Vertices of a Bipartite Graph G and the Coalitions Are the Edges of G, the Stable Marriage Theorem Says There Exists a Stable Allocation
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On Vertex Independence Number of Uniform Hypergraphs
Acta Univ. Sapientiae, Informatica, 6, 1 (2014) 132{158 On vertex independence number of uniform hypergraphs Tariq A. CHISHTI Guofei ZHOU University of Kashmir Nanjing University Department of Mathematics Department of Mathematics Srinagar, India Nanjing, China email: [email protected] email: [email protected] Shariefuddin PIRZADA Antal IVANYI´ University of Kashmir E¨otv¨osLor´andUniversity Department of Mathematics Faculty of Informatics Srinagar, India Budapest, Hungary email: email: [email protected] [email protected] Abstract. Let H be an r-uniform hypergraph with r ≥ 2 and let α(H) be its vertex independence number. In the paper bounds of α(H) are given for different uniform hypergraphs: if H has no isolated vertex, then in terms of the degrees, and for triangle-free linear H in terms of the order and average degree. 1 Introduction to independence in graphs Let n be a positive integer. A graph G on vertex set V = fv1; v2; : : : ; vng is a pair (V; E), where the edge set E is a subset of V × V. n is the order of G and jEj is the size of G. Computing Classification System 1998: G.2.2 Mathematics Subject Classification 2010: 05C30, 05C50 Key words and phrases: uniform hypergraph, independence number, lower bound DOI:10.2478/ausi-2014-0022 132 On vertex independence number of uniform hypergraphs 133 Let v 2 V and N(v) be the neighborhood of v, namely, the set of vertices x so that there is an edge which contains both v and x. Let U be a subset of V, then the subgraph of G induced by U is defined as a graph on vertex set U and edge set EU = f(u; v)ju 2 u and v 2 Ug. -
Qualitative Reasoning for Spatial Matching 1 [email protected] Qualitative Reasoning for Spatial Matching
Qualitative Reasoning for Spatial Matching 1 [email protected] Qualitative Reasoning for Spatial Matching Diedrich Wolter Universit¨atBremen, Transregional Collaborative Research Center SFB/TR 8 Spatial Cognition Abstract. Matching two representations against one another is central to many applications, e.g., matching observation against models in ob- ject or scene recognition or matching a local perception against a global map in self-localization if a mobile robot. In both examples, spatial infor- mation structures are associated, i.e., the representation includes spatial arrangement information of individual parts, either in a relative or an absolute frame of reference. In this paper the utility of qualitative configuration information for matching is investigated. Qualitative configuration information expresses relative arrangement information. It is argued that such information is helpful for making side conditions of plausible matchings explicit, which can improve matching efficiency as well as quality of the matching. Keywords. Matching, Qualitative Spatial Information 1 Introduction Many applications involve the task of matching two representations against one another. For example scene recognition involves identifying a set of spatially distributed objects. Many objects bear rich information which can be exploited the recognition task, but it is information about the spatial arrangement of the involved objects that allows for important distinctions. Consider you are observing a scene that contains dishes, glasses, and sets of cutlery. These objects, when nicely arranged on a table, gives rise to the hypothesis that you are about to enjoy a nice meal, whereas piled up dishes, glasses, and clutter of cutlery suggests dish washing to be your destiny. This article investigates into matching tasks which are involved with match- ing spatially embedded representations, i.e. -
Rainbow Matchings and Transversals∗
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 59(1) (2014), Pages 211–217 Rainbow matchings and transversals∗ Janos´ Barat´ † MTA-ELTE Geometric and Algebraic Combinatorics Research Group 1117 Budapest, P´azm´any P´eter s´et´any 1/C Hungary Ian M. Wanless School of Mathematical Sciences Monash University Clayton, Vic 3800 Australia Abstract We show that there exists a bipartite graph containing n matchings of 2 sizes mi n satisfying i mi = n + n/2−1, such that the matchings have no rainbow matching. This answers a question posed by Aharoni, Charbit and Howard. We also exhibit (n − 1) × n latin rectangles that cannot be decom- posed into transversals, and some related constructions. In the process we answer a question posed by H¨aggkvist and Johansson. Finally, we propose a Hall-type condition for the existence of a rain- bow matching. Two edges in a graph are independent if they do not share an endpoint. A matching is a set of edges that are pairwise independent. If M1,M2,...,Mn are matchings and there exist edges e1 ∈ M1, e2 ∈ M2,...,en ∈ Mn that form a matching then we say that {M1,M2,...,Mn} possesses a rainbow matching.Notethatwedo not require the matchings Mi to be disjoint, so it may even be the case that ej ∈ Mi for some j = i. A k × n matrix R, containing symbols from an alphabet Λ, can be viewed as listing edges in a bipartite graph GR. The two parts of GR correspond respectively to the columns of R, and to Λ. Each cell in R corresponds to an edge in GR that links the column and the symbol in the cell. -
Parameterized Algorithms and Kernels for Rainbow Matching
Parameterized Algorithms and Kernels for Rainbow Matching Sushmita Gupta1, Sanjukta Roy2, Saket Saurabh3, and Meirav Zehavi4 1 University of Bergen, Norway [email protected] 2 The Institute of Mathematical Sciences, HBNI, Chennai, India [email protected] 3 University of Bergen, Norway, and The Institute of Mathematical Sciences, HBNI, Chennai, India [email protected] 4 University of Bergen, Norway [email protected] Abstract In this paper, we study the NP-complete colorful variant of the classical Matching problem, namely, the Rainbow Matching problem. Given an edge-colored graph G and a positive integer k, this problem asks whether there exists a matching of size at least k such that all the edges in the matching have distinct colors. We first develop a deterministic algorithm that solves Rainbow √ k ? 1+ 5 Matching on paths in time O ( 2 ) and polynomial space. This algorithm is based on a curious combination of the method of bounded search trees and a “divide-and-conquer-like” approach, where the branching process is guided by the maintenance of an auxiliary bipartite graph where one side captures “divided-and-conquered” pieces of the path. Our second result is a randomized algorithm that solves Rainbow Matching on general graphs in time O?(2k) and polynomial-space. Here, we show how a result by Björklund et al. [JCSS, 2017] can be invoked as a black box, wrapped by a probability-based analysis tailored to our problem. We also complement our two main results by designing kernels for Rainbow Matching on general and bounded-degree graphs. Keywords and phrases Rainbow Matching, Parameterized Algorithm, Bounded Search Trees, Divide-and-Conquer, 3-Set Packing, 3-Dimensional Matching Digital Object Identifier 10.4230/LIPIcs.MFCS.2017.71 1 Introduction The classical notion of matching has been extensively studied for several decades in the area of Combinatorial Optimization [6, 14]. -
Curriculum Vitae Prof. Dr. Anand Srivastav Personal Data Name Prof
Curriculum Vitae Prof. Dr. Anand Srivastav Personal Data Name Prof. Dr. Anand Srivastav (Dipl. Math., Dipl. Phys.) Place/Date of Birth Agra (India), in 1958 Affiliation Institution Christian-Albrechts-Universität zu Kiel (CAU) Institute Department of Computer Science Address Christian-Albrechts-Platz 4, D-24118 Kiel, Germany Phone / Fax +49 431 880-7252 / -1725 E-Mail [email protected] Education 1996 Habilitation in Computer Science, Freie Universität zu Berlin, Germany 1988 Dr. rer. nat. (in Mathematics), Westfälische Wilhelms-Universität Münster 1988 Dipl. Phys. degree (Master of Physics), Westfälische Wilhelms-Universität Münster, Germany 1984 Dipl. Math. degree (Master of Mathematics), Westfälische Wilhelms- Universität Münster, Germany 1978 – 1984 Study of Physics and Mathematics at the Westfälische Wilhelms- Universität Münster, Germany 1965 – 1978 School Education in India and Germany Academic Appointments since 1997 Professor for Mathematics (Discrete Optimization), Christian-Albrechts- Universität zu Kiel 1995 – 1997 Assistent Professor at the Computer Science Department, Humboldt- Universität zu Berlin 1994 – 1995 Visiting Professor at the Computer Science Department, Freie Universität Berlin 1993 – 1994 Visiting Professor at IMA (University of Minnesota), Courant Institute for Mathematical Sciences (New York University) and Yale University, USA 1988 – 1993 Assistent Professor, Institute for Discrete Mathematics, Rheinische Friedrich- Wilhelms-Universität Bonn Important Functions since 2008 Co-Speaker of the Research -
Existences of Rainbow Matchings and Rainbow Matching Covers Lo, Allan
CORE Metadata, citation and similar papers at core.ac.uk Provided by University of Birmingham Research Portal University of Birmingham Existences of rainbow matchings and rainbow matching covers Lo, Allan DOI: 10.1016/j.disc.2015.05.015 License: Creative Commons: Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) Document Version Publisher's PDF, also known as Version of record Citation for published version (Harvard): Lo, A 2015, 'Existences of rainbow matchings and rainbow matching covers', Discrete Mathematics, vol. 338, pp. 2119-2124. https://doi.org/10.1016/j.disc.2015.05.015 Link to publication on Research at Birmingham portal Publisher Rights Statement: Eligibility for repository : checked 16/06/2015 General rights Unless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or the copyright holders. The express permission of the copyright holder must be obtained for any use of this material other than for purposes permitted by law. •Users may freely distribute the URL that is used to identify this publication. •Users may download and/or print one copy of the publication from the University of Birmingham research portal for the purpose of private study or non-commercial research. •User may use extracts from the document in line with the concept of ‘fair dealing’ under the Copyright, Designs and Patents Act 1988 (?) •Users may not further distribute the material nor use it for the purposes of commercial gain. Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. -
Matchings in Balanced Hypergraphs
Matchings in balanced hypergraphs Von der Fakultat¨ fur¨ Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Diplom-Mathematiker Robert Berthold Scheidweiler aus Duren-Birkesdorf.¨ Berichter: Universitatsprofessor¨ Dr. Eberhard Triesch Universitatsprofessor¨ Dr. Arie M.C.A. Koster Tag der mundlichen¨ Prufung:¨ 19. April 2011 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.¨ Danksagung In den letzten funf¨ Jahren habe ich am Lehrstuhl II fur¨ Mathematik der RWTH Aachen Uni- versity die vorliegende Dissertation verfasst. Einigen gebuhrt¨ fur¨ ihre Unterstutzung¨ und Hilfe wahrend¨ dieser Zeit besonderer Dank. Zuallererst mochte¨ ich mich bei dem Betreuer meiner Dissertation und meinem Chef, Eberhard Triesch, bedanken. Durch ihn habe ich das Thema dieser Arbeit erhalten, das mir sehr ans Herz gewachsen ist. Er hat mir bei meinen Forschungen immer mit Rat und Tat zur Seite gestanden und mich auch bei langer¨ andauern- den Durststrecken niemals unter Druck gesetzt. Seine positive Unterstutzung¨ und geduldige Hilfe haben mich motiviert, diese Arbeit zu vollenden. Weiterhin mochte¨ ich mich bei Arie Koster, meinem Zweitgutachter, bedanken. Mehrfach hat er im Verlauf meiner Promotion Anregungen gegeben, die dann in die Dissertation eingeflossen sind. Vor der endgultigen¨ Ab- gabe hat er durch seine Verbesserungsvorschlage,¨ fur¨ die ich sehr dankbar bin, zur jetzigen Form der Arbeit beigetragen. Danken mochte¨ ich außerdem Bert Randerath, der mir half, einige Startschwierigkeiten zu uberwinden,¨ als ich begann, die balancierten Hypergraphen zu erforschen. Hartmut Fuhr¨ hat sehr viel Zeit darauf verwendet, mir die harmonische Analysis naher¨ zu bringen. Seine Bemuhungen¨ haben meine Promotion weiter voran gebracht. -
Complexity Results for Rainbow Matchings
Complexity Results for Rainbow Matchings Van Bang Le1 and Florian Pfender2 1 Universität Rostock, Institut für Informatik 18051 Rostock, Germany [email protected] 2 University of Colorado at Denver, Department of Mathematics & Statistics Denver, CO 80202, USA [email protected] Abstract A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. We address the complexity issue of the following problem, max rainbow matching: Given an edge-colored graph G, how large is the largest rainbow matching in G? We present several sharp contrasts in the complexity of this problem. We show, among others, that max rainbow matching can be approximated by a polynomial algorithm with approxima- tion ratio 2/3 − ε. max rainbow matching is APX-complete, even when restricted to properly edge-colored linear forests without a 5-vertex path, and is solvable in polynomial time for edge-colored forests without a 4-vertex path. max rainbow matching is APX-complete, even when restricted to properly edge-colored trees without an 8-vertex path, and is solvable in polynomial time for edge-colored trees without a 7-vertex path. max rainbow matching is APX-complete, even when restricted to properly edge-colored paths. These results provide a dichotomy theorem for the complexity of the problem on forests and trees in terms of forbidding paths. The latter is somewhat surprising, since, to the best of our knowledge, no (unweighted) graph problem prior to our result is known to be NP-hard for simple paths. We also address the parameterized complexity of the problem. -
Benny Sudakov Curriculum Vitae
BENNY SUDAKOV CURRICULUM VITAE Affiliation Professor, Department of Mathematics, ETH, Zurich, 8092, Switzerland. E-mail: [email protected]. Research interests Algebraic and Probabilistic Methods in Combinatorics, Graph Theory, Extremal Combinatorics, Ramsey Theory, Random Structures, Applications of Combinatorics to Theory of Computer Science. Academic appointments 2013- : Professor, Department of Mathematics, ETH, Z¨urich, Switzerland. 2015 (Fall): Fellow of Merton College, Oxford University. 2007-2014 : Professor, Department of Mathematics, UCLA, Los Angeles, USA. 2012 (Spring): Leverhulme Visiting Professor, Cambridge University and Fellow of Trinity College. 2002-2007 : Assistant Professor, Department of Mathematics, Princeton University, USA. 1999-2002 : Veblen Instructor, Department of Mathematics, Princeton University and Institute for Advanced Study, Princeton, USA. 2003 (Spring), 2005-2006, 2007 (Fall) : Member, Institute for Advanced Study, Princeton, USA. Education 1995-1999 : Ph. D. in Mathematics (with distinction), Tel Aviv University, Tel Aviv, Israel. 1991-1993: M. Sc. in Mathematics (summa cum laude), Tel Aviv University, Tel Aviv, Israel. 1990: B. Sc. in Mathematics (summa cum laude), Georgian State University, Tbilisi, USSR. Honors and Awards 2019: Member of the Academia Europaea. 2014: Humboldt Research Award, Germany. 2013: Fellow of the American Mathematical Society, Inaugural Class. 2010: Invited speaker, Combinatorics section, International Congress (ICM), Hyderabad, India. 2007-2011: David Saxon Presidential Term Chair in Mathematics, UCLA, USA. 2006-2011: NSF CAREER award. 2004-2006: Alfred P. Sloan Fellowship, USA. Grants 2013-2016, 2017-2020, 2020-2024: Swiss National Science Foundation grants. 1 2001-2004, 2004-2007, 2006-2011, 2011-2014: USA National Science Foundation grants. 2005-2009, 2009-2013: USA-Israel Binational Science Foundation grants. Ph. -
Learning from Networked Examples
Learning from Networked Examples∗ Yuyi Wang ETH Zurich,¨ Switzerland Zheng-Chu Guo Zhejiang University, P.R. China Jan Ramon Inria, France & KULeuven, Belgium Editor: The correct dates will be entered by the editor Abstract Many machine learning algorithms are based on the assumption that training examples are drawn independently. However, this assumption does not hold anymore when learning from a networked sample because two or more training examples may share some common objects, and hence share the features of these shared objects. We show that the classic approach of ignoring this problem potentially can have a harmful effect on the accuracy of statistics, and then consider alternatives. One of these is to only use independent examples, discarding other information. However, this is clearly suboptimal. We analyze sample error bounds in this networked setting, providing signifi- cantly improved results. An important component of our approach is formed by efficient sample weighting schemes, which leads to novel concentration inequalities. Keywords: Networked Examples, Non-i.i.d. Sample, Concentration Inequality, Sample Error Bound, Generalization Error Bound 1. Introduction Recently, there has been an increasing interest in network-structured data, such as data in social net- works (Leinhardt, 2013), economic networks (Knoke, 2014), citation networks (Nykl et al., 2014) and chemical interaction networks (Szklarczyk et al., 2014). An important challenge is that data concerning related objects cannot be assumed to be independent. More precisely, a key assump- arXiv:1405.2600v4 [cs.AI] 3 Jun 2017 tion made by many approaches in the field of statistics and machine learning is that observations are drawn independently and identically (i.i.d.) from a fixed distribution. -
Deterministic Distributed Edge-Coloring Via Hypergraph Maximal Matching
Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching Manuela Fischer Mohsen Ghaffari Fabian Kuhn ETH Zurich ETH Zurich University of Freiburg manuela.fi[email protected] ghaff[email protected] [email protected] Abstract We present a deterministic distributed algorithm that computes a (2∆ 1)-edge-coloring, or even list-edge-coloring, in any n-node graph with maximum degree ∆, in O(log− 7 ∆ log n) rounds. This answers one of the long-standing open questions of distributed graph algorithms· from the late 1980s, which asked for a polylogarithmic-time algorithm. See, e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and Elkin. The previous best round complexities O(√log n) were 2 by Panconesi and Srinivasan [STOC’92] and O˜(√∆) + O(log∗ n) by Fraigniaud, Heinrich, and Kosowski [FOCS’16]. A corollary of our deterministic list-edge-coloring also improves the randomized complexity of (2∆ 1)-edge-coloring to poly(log log n) rounds. − The key technical ingredient is a deterministic distributed algorithm for hypergraph maximal matching, which we believe will be of interest beyond this result. In any hypergraph of rank r — where each hyperedge has at most r vertices — with n nodes and maximum degree ∆, this algorithm computes a maximal matching in O(r5 log6+log r ∆ log n) rounds. · This hypergraph matching algorithm and its extensions also lead to a number of other results. In particular, we obtain a polylogarithmic-time deterministic distributed maximal independent set (MIS) algorithm for graphs with bounded -
On Latin Squares and Avoidable Arrays for Daniel, and for What Springtime Brings
On Latin squares and avoidable arrays For Daniel, and for what springtime brings Lina J. Andr´en: On Latin squares and avoidable arrays c 2010 Lina J. Andr´en Tryck: Print & Media, Ume˚auniversitet, Ume˚a isbn 978-91-7459-060-9 issn 1102-8300 Doctoral thesis No. 46, 2010, Department of Mathematics and Mathematical statistics, Ume˚auniversity On Latin squares and avoidable arrays Lina J. Andr´en Doctoral thesis no. 46 Department of Mathematics and Mathematical statistics Ume˚aUniversity, 2010 Contents 1 Summary of papers 1 2 Introduction 3 3 Basic definitions and tools 5 3.1 Latin squares and arrays . 5 3.2 Graphs and colorings . 10 4 Problems and known results 13 5 Structural properties of Latin squares 21 5.1 Cycles and homogeneity . 21 5.2 Transversals and rainbow matchings . 22 6 Research on avoidability and list-edge colorings 25 6.1 Someusefultechniques. 25 Bibliography 33 v Abstract This thesis consists of the four papers listed below and a survey of the research area. I Lina J. Andr´en: Avoiding (m, m, m)-arrays of order n =2k II Lina J. Andr´en: Avoidability of random arrays III Lina J. Andr´en: Avoidability by Latin squares of arrays with even order IV Lina J. Andr´en, Carl Johan Casselgren and Lars-Daniel Ohman:¨ Avoiding arrays of odd order by Latin squares Papers I, III and IV are all concerned with a conjecture by H¨aggkvist saying that there is a constant c such that for any positive integer n,ifm cn, then for every n n array A of subsets of 1,...,n such that no cell contains≤ a set of size greater× than m, and none of the{ elements} 1,...,nbelongs to more than m of the sets in any row or any column of A, there is a Latin square L on the symbols 1,...,n such that there is no cell in L that contains a symbol that belongs to the set in the corresponding cell of A.