1. CNRS-Laboratoire de Physique Théorique, 24, Rue Lhomond, 75231 Paris Cedex, France
2. The Abdus Salam International Centre for Theoretical Physics, Strada Costieri 11, 34100 Trieste, Italy
3. IBM, Thomas J. Watson Research Center, Yorktown Heights, New York 10598, USA
4. Computer Science Department, Cornell University, Ithaca, New York 14853, USA
5. Institute of Computer Science, The Hebrew University, Jerusalem 91904, Israel

Correspondence to: Scott Kirkpatrick³ Correspondence and requests for materials should be addressed to S.K.
(e-mail: Email: kirk@watson.ibm.com).

Abstract

Non-deterministic polynomial time (commonly termed 'NP-complete') problems are relevant to many computational tasks of practical interest—such as the 'travelling salesman problem'—but are difficult to solve: the computing time grows exponentially with problem size in the worst case. It has recently been shown that these problems exhibit 'phase boundaries', across which dramatic changes occur in the computational difficulty and solution character—the problems become easier to solve away from the boundary. Here we report an analytic solution and experimental investigation of the phase transition in K -satisfiability, an archetypal NP-complete problem. Depending on the input parameters, the computing time may grow exponentially or polynomially with problem size; in the former case, we observe a discontinuous transition, whereas in the latter case a continuous (second-order) transition is found. The nature of these transitions may explain the differing computational costs, and suggests directions for improving the efficiency of search algorithms. Similar types of transition should occur in other combinatorial problems and in glassy or granular materials, thereby strengthening the link between computational models and properties of physical systems.