Monotone DNF Formula That Has a Minimal or Maximal Number of Satisfying Assignments

  • Authors:

  • Takayuki Sato;Kazuyuki Amano;Eiji Takimoto;Akira Maruoka

  • Affiliations:

  • Dept. of Information Engineering, Sendai National College of Technology, Ayashi, Aoba, Japan 989-3128;Department of Computer Science, Gunma University, Kiryu, Japan 376-8515;Graduate School of Information Sciences, Tohoku University, Aramaki, Japan 980-8579;Dept. of Information Technology and Electronics, Ishinomaki Senshu University,

  • Venue:
  • COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
  • Year:
  • 2008

Quantified Score

Hi-index 0.00

Visualization

Abstract

We consider the following extremal problem: Given three natural numbers n, mand l, what is the monotone DNF formula that has a minimal or maximal number of satisfying assignments over all monotone DNF formulas on nvariables with mterms each of length l? We first show that the solution to the minimization problem can be obtained by the Kruskal-Katona theorem developed in extremal set theory. We also give a simple procedure that outputs an optimal formula for the more general problem that allows the lengths of terms to be mixed. We then show that the solution to the maximization problem can be obtained using the result of Bollobás on the number of complete subgraphs when l= 2 and the pair (n,m) satisfies a certain condition. Moreover, we give the complete solution to the problem for the case l= 2 and m≤ n, which cannot be solved by direct application of Bollobás's result. For example, when n= m, an optimal formula is represented by a graph consisting of $\lfloor{n/3}\rfloor-1$ copies of C3and one $C_{3+(n \mbox{\scriptsize \ mod\ } 3)}$, where Ckdenotes a cycle of length k.