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

  • Takayuki Sato
  • Kazuyuki Amano
  • Eiji Takimoto
  • Akira Maruoka
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5092)


We consider the following extremal problem: Given three natural numbers n, m and l, what is the monotone DNF formula that has a minimal or maximal number of satisfying assignments over all monotone DNF formulas on n variables with m terms 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 C 3 and one \(C_{3+(n \mbox{\scriptsize \ mod\ } 3)}\), where C k denotes a cycle of length k.


Connected Graph Disjunctive Normal Form Satisfying Assignment Complete Subgraph Disjoint Copy 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Takayuki Sato
    • 1
  • Kazuyuki Amano
    • 2
  • Eiji Takimoto
    • 3
  • Akira Maruoka
    • 4
  1. 1.Dept. of Information EngineeringSendai National College of TechnologyAyashi, AobaJapan
  2. 2.Department of Computer ScienceGunma UniversityKiryuJapan
  3. 3.Graduate School of Information SciencesTohoku UniversityAramakiJapan
  4. 4.Dept. of Information Technology and ElectronicsIshinomaki Senshu University 

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