Abstract
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 \({\mathbb {F}}_2\) each of which is a product of affine forms. We focus on the case of k-CNF formulas (the \({k}-\textsc {Sub}-\textsc {Sat}\) problem). Clearly, \({k}-\textsc {Sub}-\textsc {Sat}\) is no easier than k-SAT, and might be harder. Indeed, via simple reductions we show that \({2}-\textsc {Sub}-\textsc {Sat}\) is NP-hard, and \({\small \mathrm {W}}[1]\)-hard when parameterized by the co-dimension of the subspace. We also prove that the optimization version Max-\({2}-\textsc {Sub}-\textsc {Sat}\) 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 running time \(O^*(1.5)^r\) for \({2}-\textsc {Sub}-\textsc {Sat}\), where r is the subspace dimension, as well as an \(O^*(1.4312)^n\) time algorithm where n is the number of variables. Turning to \({k}-\textsc {Sub}-\textsc {Sat}\) for \(k \geqslant 3\), while known algorithms for solving a system of degree k polynomial equations already imply a solution with running time \(\approx 2^{r(1-1/2k)}\), 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 running time \(\approx {n\atopwithdelims (){\leqslant t}} 2^{n-n/k}\) 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 combinatorial approach also achieves polynomial space in contrast to the algebraic approach that uses exponential space. We also give a PPZ-style algorithm for \({k}-\textsc {Sub}-\textsc {Sat}\) with running time \(\approx 2^{n-n/2k}\). 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 \({\mathbb {F}}_2\), 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.
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Notes
The NP-hardness would also follow from Schaefer’s dichotomy theorem for Boolean CSP [23], though that is an overkill hammer for this result.
The notation \(O^*(f(n))\) for running time bounds suppresses polynomial factors.
For nonnegative integers n, t, the notation \({n \atopwithdelims (){\leqslant t}}\) stands for \(\sum _{i=0}^t {n \atopwithdelims ()i}\).
Of course, there is also a trivial \(O^*(2^{n-t})\) time brute force algorithm.
From a solution to \((\Phi _U,A_U)\) we can reconstruct the solution to \((\Phi ,A)\) as the values to variables in U are uniquely determined via the linear equations from the values to the other variables.
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We thank anonymous reviewers for their useful comments and pointers to the literature.
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A conference version of this work was presented at the 2021 International Symposium on Parameterized and Exact Computation (IPEC). 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.
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Arvind, V., Guruswami, V. CNF Satisfiability in a Subspace and Related Problems. Algorithmica 84, 3276–3299 (2022). https://doi.org/10.1007/s00453-022-00958-4
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DOI: https://doi.org/10.1007/s00453-022-00958-4