About a surprising computer program of Matthias M¨ uller Mihai Prunescu * Abstract Matthias M¨ uller published on the net a C++ code and a text describing the implemented algorithm. He claimed that the algorithm ”maybe” solves the NP-complete problem 3-SAT in polynomial time. The program decided correctly so far the solvability for all instances that have been checked. This intriguing fact must be understood. In order to achieve this goal, we introduce the graph of all possible k-SAT clauses with edges connecting every two non-conflicting clauses. We prove that a k-SAT instance I is satisfiable if and only if there is a maximal clique of the clause graph that does not intersect I . Key Words: computational complexity, 3-SAT, clause table, polynomial time, P = NP. A.M.S.-Classification: 03D15, 03D55, 03D80. 1 Introduction Matthias M¨ uller published on the net a C++ code and a text describing the implemented algo- rithm, see [1] (started in December 2013 and updated some times). He claimed that the algorithm ”maybe” solves the NP-complete problem 3-SAT in polynomial time. All objects and a lot of the ideas below are suggested by Matthias M¨ uller’s program and companion text. However, objects and ideas are presented by him only intuitively, and proofs are reduced to arguments on some examples. I consider that his approach is more credible if it is described in a better founded mathematical language and with rigorous proofs. This is the goal of the present paper. In order to achieve this goal, we introduce the graph of all possible k-SAT instances with edges connecting every two non-conflicting clauses. We prove that a k-SAT instance I is satisfiable if and only if there is a maximal clique of the clause graph that does not intersect I . 2 General definitions The problem we are concerned here is k-SAT with exactly k variables per clause, seen as decision problem. The most important case is 3-SAT, because for k = 3 it is known that the corresponding problem is NP-complete. See [2]. Definition 2.1 An instance of k-SAT is a formula written in conjunctive normal form with exactly k different variables in every disjunctive clause: I = m  i=1 (ǫ i,1 x i,1 ∨ ǫ i,2 x i,2 ∨···∨ ǫ i,k x i,k ), where k ≥ 1 and for all i =1,...,m, the variables x i,1 ,x i,2 ,...,x i,k are pairwise distinct, and every ǫ i,j means a negation or its absence. In fact, it is useful to consider that ǫ i,j ∈{0, 1} such * Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit 5, P. O. Box 1-764, RO- 014700 Bucharest, Romania. mihai.prunescu@imar.ro, mihai.prunescu@gmail.com 1