Abstract
Given a CNF formula Φ with clauses C1,…,Cm and variables V={x1,…,xn}, a truth assignment a:V→{0,1} of Φ leads to a clause sequence σΦ(a)=(C1(a),…,Cm(a))∈{0,1}m where Ci(a)=1 if clause Ci evaluates to 1 under assignment a, otherwise Ci(a)=0. The set of all possible clause sequences carries a lot of information on the formula, e.g. SAT, MAX-SAT and MIN-SAT can be encoded in terms of finding a clause sequence with extremal properties.We consider a problem posed at Dagstuhl Seminar 19211 “Enumeration in Data Management” (2019) about the generation of all possible clause sequences of a given CNF with bounded dimension. We prove that the problem can be solved in incremental polynomial time. We further give an algorithm with polynomial delay for the class of tractable CNF formulas. We also consider the generation of maximal and minimal clause sequences, and show that generating maximal clause sequences is NP-hard, while minimal clause sequences can be generated with polynomial delay.
Highlights
The concept of well-designed pattern trees was introduced by Letelier et al [1] as a convenient graphic representation of conjunctive queries extended by the optional operator
In this paper we show that all signatures of a given CNF with a bounded dimension can be generated in incremental polynomial time, answering an open problem posed by Kröll [8, Problem 4.7]
A faster incremental polynomial algorithm is provided for the class of formulas where both the dimension and the occurrence are bounded
Summary
The concept of well-designed pattern trees was introduced by Letelier et al [1] as a convenient graphic representation of conjunctive queries extended by the optional operator. Previous work The generation problem was studied for First-Order and Conjunctive Queries [2,3,4,5] and for well-designed pattern trees [1]. In [6], the complexity of the generation problem for the class of well-designed pattern trees falling globally in the class of queries of treewidth at most k and having c-semi-bounded interface was left open (see [6, Table 1 on page 16]).
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