，其中$n$是变量的数量。当k大于2时，虽然已知的求解$k$多项式方程组的算法已经暗示了运行时间为2^{r（1-1/2k）}的解，但我们探索了一种更具组合性的方法。例如，基于临界变量的概念，我们给出了一个运行时间为${n\choose {\le t}} 2^{n-n/k}$的算法，其中$n$是变量的个数，$t$是子空间的余维数。这改善了小余维的多项式方程方法的运行时间。我们的算法还实现了多项式空间，而不是使用指数空间的代数方法。

英文摘要：

We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero of a system of polynomials over $\F_2$ each of which is a product of affine forms. We focus on the case of k-CNF formulas (the k-SUB-SAT problem). Clearly, it is no easier than k-SAT, and might be harder. Indeed, via simple reductions we show NP-hardness for k=2 and W[1]-hardness parameterized by the co-dimension of the subspace. We also prove that the optimization version Max-2-SUB-SAT is NP-hard to approximate better than the trivial 3/4 ratio even on satisfiable instances. On the algorithmic front, we investigate fast exponential algorithms which give non-trivial savings over brute-force algorithms. We give a simple branching algorithm with runtime 1.5^r for 2-SUB-SAT, where $r$ is the subspace dimension and an O^*(1.4312)^n time algorithm where $n$ is the number of variables. For k more than 2, while known algorithms for solving a system of degree $k$ polynomial equations already imply a solution with runtime 2^{r(1-1/2k)}, we explore a more combinatorial approach. For instance, based on the notion of critical variables, we give an algorithm with running time ${n\choose {\le t}} 2^{n-n/k}$, where $n$ is the number of variables and $t$ is the co-dimension of the subspace. This improves upon the running time of the polynomial equations approach for small co-dimension. Our algorithm also achieves polynomial space in contrast to the algebraic approach that uses exponential space.

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英文摘要：

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