On the readability of monotone Boolean formulae

Golumbic et al. (Discrete Appl. Math. 154:1465–1477, 2006 ) defined the readability of a monotone Boolean function f to be the minimum integer k such that there exists an ∧ − ∨ -formula equivalent to f in which each variable appears at most k times. They asked whether there exists a polynomial-time...

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Veröffentlicht in:Journal of combinatorial optimization Jg. 22; H. 3; S. 293 - 304
Hauptverfasser: Elbassioni, Khaled, Makino, Kazuhisa, Rauf, Imran
Format: Journal Article Tagungsbericht
Sprache:Englisch
Veröffentlicht: Boston Springer US 01.10.2011
Springer Science + Business Media
Springer
Schlagworte:
Applied sciences
Combinatorics
Computer science; control theory; systems
Convex and Discrete Geometry
Exact sciences and technology
Logical, boolean and switching functions
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Theoretical computing
Theory of Computation
Complexity of monotone Boolean functions
Readability
Read-once functions
Monotone Boolean functions
Logarithmic function
Boolean function
Factorization
Legibility
Polynomial time
Monotone Boolean function
Upper bound
Boolean logic
Disjunctive form
NP hard problem
Conjunctive normal form
ISSN:1382-6905, 1573-2886
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Zusammenfassung:Golumbic et al. (Discrete Appl. Math. 154:1465–1477, 2006 ) defined the readability of a monotone Boolean function f to be the minimum integer k such that there exists an ∧ − ∨ -formula equivalent to f in which each variable appears at most k times. They asked whether there exists a polynomial-time algorithm, which given a monotone Boolean function f , in CNF or DNF form, checks whether f is a read- k function, for a fixed k . In this paper, we partially answer this question already for k =2 by showing that it is NP-hard to decide if a given monotone formula represents a read-twice function. It follows also from our reduction that it is NP-hard to approximate the readability of a given monotone Boolean function f :{0,1} n →{0,1} within a factor of . We also give tight sublinear upper bounds on the readability of a monotone Boolean function given in CNF (or DNF) form, parameterized by the number of terms in the CNF and the maximum size in each term, or more generally the maximum number of variables in the intersection of any constant number of terms. When the variables of the DNF can be ordered so that each term consists of a set of consecutive variables, we give much tighter logarithmic bounds on the readability.
ISSN:1382-6905
1573-2886
DOI:10.1007/s10878-009-9283-0