Complexity and Algorithms for Well-Structured k-SAT Instances

This paper consists of two conceptually related but independent parts. In the first part we initiate the study of k-SAT instances of bounded diameter. The diameter of an ordered CNF formula is defined as the maximum difference between the index of the first and the last occurrence of a variable. We investigate the relation between the diameter of a formula and the tree-width and the path-width of its corresponding incidence graph. We show that under highly parallel and efficient transformations, diameter and path-width are equal up to a constant factor. Our main result is that the computational complexity of SAT, Max-SAT, #SAT grows smoothly with the diameter (as a function of the number of variables). Our focus is in providing space efficient and highly parallel algorithms, while the running time of our algorithms matches previously known results. Our results refer to any diameter, whereas for the special case where the diameter is O(log n) we show NL-completeness of SAT and NC algorithms for Max-SAT and #SAT. In the second part we deal directly with k-CNF formulas of bounded tree-width. We describe algorithms in an intuitive but not-so-standard model of computation. Then we apply constructive theorems from computational complexity to obtain deterministic time-efficient and simultaneously space-efficient algorithms for k-SAT as asked by Alekhnovich and Razborov [1].