Default Logic and Bounded Treewidth

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

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More on Modal Aspects of Default Logic1

Fundamenta Informaticae ◽

10.3233/fi-1992-171-207 ◽

1992 ◽

Vol 17 (1-2) ◽

pp. 99-116

Author(s):

V. Wiktor Marek ◽

Miroslaw Truszczynski

Keyword(s):

Modal Logic ◽

Default Logic ◽

Nonmonotonic Logic ◽

Default Reasoning ◽

Default Theory ◽

Autoepistemic Logic ◽

The Relationship ◽

Natural Classes ◽

Weak Extension

Investigations of default logic have been so far mostly concerned with the notion of an extension of a default theory. It turns out, however, that default logic is much richer. Namely, there are other natural classes of objects that might be associated with default reasoning. We study two such classes of objects with emphasis on their relations with modal nonmonotonic formalisms. First, we introduce the concept of a weak extension and study its properties. It has long been suspected that there are close connections between default and autoepistemic logics. The notion of weak extension allows us to precisely describe the relationship between these two formalisms. In particular, we show that default logic with weak extensions is essentially equivalent to autoepistemic logic, that is, nonmonotonic logic KD45. In the paper we also study the notion of a set of formulas closed under a default theory. These objects are shown to correspond to stable theories and to modal logic S5. In particular, we show that skeptical reasoning with sets closed under default theories is closely related with provability in S5. As an application of our results we determine the complexity of reasoning with weak extensions and sets closed under default theories.

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Default Logic Models of Certain Inheritance Systems with Exceptions

Fundamenta Informaticae ◽

10.3233/fi-1993-182-404 ◽

1993 ◽

Vol 18 (2-4) ◽

pp. 129-149

Author(s):

Serge Garlatti

Keyword(s):

Hierarchical Structure ◽

Formal Semantics ◽

Sufficient Conditions ◽

Nonmonotonic Reasoning ◽

Default Logic ◽

Representation Systems ◽

The Hierarchical Structure ◽

Necessary And Sufficient ◽

Structure Of Knowledge ◽

Made In

Representation systems based on inheritance networks are founded on the hierarchical structure of knowledge. Such representation is composed of a set of objects and a set of is-a links between nodes. Objects are generally defined by means of a set of properties. An inheritance mechanism enables us to share properties across the hierarchy, called an inheritance graph. It is often difficult, even impossible to define classes by means of a set of necessary and sufficient conditions. For this reason, exceptions must be allowed and they induce nonmonotonic reasoning. Many researchers have used default logic to give them formal semantics and to define sound inferences. In this paper, we propose a survey of the different models of nonmonotonic inheritance systems by means of default logic. A comparison between default theories and inheritance mechanisms is made. In conclusion, the ability of default logic to take some inheritance mechanisms into account is discussed.

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Complete-Subgraph-Transversal-Sets problem on bounded treewidth graphs

Journal of Combinatorial Optimization ◽

10.1007/s10878-021-00703-7 ◽

2021 ◽

Author(s):

Ke Liu ◽

Mei Lu

Keyword(s):

Complete Subgraph ◽

Bounded Treewidth

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C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

Algorithmica ◽

10.1007/s00453-021-00839-2 ◽

2021 ◽

Author(s):

Giordano Da Lozzo ◽

David Eppstein ◽

Michael T. Goodrich ◽

Siddharth Gupta

Keyword(s):

Dual Graph ◽

Graph Visualization ◽

Closed Disk ◽

Running Time ◽

Planar Embedding ◽

Bounded Treewidth ◽

Fpt Algorithm ◽

Vertex Set ◽

The One ◽

Clustered Planarity

AbstractFor a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that (1) the subgraph induced by each cluster is drawn in the interior of the corresponding disk, (2) each edge intersects any disk at most once, and (3) the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA’95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA’20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT (resp., XP) algorithm for embedded flat (resp., non-flat) clustered graphs, when parameterized by the carving-width of the dual graph of the input. These are the first FPT and XP algorithms for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. In particular, our algorithm runs in quadratic time for flat instances of bounded treewidth and bounded face size. To further strengthen the relevance of this result, we show that an algorithm with running time O(r(n)) for flat instances whose underlying graph has pathwidth 1 would result in an algorithm with running time O(r(n)) for flat instances and with running time $$O(r(n^2) + n^2)$$ O ( r ( n 2 ) + n 2 ) for general, possibly non-flat, instances.

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