Monotone DNF formula that has a minimal or maximal number of satisfying assignments

Takayuki Sato, Kazuyuki Amano, Eiji Takimoto, Akira Maruoka

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

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 copies of C 3 and one , where C k denotes a cycle of length k.

Original languageEnglish
Title of host publicationComputing and Combinatorics - 14th Annual International Conference, COCOON 2008, Proceedings
Pages191-203
Number of pages13
DOIs
Publication statusPublished - Aug 4 2008
Externally publishedYes
Event14th Annual International Conference on Computing and Combinatorics, COCOON 2008 - Dalian, China
Duration: Jun 27 2008Jun 29 2008

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5092 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other14th Annual International Conference on Computing and Combinatorics, COCOON 2008
CountryChina
CityDalian
Period6/27/086/29/08

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All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Sato, T., Amano, K., Takimoto, E., & Maruoka, A. (2008). Monotone DNF formula that has a minimal or maximal number of satisfying assignments. In Computing and Combinatorics - 14th Annual International Conference, COCOON 2008, Proceedings (pp. 191-203). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5092 LNCS). https://doi.org/10.1007/978-3-540-69733-6_20