TY - GEN

T1 - Monotone DNF formula that has a minimal or maximal number of satisfying assignments

AU - Sato, Takayuki

AU - Amano, Kazuyuki

AU - Takimoto, Eiji

AU - Maruoka, Akira

N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

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U2 - 10.1007/978-3-540-69733-6_20

DO - 10.1007/978-3-540-69733-6_20

M3 - Conference contribution

AN - SCOPUS:48249128688

SN - 3540697322

SN - 9783540697329

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 191

EP - 203

BT - Computing and Combinatorics - 14th Annual International Conference, COCOON 2008, Proceedings

T2 - 14th Annual International Conference on Computing and Combinatorics, COCOON 2008

Y2 - 27 June 2008 through 29 June 2008

ER -