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The stable roommates problem with ties

JOURNAL OF algorithmS 2002年第1期43卷 85-105页

作者： Irving, RW Manlove, DF Univ Glasgow Dept Comp Sci Glasgow G12 8QQ Lanark Scotland

We study the variant of the well-known stable roommates problem in which participants are permitted to express ties in their preference lists. In this setting, more than one definition of stability is possible. Here we consider two of these stability criteria, so-called super-stability and weak stability. We present a linear-time algorithm for finding a super-stable matching if one exists, given a stable roommates instance with ties. This contrasts with the known NP-hardness of the analogous problem under weak stability. We also extend our algorithm to cope with preference lists that are incomplete and/or partially ordered. On the other hand, for a given stable roommates instance with ties and incomplete lists, we show that the weakly stable matchings may be of different sizes and the problem of finding a maximum cardinality weakly stable matching is NP-hard, though approximable within a factor of 2. (C) 2002 Elsevier Science (USA).

关键词： stable matching problem indifference super-stability weak stability linear-time algorithm NP-completeness approximation algorithm

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A note on the subgraphs of the (2 x ∞)-grid

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DISCRETE MATHEMATICS 2010年第3期310卷 531-536页

作者： Diaz, Josep Kaminski, Marcin Thilikos, Dimitrios M. Univ Politecn Cataluna Dept Llenguatges & Sistemes Informat ES-08034 Barcelona Spain Univ Libre Bruxelles Dept Comp Sci Brussels Belgium Univ Athens Dept Math GR-15784 Athens Greece

We give a linear-time algorithm checking whether a graph is a subgraph of the (2 x k)-grid for some value of k. Our algorithm is based on a detailed characterization of the structure of such graphs. (C) 2009 Elsevier ... 详细信息

关键词： Partial grids linear-time algorithm Biconnected components

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The algorithmic use of hypertree structure and maximum neighbourhood orderings

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DISCRETE APPLIED MATHEMATICS 1998年第1-3期82卷 43-77页

作者： Brandstadt, A Chepoi, VD Dragan, FF Univ Rostock Fachbereich Informat D-18051 Rostock Germany Moldavian State Univ Dept Math & Cybern Kishinev 277009 Moldova

The use of (generalized) tree structure in graphs is one of the main topics in the field of efficient graph algorithms. The well-known partial k-tree (resp. treewidth) approach belongs to this kind of research and bases on a tree structure of constant-size bounded maximal cliques. Without size bound on the cliques this tree structure of maximal cliques characterizes chordal graphs which are known to be important also in connection with relational database schemes where hypergraphs with tree structure (acyclic hypergraphs) and their elimination orderings (perfect elimination orderings for chordal graphs, Graham-reduction for acyclic hypergraphs) are studied. We consider here graphs with a tree structure which is dual (in the sense of hypergraphs) to that one of chordal graphs (therefore we call these graphs dually chordal). The corresponding vertex elimination orderings of these graphs are the maximum neighbourhood orderings. These orderings were studied recently in several papers and some of the algorithmic consequences of such orderings are given. The aim of this paper is a systematic treatment of the algorithmic use of maximum neighbourhood orderings. These orderings are useful especially for dominating-like problems (including Steiner tree) and distance problems. Many problems efficiently solvable for strongly chordal and doubly chordal graphs remain efficiently solvable for dually chordal graphs too. Our results on dually chordal graphs not only generalize, but also improve and extend the corresponding results on strongly chordal and doubly chordal graphs, since a maximum neighbourhood ordering (if it exists) can be constructed in linear time and we consequently use the underlying structure properties of dually chordal graphs closely connected to hypergraphs. Furthermore, a collection of problems remaining NP-complete on dually chordal graphs is given. (C) 1998 Elsevier Science B.V. All rights reserved.

关键词： tree structure hypertree disk hypergraph dually chordal graph strongly chordal graph maximum neighbourhood ordering domination duality location problem Steiner tree linear-time algorithm

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Sharp threshold for embedding balanced spanning trees in random geometric graphs

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JOURNAL OF GRAPH THEORY 2024年第1期107卷 107-125页

作者： Diaz, Alberto Espuny Lichev, Lyuben Mitsche, Dieter Wesolek, Alexandra Tech Univ Ilmenau Inst Math Ilmenau Germany Univ Jean Monnet Inst Camille Jordan St Etienne France Pont Univ Catolica IMC Santiago Chile Simon Fraser Univ Dept Math Burnaby BC Canada

A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph G(n,r,d) ${\mathscr{G}}(n,r,d)$. In particular, we find the sharp threshold for balanced binary trees. More generally, we show that all sequences of balanced trees with uniformly bounded degrees and height tending to infinity appear above a sharp threshold, and none of these appears below the same value. Our results hold more generally for geometric graphs satisfying a mild condition on the distribution of their vertex set, and we provide a polynomial time algorithm to find such trees.

关键词： linear-time algorithm random geometric graphs sharp threshold spanning trees

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A generic model decomposition technique and its application to the Eclipse modeling framework

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SOFTWARE AND SYSTEMS MODELING 2015年第2期14卷 921-952页

作者： Ma, Qin Kelsen, Pierre Glodt, Christian Univ Luxembourg FSTC Richard Coudenhove Kaler Luxembourg Univ Luxembourg SnT Richard Coudenhove Kaler Luxembourg

Model-driven software development aims at easing the process of software development by using models as primary artifacts. Although less complex than the real systems, they are based on models tend to be complex nevertheless, thus making the task of handling them non-trivial in many cases. In this paper, we propose a generic model decomposition technique to facilitate model management by decomposing complex models into smaller sub-models that conform to the same metamodel as the original model. The technique is based upon a formal foundation that consists of a formal capturing of the concepts of models, metamodels, and model conformance;a formal constraint language based on EssentialOCL;and a set of formally proved properties of the technique. We organize the decomposed sub-models in a mathematical structure as a lattice, and design a linear-time algorithm for constructing this decomposition. The generic model decomposition technique is applied to the Eclipse modeling framework, and the result is used to build a solution to a specific model comprehension problem of Ecore models based upon model pruning. We report two case studies of the model comprehension method: one in BPMN and the other in fUML.

关键词： MDE EMF Model decomposition Model comprehension linear-time algorithm Sub-model lattice OCL EssentialOCL BPMN fUML

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Note: An algorithm for contour-based region filling

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COMPUTERS & GRAPHICS-UK 2005年第3期29卷 441-450页

作者： Codrea, MC Nevalainen, OS Univ Turku Turku Ctr Comp Sci FIN-20520 Turku Finland Univ Turku Dept Informat Technol FIN-20520 Turku Finland

A linear-time algorithm for filling regions defined by closed contours in raster format is proposed. The algorithm relies on a single pass contour labeling and the actual filling is done in a scan-line manner, visiting the interior pixels only once. The interior endpoints of the scan-lines are discovered solely on the basis of the labeled contour. Despite its simplicity and low computation cost, the proposed method fills-in arbitrary shapes correctly, with no obvious exceptions. Possible special cases may arise from the 8-connectivity, inner contour detection. However, in extensive tests on complicated images, the proposed labeling scheme has given correct results. (c) 2005 Elsevier Ltd. All rights reserved.

关键词： Raster region filling linear-time algorithm contour labeling scan-line method

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An O(n) algorithm for projecting a vector on the intersection of a hyperplane and a box in Rⁿ

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2003年第3期117卷 553-574页

作者： Maculan, N Santiago, CP Macambira, EM Jardim, MHC Univ Fed Rio de Janeiro COPPE PESC BR-21945 Rio De Janeiro Brazil Univ Fed Rio de Janeiro IM DCC Rio De Janeiro Brazil

We consider the problem of projecting a vector on the intersection of a hyperplane and a box in R-n. This paper extends a previous result of Maculan, Minoux, and Plateau (Ref. 1) concerning the projection of a vector on the intersection of a hyperplane and R-+(n). We present an O(n) time algorithm based on the linear-time median-finding algorithm. This algorithm determines the median of the components of the vector to be projected. Computational results are also presented in order to evaluate the algorithm and its time complexity. We consider two sets of instances which are randomly generated for any given n. The algorithm was successful in solving all the instances in a reasonable time.

关键词： orthogonal projections linear-time algorithm computational linear algebra subgradient methods

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Hardness and algorithms of equitable tree-coloring problem in chordal graphs

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THEORETICAL COMPUTER SCIENCE 2021年 857卷 8-15页

作者： Niu, Bei Li, Bi Zhang, Xin Xidian Univ Sch Math & Stat Xian 710071 Peoples R China

An equitable tree-k-coloring of a graph is a vertex k-coloring such that each color class induces a forest and the size of any two color classes differs by at most one. In this work, we show that every interval graph G has an equitable tree-k-coloring for any integer k >= (Delta(G) + 1)/2, solving a conjecture of Wu, Zhang and Li (2013) for interval graphs, and furthermore, give a linear-time algorithm for determining whether a proper interval graph admits an equitable tree -k-coloring for a given integer k. For disjoint union of split graphs, or K-1,K-r-free interval graphs with r >= 4, we prove that it is W[1]-hard to decide whether there is an equitable tree -k-coloring when parameterized by number of colors, or by treewidth, number of colors and maximum degree, respectively. On the positive side, we propose a quadratic 2-approximation algorithm for the equitable tree-coloring problem in chordal graphs. Moreover, it is proved that there is no alpha-approximation algorithm for any alpha < 3/2 . (c) 2020 Elsevier B.V. All rights reserved.

关键词： W[1]-hardness linear-time algorithm Approximation algorithm Equitable tree-coloring Chordal graphs

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Hierarchical Gaussian Markov Random Field for Image Denoising

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IEICE TRANSACTIONS ON INFORMATION AND SYSTEMS 2022年第3期E105D卷 689-699页

作者： Monma, Yuki Aro, Kan Yasuda, Muneki Yamagata Univ Grad Sch Sci & Engn Yonezawa Yamagata 9928510 Japan Diamond Head Co Ltd Sapporo Hokkaido 0600001 Japan

In this study, Bayesian image denoising, in which the prior distribution is assumed to be a Gaussian Markov random field (GMRF), is considered. Recently, an effective algorithm for Bayesian image denoising with a standard GMRF prior has been proposed, which can help implement the overall procedure and optimize its parameters in O(n)-time, where n is the size of the image. A new GMRF-type prior, referred to as a hierarchical GMRF (HGMRF) prior, is proposed, which is obtained by applying a hierarchical Bayesian approach to the standard GMRF prior;in addition, an effective denoising algorithm based on the HGMRF prior is proposed. The proposed HGMRF method can help implement the overall procedure and optimize its parameters in 0(n)-time, as well as the previous GMRF method. The restoration quality of the proposed method is found to be significantly higher than that of the previous GMRF method as well as that of a non-local means filter in several cases. Furthermore, numerical evidence implies that the proposed HGMRF prior is more suitable for the image prior than the standard GMRF prior.

关键词： Bayesian image denoising Gaussian Markov random field hierarchical Bayes method linear-time algorithm

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Efficient image sharpening and denoising using adaptive guided image filtering

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IET IMAGE PROCESSING 2015年第1期9卷 71-79页

作者： Cuong Cao Pham Jeon, Jae Wook Sungkyunkwan Univ Coll Informat & Commun Engn Suwon South Korea

Enhancing the sharpness and reducing the noise of blurred, noisy images are crucial functions of image processing. Widely used unsharp masking filter-based approaches suffer from halo-artefacts and/or noise amplification, while noise-and halo-free adaptive bilateral filtering (ABF) is computationally intractable. In this study, the authors present an efficient sharpening algorithm inspired by guided image filtering (GF). The author's proposed adaptive GF (AGF) integrates the shift-variant technique, a part of ABF, into a guided filter to render crisp and sharpened outputs. Experiments showed the superiority of their proposed algorithm to existing algorithms. The proposed AGF sharply enhances edges and textures without causing halo-artefacts or noise amplification, and it is efficiently implemented using a fast linear-time algorithm.

关键词： image denoising edge detection image enhancement image texture adaptive filters interference suppression rendering (computer graphics) image sharpening enhancement image denoising image noise reduction image processing unsharp masking filter-based approach halo artefact noise amplification halo free ABF adaptive bilateral filtering guided image filtering adaptive GF shift variant technique crisp output rendering sharpened output rendering AGF edge enhancement texture enhancement linear-time algorithm

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