CNF Satisfiability in a Subspace and Related Problems

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 F2 each of which is a product of affine forms. We focus on the case of k-CNF formulas (the k-SUB-SAT problem). Clearly, k-SUB-SAT is no easier than k-SAT, and might be harder. Indeed, via simple reductions we show that 2-SUB-SAT is NP-hard, and W[1]-hard when 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) for 2-SUB-SAT, where r is the subspace dimension, as well as a (1.4312) time algorithm where n is the number of variables. Turning to k-SUB-SAT for k > 3, while known algorithms for solving a system of degree k polynomial equations already imply a solution with runtime ≈ 2r(1−1/2k), we explore a more combinatorial approach. Based on an analysis of critical variables (a key notion underlying the randomized k-SAT algorithm of Paturi, Pudlak, and Zane), we give an algorithm with runtime ≈ ( n 6t ) 2n−n/k where n is the number of variables and t is the co-dimension of the subspace. This improves upon the runtime of the polynomial equations approach for small co-dimension. Our combinatorial approach also achieves polynomial space in contrast to the algebraic approach that uses exponential space. We also give a PPZ-style algorithm for k-SUB-SAT with runtime ≈ 2n−n/2k. This algorithm is in fact oblivious to the structure of the subspace, and extends when the subspace-membership constraint is replaced by any constraint for which partial satisfying assignments can be efficiently completed to a full satisfying assignment. Finally, for systems of O(n) polynomial equations in n variables over F2, we give a fast exponential algorithm when each polynomial has bounded degree irreducible factors (but can otherwise have large degree) using a degree reduction trick. ∗Institute of Mathematical Sciences (HBNI), Chennai, India. Email: arvind@imsc.res.in †Computer Science Department, Carnegie Mellon University, Pittsburgh, USA. Email: venkatg@cs.cmu.edu. Portions of this work were done during visits to the Institute of Mathematical Sciences, Chennai. Research supported in part by the US National Science Foundation grant CCF-1908125 and a Simons Investigator Award. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 112 (2021)