The complexity of Boolean functions
The complexity of Boolean functions
The subchromatic number of a graph
Discrete Mathematics - Graph colouring and variations
Learning read-once formulas with queries
Journal of the ACM (JACM)
On generating the irredundant conjunctive and disjunctive normal forms of monotone Boolean functions
Discrete Applied Mathematics - Special issue on the satisfiability problem and Boolean functions
A characterization of span program size and improved lower bounds for monotone span programs
Computational Complexity
Theoretical Computer Science
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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 驴驴驴-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 驴{0,1} within a factor of $\mathcal{O}(n)$ . 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.