Complexity and algorithms for well-structured k -SAT instances Konstantinos Georgiou * and Periklis A. Papakonstantinou *† * Department of Computer Science, University of Toronto, 10 King’s College Road, Toronto, ON M5S 3G4, Canada † Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, M5S 2E4, Canada {cgeorg,papakons}@cs.toronto.edu January 20, 2008 Abstract This paper initiates the study of 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 study the complexity of the satisfiability, the counting and the maximization problems for formulas of bounded diameter. We investigate the relation between the diameter of a formula, and the tree-width and the path-width of its corresponding incidence graph, and show that under highly parallel and efficient transformations, diameter and path-width are equal up to a constant factor. Our main technical contribution is that the computational complexity of SAT, Max-SAT, #SAT grows smoothly with the diameter (as a function of the number of variables). Our main focus is in providing space efficient and highly parallel algorithms, while the running time of our algorithms matches previously known results. Among others, we show NL-completeness of SAT and NC 2 algorithms for Max-SAT, #SAT when diameter is O(log n). Given the tree decomposition of a formula, we further improve on the space efficiency to decide SAT as asked by Alekhnovich and Razborov [1]. 1 Introduction SAT, Max-SAT and #SAT are among the most fundamental and well-studied problems in the- oretical computer science, all intractable in the most general case: SAT is NP-complete [14], Max-SAT is NP-hard to approximate within some constant [5], while #SAT is hard for #P [39]. The intractability of SAT, Max-SAT and #SAT soon led to the study of restricted versions. Many of them are based on hidden structures of formulas and in particular on the so-called width restrictions. In this direction, we introduce the diameter of a formula, a natural structural re- striction which unlike width restrictions, it is defined directly on k-CNF formulas. For a CNF formula ordered as a sequence of clauses and for every variable consider the distance between the clause-indices where this variable appears. The maximum such distance is the diameter of the ordered formula. In this work, we consider SAT, Max-SAT and #SAT and parameterize them with respect to diameter. 1