On Compiling CNFs into Structured Deterministic DNNFs

Bova, Simone; Capelli, Florent; Mengel, Stefan; Slivovsky, Friedrich

doi:10.1007/978-3-319-24318-4_15

Cited by 11 publications

(34 citation statements)

References 10 publications

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“…Golovach et al [18] applied linear mim-width in the context of enumeration algorithms, showing that all minimal dominating sets of graphs of bounded linear mim-width can be enumerated with polynomial delay using polynomial space. The graph parameter mim-width has also been used for knowledge compilation and model counting in the field of satisfiability [7,27,15]. The LC-VSVP problems include the class of domination-type problems known as (σ, ρ)-Domination problems, whose intractability on chordal graphs is well known [6].…”

Section: Introductionmentioning

confidence: 99%

A Width Parameter Useful for Chordal and Co-comparability Graphs

Kang

Kwon

Strømme

et al. 2017

WALCOM: Algorithms and Computation

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Belmonte and Vatshelle (TCS 2013) used mim-width, a graph width parameter bounded on interval graphs and permutation graphs, to explain existing algorithms for many domination-type problems on those graph classes. We investigate new graph classes of bounded mim-width, strictly extending interval graphs and permutation graphs. The graphs K t K t and K t S t are graphs obtained from the disjoint union of two cliques of size t, and one clique of size t and one independent set of size t respectively, by adding a perfect matching. We prove that :• interval graphs are (K 3 S 3 )-free chordal graphs; and (K t S t )-free chordal graphs have mim-width at most t − 1,• permutation graphs are (K 3 K 3 )-free co-comparability graphs; and (K t K t )-free cocomparability graphs have mim-width at most t − 1,• chordal graphs and co-comparability graphs have unbounded mim-width in general.We obtain several algorithmic consequences; for instance, while Minimum Dominating Set is NP-complete on chordal graphs, it can be solved in time n O(t) on (K t S t )-free chordal graphs. The third statement strengthens a result of Belmonte and Vatshelle stating that either those classes do not have constant mim-width or a decomposition with constant mim-width cannot be computed in polynomial time unless P = N P .We generalize these ideas to bigger graph classes. We introduce a new width parameter simwidth, of stronger modelling power than mim-width, by making a small change in the definition of mim-width. We prove that chordal graphs and co-comparability graphs have sim-width at most 1. We investigate a way to bound mim-width for graphs of bounded sim-width by excluding K t K t and K t S t as induced minors or induced subgraphs, and give algorithmic consequences. Lastly, we show that circle graphs have unbounded sim-width, and thus also unbounded mim-width.

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Section: Introductionmentioning

confidence: 99%

A Width Parameter Useful for Chordal and Co-comparability Graphs

Kang

Kwon

Strømme

et al. 2017

WALCOM: Algorithms and Computation

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“…This paper should be seen as complementing the findings of [1]: In that paper, algorithms compiling CNF formulas with restricted underlying graph structure were presented, showing that popular graph width measures like treewidth and cliquewidth can be used in knowledge compilation. More specifically, every CNF formula of incidence treewidth k and size n can be compiled into a DNNF of size 2 O(k) n. Moreover, if k is the incidence cliquewidth, the size bound on the encoding becomes n O(k) .…”

Section: Introductionmentioning

confidence: 61%

“…[14]. Consequently, the results of [1] leave open the question if the algorithm for clique-width based compilation of CNF formulas can be improved.…”

Section: Introductionmentioning

confidence: 99%

Parameterized Compilation Lower Bounds for Restricted CNF-Formulas

Mengel

2016

Theory and Applications of Satisfiability Testing – SAT 2016

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We show unconditional parameterized lower bounds in the area of knowledge compilation, more specifically on the size of circuits in decomposable negation normal form (DNNF) that encode CNF-formulas restricted by several graph width measures. In particular, we show that• there are CNF formulas of size n and modular incidence treewidth k whose smallest DNNF-encoding has size n Ω(k) , and• there are CNF formulas of size n and incidence neighborhood diversity k whose smallest DNNF-encoding has size n Ω( √ k) .These results complement recent upper bounds for compiling CNF into DNNF and strengthen-quantitatively and qualitatively-known conditional lower bounds for cliquewidth. Moreover, they show that, unlike for many graph problems, the parameters considered here behave significantly differently from treewidth.

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“…Pushing the dependency of the compilation size down from m (the size of the given circuit, as in (3)) to n (the number of its inputs, as in (4)) required an entirely new compilation idea. 5 The idea used by Petke and Razgon [31] to obtain (3) was the following. Given a circuit C(X) of n = |X| variables and m = |Z| gates, first compute its Tseitin CNF T (X, Z); the circuit treewidth of the latter is (linearly) related to the former.…”

Section: Discussionmentioning

confidence: 99%

Circuit Treewidth, Sentential Decision, and Query Compilation

Bova

Szeider

2017

Proceedings of the 36th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems

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The evaluation of a query over a probabilistic database boils down to computing the probability of a suitable Boolean function, the lineage of the query over the database. The method of query compilation approaches the task in two stages: first, the query lineage is implemented (compiled) in a circuit form where probability computation is tractable; and second, the desired probability is computed over the compiled circuit. A basic theoretical quest in query compilation is that of identifying pertinent classes of queries whose lineages admit compact representations over increasingly succinct, tractable circuit classes.Fostering previous work by Jha and Suciu [20] and Petke and Razgon [31], we focus on queries whose lineages admit circuit implementations with small treewidth, and investigate their compilability within tame classes of decision diagrams. In perfect analogy with the characterization of bounded circuit pathwidth by bounded OBDD width [20], we show that a class of Boolean functions has bounded circuit treewidth if and only if it has bounded SDD width. Sentential decision diagrams (SDDs) are central in knowledge compilation, being essentially as tractable as OBDDs [13] but exponentially more succinct [4]. By incorporating constant width SDDs and polynomial size SDDs, we refine the panorama of query compilation for unions of conjunctive queries with and without inequalities [19,20]. ContributionIn the first part of the article (Section 3), we show that a class of Boolean functions has bounded circuit treewidth if and only if it has bounded SDD width, which perfectly complements the aforementioned characterization of circuit pathwidth via OBDD width by Jha and Suciu.More precisely, we prove the following (Theorem 4 and surrounding discussion).1 OBDDs are deterministic decomposable circuits. 2 The lower bound holds even for primal treewidth, which is (unboundedly) larger than circuit treewidth. 3 A special case where such a compilation is available is that of lineages of an MSO query over databases of bounded treewidth, which have linear size deterministic decomposable forms [1, If (a, i) gives {z 2i−1 , z 2i } ⊆ {x 2 k ,1 , . . . , x 2 k ,m−2 }, then the corresponding subdisjunction reduces to one disjunct only; and, if (a, i) gives {z 2i−1 , z 2i } = {x 2 k ,m−1 , x 2 k ,m }, then the corresponding subdisjunction reduces to two disjuncts only, as the following example illustrates.Example 6 (k = 2, m = 4). Continuing Example 5, the following lists the one disjunct corresponding to "a orbits on 7": z 13 z 14 z 15 ∧ *

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On Compiling CNFs into Structured Deterministic DNNFs

Cited by 11 publications

References 10 publications

A Width Parameter Useful for Chordal and Co-comparability Graphs

A Width Parameter Useful for Chordal and Co-comparability Graphs

Parameterized Compilation Lower Bounds for Restricted CNF-Formulas

Circuit Treewidth, Sentential Decision, and Query Compilation

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