On the average similarity degree between solutions of random k-SAT and random CSPs

  • Authors:

  • Ke Xu;Wei Li

  • Affiliations:

  • National Laboratory of Software Development Environment, Department of Computer Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, People's Republic of Ch ...;National Laboratory of Software Development Environment, Department of Computer Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, People's Republic of Ch ...

  • Venue:
  • Discrete Applied Mathematics - Discrete mathematics and theoretical computer science (DMTCS)
  • Year:
  • 2004

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Abstract

To study the structure of solutions for random k-SAT and random CSPs, this paper introduces the concept of average similarity degree to characterize how solutions are similar to each other. It is proved that under certain conditions, as r (i.e. the ratio of constraints to variables) increases, the limit of average similarity degree when the number of variables approaches infinity exhibits phase transitions at a threshold point, shifting from a smaller value to a larger value abruptly. For random k-SAT this phenomenon will occur when k ≥ 5. It is further shown that this threshold point is also a singular point with respect to r in the asymptotic estimate of the second moment of the number of solutions. Finally, we discuss how this work is helpful to understand the hardness of solving random instances and a possible application of it to the design of search algorithms.