Satisfiability with Index Dependency

Abstract

We study the Boolean Satisfiability Problem (SAT) restricted on input formulas for which there are linear arithmetic constraints imposed on the indices of variables occurring in the same clause. This can be seen as a structural counterpart of Schaefer’s dichotomy theorem which studies the SAT problem with additional constraints on the assigned values of variables in the same clause. More precisely, let k-SAT(\(m,\mathcal{A}\)) denote the SAT problem restricted on instances of k-CNF formulas, in every clause of which the indices of the last k − m variables are totally decided by the first m ones through some linear equations chosen from \(\mathcal{A}\). For example, if \(\mathcal{A}\) contains i 3 = i 1 + 2i 2 and i 4 = i 2 − i 1 + 1, then a clause of the input to 4-SAT(\(2,\mathcal{A}\)) has the form \(y_{i_1}\lor y_{i_2} \lor y_{i_1+2i_2} \lor y_{i_2-i_1+1}\), with y i being x i or \(\overline{x_i}\). We obtain the following results: