Treewidth and Minimum Fill-in on d-Trapezoid Graphs

Bodlaender, HL Hans; Kloks, Ajj Ton; Kratsch, Dieter; Müller, Haiko

doi:10.1142/9789812777638_0008

Cited by 11 publications

(6 citation statements)

References 40 publications

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“…To deal with this difficulty of computation, the domain has developed a number of approaches. This includes approximation [45], restricted input [8,9,12,38,39,44], parameterization [13,22,35,43,54] and inclusion-minimal completions. In the latter approach, one does not ask for a completion having the minimum number of fill edges but only ask for a set of fill edges which is minimal for inclusion, i.e.…”

Section: Introductionmentioning

confidence: 99%

Faster and Enhanced Inclusion-Minimal Cograph Completion

Crespelle

Lokshtanov

Phan

et al. 2017

Combinatorial Optimization and Applications

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We design two incremental algorithms for computing an inclusion-minimal completion of an arbitrary graph into a cograph. The first one is able to do so while providing an additional property which is crucial in practice to obtain inclusion-minimal completions using as few edges as possible : it is able to compute a minimum-cardinality completion of the neighbourhood of the new vertex introduced at each incremental step. It runs in O(n + m ) time, where m is the number of edges in the completed graph. This matches the complexity of the algorithm in [41] and positively answers one of their open questions. Our second algorithm improves the complexity of inclusion-minimal completion to O(n + m log 2 n) when the additional property above is not required. Moreover, we prove that many very sparse graphs, having only O(n) edges, require Ω(n 2 ) edges in any of their cograph completions. For these graphs, which include many of those encountered in applications, the improvement we obtain on the complexity scales as O(n/ log 2 n).

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Section: Introductionmentioning

confidence: 99%

Faster and Enhanced Inclusion-Minimal Cograph Completion

Crespelle

Lokshtanov

Phan

et al. 2017

Combinatorial Optimization and Applications

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“…A large stream of research in the computer science community has focused on the identification of polynomial-time algorithms for structured classes of graphs, such as the works by Chang (1996), Broersma et al (1997), Kloks et al (1998), and Bodlaender et al (1998). For general graphs, Kaplan et al (1999) proposed the first exact fixed-parameter tractable algorithm for the MCCP, and recently Fomin and Villanger (2012) presented a substantially faster procedure with subexponential parameterized time complexity.…”

Section: Introductionmentioning

confidence: 99%

A Benders Approach to the Minimum Chordal Completion Problem

Bergman

Raghunathan

2015

Integration of AI and OR Techniques in Constraint Programming

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A graph is chordal if every cycle of length at least four contains a chord, that is, an edge connecting two nonconsecutive vertices of the cycle. Several classical applications in sparse linear systems, database management, computer vision, and semidefinite programming can be reduced to finding the minimum number of edges to add to a graph so that it becomes chordal, known as the minimum chordal completion problem (MCCP). In this article we propose a new formulation for the MCCP which does not rely on finding perfect elimination orderings of the graph, as has been considered in previous work. We introduce several families of facet-defining inequalities for cycle subgraphs and investigate the underlying separation problems, showing that some key inequalities are NP-Hard to separate. We also show general properties of the proposed polyhedra, indicating certain conditions and methods through which facets and inequalities associated with the polytope of a certain graph can be adapted in order to become valid and eventually facet-defining for some of its subgraphs or supergraphs. Numerical studies combining heuristic separation methods based on a threshold rounding and lazy-constraint generation indicate that our approach substantially outperforms existing methods for the MCCP, solving many benchmark graphs to optimality for the first time.

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“…Chess variants may vary from the standard game in terms of factors like the types of pieces used, the shape and size of the board, and particularly the rules [1,2]. The following provides a succinct description.…”

Section: Introductionmentioning

confidence: 99%

“…And most have been designed by creative people who like to try out new pieces, new rules, or new ideas." [2] Our purpose was to challenge the field of artificial intelligence (AI) with a new Drosophila [3]. One of the problems with standard chess AI today is that computers are already able to play it well enough using efficient search techniques and well-designed heuristics.…”

Section: Introductionmentioning

confidence: 99%

A New Chess Variant for Gaming AI

Iqbal

2013

Entertainment Computing – ICEC 2013

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Abstract. In this article, we describe a newly-invented chess variant called Switch-Side Chain-Chess that is demonstrably more challenging for humans and computers than the standard, international version of the game. A new rule states that players have the choice to switch sides with each other if a continuous link of pieces is created on the board. This simple rule increases significantly the complexity of chess, as perceived by the players, but not the actual size of its game tree. The new variant therefore more easily allows board game researchers to focus on the 'higher level' aspects of intelligence such as perception and intuition without being constrained by a larger search space as they would be if using a game like Go or Arimaa. They can also immediately build upon the tried and tested approaches already being used in strong chess engines instead of having to start from scratch or a lower level of progress as is the case with other games of this type.

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Treewidth and Minimum Fill-in on d-Trapezoid Graphs

Cited by 11 publications

References 40 publications

Faster and Enhanced Inclusion-Minimal Cograph Completion

Faster and Enhanced Inclusion-Minimal Cograph Completion

A Benders Approach to the Minimum Chordal Completion Problem

A New Chess Variant for Gaming AI

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