Exploiting Treewidth for Projected Model Counting and Its Limits

Fichte, Johannes Klaus; Hecher, Markus; Morak, Michael; Woltran, Stefan

doi:10.1007/978-3-319-94144-8_11

Cited by 21 publications

(39 citation statements)

References 34 publications

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“…Assume for proof by contradiction that there is such an algorithm. We show that this contradicts a very recent result [21,35], which states that one cannot decide the validity of a QBF ∀V 1…”

Section: Dynamic Programming For #Pascontrasting

confidence: 89%

“…Next, we provide a notion to reconstruct answer sets from a computed TTD, which allows for computing for a given row its predecessor rows in the corresponding child tables, c.f., [21]. Let Π be a program, T = (T, χ, τ ) be an A-TTD of G Π , and t be a node of T where children(t, T ) = t 1 , .…”

Section: Dynamic Programming On Tdsmentioning

confidence: 99%

“…However, we aim for the projected answer sets count (pasc), whose computation requires a few additional definitions. Therefore, we can simply widen definitions from very recent work [21].…”

Section: Dynamic Programming For #Pasmentioning

confidence: 99%

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Treewidth and Counting Projected Answer Sets

Fichte

Hecher

2019

Logic Programming and Nonmonotonic Reasoning

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In this paper, we introduce novel algorithms to solve projected answer set counting (#PAs). #PAs asks to count the number of answer sets with respect to a given set of projected atoms, where multiple answer sets that are identical when restricted to the projected atoms count as only one projected answer set. Our algorithms exploit small treewidth of the primal graph of the input instance by dynamic programming (DP).We establish a new algorithm for head-cycle-free (HCF) programs and lift very recent results from projected model counting to #PAs when the input is restricted to HCF programs. Further, we show how established DP algorithms for tight, normal, and disjunctive answer set programs can be extended to solve #PAs. Our algorithms run in polynomial time while requiring double exponential time in the treewidth for tight, normal, and HCF programs, and triple exponential time for disjunctive programs.Finally, we take the exponential time hypothesis (ETH) into account and establish lower bounds of bounded treewidth algorithms for #PAs. Under ETH, one cannot significantly improve our obtained worst-case runtimes. * This work extends an abstract [18] explaining only concepts, and a preliminary workshop paper [17], and has beento count the answer sets of a disjunctive program with respect to a given set of projected atoms (#PAs). Particularly, multiple answer sets that are identical when reduced to the projected atoms are considered as only one solution. Intuitively, #PAs is needed to count answer sets without counting functionally independent auxiliary atoms. Under standard assumptions the problem #PAs is complete for the class #·Σ 2 P . However, if we take all atoms as projected, then #PAs is again #·coNP-complete and if there are no projected atoms then it is simply Σ p 2 -complete. But some fragments of ASP have lower complexity. A prominent example is the class of head-cycle-free (HCF) programs [4], which requires the absence of cycles in a certain graph representation of the program. Deciding whether a HCF program has an answer set is NP-complete.A way to solve computationally hard problems is to employ parameterized algorithmics [12], which exploits certain structural restrictions in a given input instance. Because structural properties of an input instance often allow for algorithms that solve problems in polynomial time in the size of the input and exponential time in a measure of the structure, whereas under standard assumptions an efficient algorithm is not possible if we consider only the size of the input. In this paper, we consider the treewidth of a graph representation associated with the given input program as structural restriction, namely the treewidth of the primal graph [30]. Generally speaking, treewidth 1 measures the closeness of a graph to a tree, based on the observation that problems on trees are often easier to solve than on arbitrary graphs.Our results are as follows: We establish the classical complexity of #PAs and a novel algorithm that solves ASP problems by exploiting treewidth when ...

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Section: Dynamic Programming For #Pascontrasting

confidence: 89%

Section: Dynamic Programming On Tdsmentioning

confidence: 99%

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Treewidth and Counting Projected Answer Sets

Fichte

Hecher

2019

Logic Programming and Nonmonotonic Reasoning

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“…We presented a generic system dpdb for explicitly exploiting treewidth by means of dynamic programming on databases. The idea of dpdb is to use database solver 0-20 21 management systems (DBMS) for table manipulation, which makes it (1) easy and elegant to perform rapid prototyping for problems, and (2) allows to leverage from decades of database theory and database system tuning. It turned out that all the cases that occur in dynamic programming can be handled quite elegantly with plain SQL queries.…”

Section: Final Discussion and Conclusionmentioning

confidence: 99%

“…Another interesting research direction is to study whether efficient data representation techniques on DBMS can be combined with dynamic programming in order to lift our solver to quantified Boolean formulas. Finally, we are also interested in extending this work to projected model counting [21].…”

Section: Final Discussion and Conclusionmentioning

confidence: 99%

Exploiting Database Management Systems and Treewidth for Counting

Fichte

Hecher

Thier

et al. 2020

Lecture Notes in Computer Science

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Bounded treewidth is one of the most cited combinatorial invariants, which was applied in the literature for solving several counting problems efficiently. A canonical counting problem is #Sat, which asks to count the satisfying assignments of a Boolean formula. Recent work shows that benchmarking instances for #Sat often have reasonably small treewidth. This paper deals with counting problems for instances of small treewidth. We introduce a general framework to solve counting questions based on state-of-the-art database management systems (DBMS). Our framework takes explicitly advantage of small treewidth by solving instances using dynamic programming (DP) on tree decompositions (TD). Therefore, we implement the concept of DP into a DBMS (PostgreSQL), since DP algorithms are already often given in terms of table manipulations in theory. This allows for elegant specifications of DP algorithms and the use of SQL to manipulate records and tables, which gives us a natural approach to bring DP algorithms into practice. To the best of our knowledge, we present the first approach to employ a DBMS for algorithms on TDs. A key advantage of our approach is that DBMS naturally allow to deal with huge tables with a limited amount of main memory (RAM), parallelization, as well as suspending computation.

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