Computing secure sets in graphs using answer set programming

Abseher, Michael; Bliem, Bernhard; Charwat, Günther; Dusberger, Frederico; Woltran, Stefan

doi:10.1093/logcom/exv060

Cited by 6 publications

(18 citation statements)

References 14 publications

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“…However, non-convex aggregations may arise in several contexts while modeling complex knowledge (Eiter et al 2008;Eiter et al 2012;Abseher et al 2014). A minimalistic example is provided by the Σ P 2 -complete problem called Generalized Subset Sum (Berman et al 2002), where two vectors u and v of integers as well as an integer b are given, and the task is to decide whether the formula ∃x∀y(ux + vy = b) is true, where x and y are vectors of binary variables of the same length as u or v, respectively.…”

Section: Introductionmentioning

confidence: 99%

Rewriting recursive aggregates in answer set programming: back to monotonicity

Alviano

Faber

Gebser

2015

Theory and Practice of Logic Programming

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Aggregation functions are widely used in answer set programming for representing and reasoning on knowledge involving sets of objects collectively. Current implementations simplify the structure of programs in order to optimize the overall performance. In particular, aggregates are rewritten into simpler forms known as monotone aggregates. Since the evaluation of normal programs with monotone aggregates is in general on a lower complexity level than the evaluation of normal programs with arbitrary aggregates, any faithful translation function must introduce disjunction in rule heads in some cases. However, no function of this kind is known. The paper closes this gap by introducing a polynomial, faithful, and modular translation for rewriting common aggregation functions into the simpler form accepted by current solvers. A prototype system allows for experimenting with arbitrary recursive aggregates, which are also supported in the recent version 4.5 of the grounder GRINGO, using the methods presented in this paper.

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Section: Introductionmentioning

confidence: 99%

Rewriting recursive aggregates in answer set programming: back to monotonicity

Alviano

Faber

Gebser

2015

Theory and Practice of Logic Programming

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“…ASP solvers can only process sums of the form (1) in which all numbers are nonnegative integers, and the comparison operator is ≥. This is due to the numeric format encoding the propositional program produced by the grounder.…”

Section: Non-convex Aggregates Eliminationmentioning

confidence: 99%

“…However, non-convex aggregations may arise in several contexts while modeling complex knowledge [1,11,13], and there are also minimalistic examples that are easily encoded in ASP using recursive non-convex aggregates, while alternative encodings not using aggregates are not so obvious. One of such examples is provided by the Σ P 2complete problem called Generalized Subset Sum [6].…”

Section: Introductionmentioning

confidence: 99%

“…In this problem, two vectors u and v of integers as well as an integer b are given, and the task is to decide whether the formula ∃x∀y(ux + vy = b) is true, where x and y are vectors of binary variables of the same length as u and v, respectively. For example, for u = [1,2], v = [2,3], and b = 5, the task is to decide whether the following formula is true:…”

Section: Introductionmentioning

confidence: 99%

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Evaluating Answer Set Programming with Non-Convex Recursive Aggregates

Alviano

2016

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Aggregation functions are widely used in answer set programming (ASP) for representing and reasoning on knowledge involving sets of objects collectively. These sets may also depend recursively on the results of the aggregation functions, even if so far the support for such recursive aggregations was quite limited in ASP systems. In fact, recursion over aggregates was restricted to convex aggregates, i.e., aggregates that may have only one transition from false to true, and one from true to false, in this specific order. Recently, such a restriction has been overcome, so that the user can finally use non-convex recursive aggregates in ASP programs, either on purpose or accidentally. A preliminary evaluation of ASP programs with non-convex recursive aggregates is reported in this paper.

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“…Unfortunately, the exact complexity of this problem has so far remained unresolved. This is an unsatisfactory state of affairs because it leaves the possibility open that existing approaches for solving the problem (e.g., [1]) are suboptimal in that they employ unnecessarily powerful programming techniques. Hence we require a precise complexity-theoretic classification of the problem.…”

Section: Introductionmentioning

confidence: 99%

Complexity of Secure Sets

Bliem

Woltran

2017

Algorithmica

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A secure set S in a graph is defined as a set of vertices such that for any the majority of vertices in the neighborhood of X belongs to S. It is known that deciding whether a set S is secure in a graph is -complete. However, it is still open how this result contributes to the actual complexity of deciding whether for a given graph G and integer k, a non-empty secure set for G of size at most k exists. In this work, we pinpoint the complexity of this problem by showing that it is -complete. Furthermore, the problem has so far not been subject to a parameterized complexity analysis that considers structural parameters. In the present work, we prove that the problem is -hard when parameterized by treewidth. This is surprising since the problem is known to be FPT when parameterized by solution size and “subset problems” that satisfy this property usually tend to be FPT for bounded treewidth as well. Finally, we give an upper bound by showing membership in , and we provide a positive result in the form of an FPT algorithm for checking whether a given set is secure on graphs of bounded treewidth.

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Computing secure sets in graphs using answer set programming

Cited by 6 publications

References 14 publications

Rewriting recursive aggregates in answer set programming: back to monotonicity

Rewriting recursive aggregates in answer set programming: back to monotonicity

Evaluating Answer Set Programming with Non-Convex Recursive Aggregates

Complexity of Secure Sets

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