DOCSLIB.ORG

Approximation Algorithms for Routing and Related Problems on Directed Minor-Free Graphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Approximation Algorithms for Routing and Related Problems on Directed Minor-Free Graphs Approximation algorithms for routing and related problems on directed minor-free graphs Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ario Salmasi, Graduate Program in Department of Computer Science and Engineering The Ohio State University 2019 Dissertation Committee: Yusu Wang, Co-Advisor Anastasios Sidiropoulos, Co-Advisor Srinivasan Parthasarathy Rephael Wenger c Copyright by Ario Salmasi 2019 Abstract This thesis addresses two of the fundamental routing problems in directed minor-free graphs. Broadly speaking, minor-free graphs are a generalization of planar graphs, and their structural and topological properties can be used in order to achieve efficient algorithms. A graph H is called a minor of a given graph G, if it can be obtained from G by deleting vertices and edges, and contracting edges. For any graph H, the family of H-minor-free graphs is the set of all graphs excluding H as a minor. The graph structure theorem by Robertson and Seymour determines the rough structure of any family of minor-free graphs. By using the graph structure theorem, we are able to exploit the structural and topological features of a minor-free graph, in order to obtain better approximation algorithms for several problems. As mentioned, we study two of the fundamental routing problems in different families of minor-free graphs. We first study the Asymmetric Traveling Salesman Problem (ATSP). In this problem, the goal is to find a closed walk of minimum cost in a directed graph visiting every vertex. We consider this problem on nearly-embeddable graphs, and show that it admits a constant factor approximation. More precisely, we show that for any fixed integer k ¥ 0, there exist α; β ¡ 0, such that ATSP on n-vertex k-nearly-embeddable graphs admits an α- approximation in time Opnβq. The class of k-nearly-embeddable graphs contains graphs with at most k apices, k vortices of width at most k, and an underlying surface of either orientable or non-orientable genus at most k. ii We complement our upper bound by showing that solving ATSP exactly on graphs of pathwidth k (and hence on k-nearly embeddable graphs) requires time nΩpkq, assuming the Exponential-Time Hypothesis (ETH). This is surprising in light of the fact that both TSP on undirected graphs and Minimum Cost Hamiltonian Cycle on directed graphs are FPT parameterized by treewidth. We further study the routing problem in graphs using node-disjoint paths. This problem has received a lot of attention recently, and has several applications in both the real world and graph theory. For instance, it can be used to approximate directed treewidth, which is a fundamental property of a directed graph. For routing problems using node-disjoint paths, a polylogarithmic approximation algo- rithm with constant congestion is known for the case of undirected graphs. However, it is known that the problem is hard to approximate within polynomial factors on directed graphs. We focus on the special case of symmetric demands on directed graphs, and show that it admits a polylogarithmic approximation with constant congestion on arbitrary di- rected minor-free graphs. iii Acknowledgments I am deeply indebted to my research advisor, Anastasios Sidiropoulos, for unwavering support and guidance. It has been my honor. I would also like to extend my deepest gratitude to my academic advisor, Yusu Wang, for her relentless support and advice. Thanks should also go to my dissertation committee, Srinivasan Parthasarathy and Rephael Wenger. I am grateful to Raef Bassily, Mikhail Belkin, and Tamal Dey for serving on my candidacy. I would also like to thank my co-authors, Spyros Blanas, Timothy Carpenter, Feilong Liu, D´anielMarx, and Vijay Sridhar for their knowledge, hard-work, and guidance. Finally, I would like to thank my friends and family for their love and support. I am grateful to Sonya, Fardad, my mother Kathy, and my father Amir. This thesis is dedicated to them. iv Vita 2014 . .B.S. Computer Science, Sharif University of technology. 2014 . .B.S. Mathematics, Sharif University of technology. 2018 . .M.S. Computer Science and Engineer- ing, The Ohio State University 2014-2019 . Graduate Research Associate, Computer Science and Engineering, The Ohio State University. v Publications Daniel Marx, Ario Salmasi, and Anastasios Sidiropoulos. Constant-factor approximations for asymmetric TSP on nearly-embeddable graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX, 2016. Ario Salmasi, Anastasios Sidiropoulos, and Vijay Sridhar. On constant multi-commodityflow- cut gaps for families of directed minor-free graphs. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA , 2019. Feilong Liu, Ario Salmasi, Spyros Blanas, and Anastasios Sidiropoulos. Chasing Similar- ity: Distribution-aware Aggregation Scheduling. In Proceedings of the VLDB Endowment, PVLDB, 2019. Timothy Carpenter, Ario Salmasi, and Anastasios Sidiropoulos. Routing Symmetric De- mands in Directed Minor-Free Graphs with Constant Congestion. In Approximation, Ran- domization, and Combinatorial Optimization. Algorithms and Techniques, APPROX, 2019. Fields of Study Major Field: Computer Science and Engineering vi Table of Contents Page Abstract . ii Acknowledgments . iv Vita.............................................v List of Figures . ix 1. Introduction . .1 1.1 Traveling Salesman Problem . .3 1.1.1 Motivation . .4 1.1.2 Contribution . .5 1.1.3 Overview of the algorithm . .6 1.2 Routing Symmetric Demands . .8 1.2.1 Motivation . .9 1.2.2 Contribution . 10 2. Notation and Preliminaries . 13 3. Traveling Salesman Problem . 17 3.1 The Held-Karp LP . 18 3.2 An approximation algorithm for nearly-embeddable graphs . 19 3.3 Combining the Held-Karp LP with the dynamic program . 22 3.4 Thin trees in 1-apex graphs . 24 3.4.1 Analysis . 26 3.5 Thin forests in graphs with many apices . 31 3.5.1 Analysis . 34 3.6 Thin forests in higher genus graphs with many apices . 37 vii 3.6.1 Analysis . 39 3.7 Thin subgraphs in nearly-embeddable graphs . 42 3.7.1 p0; g; 1; pq-nearly embeddable graphs . 42 3.7.2 pa; g; 1; pq-nearly embeddable graphs . 46 3.8 A preprocessing step for the dynamic program . 48 3.9 Uncrossing an optimal walk traversing a vortex . 53 3.9.1 The structure of an optimal solution . 53 3.10 The dynamic program for traversing a vortex in a planar graph . 56 3.10.1 The dynamic program . 57 3.10.2 Analysis . 64 3.11 The dynamic program for traversing a vortex in a bounded genus graph . 78 3.11.1 The dynamic program . 78 3.11.2 Analysis . 80 3.12 The algorithm for traversing a vortex in a nearly-embeddable graph . 97 3.13 The lower bound for graphs of bounded pathwidth . 98 3.13.1 Edge Balancing ........................... 98 3.13.2 Constrained Closed Walk and ATSP ............. 102 4. Routing Symmetric Demands . 113 4.1 The Algorithm for Minor-Free Graphs . 113 4.2 The Crossbar Construction . 116 4.3 Graphs of Bounded Genus . 119 4.4 Minor Free Graphs . 127 4.5 Nearly Embeddable Graphs . 127 4.5.1 Dealing with clique-sums . 129 5. Conclusion . 130 5.1 Main results . 130 5.2 Work by others . 131 5.3 Future work and open problems . 132 Bibliography . 134 viii List of Figures Figure Page 1.1 An illustration of minor-free graphs (Figure by Felix Reidl) . .2 2.1 An example of a 1-apex graph (Figure from Wikipedia) . 15 3.1 Graph of components . 25 3.2 Modified ribbon contraction . 43 3.3 Additional edges in the preprocessing step . 49 3.4 Normalization step . 50 3.5 Construction of P where x and y are in opposite sides . 53 3.6 Updating the grip using concatenation . 60 3.7 Crossing grip . 61 3.8 Illustration of the sixth step . 62 3.9 Example of a basic path. 65 3.10 Part of the collection of walks in W (left) and the corresponding P -facial restriction of W depicted in bold (right). 66 1 1 3.11 Construction of R1 and R2 ............................ 73 3.12 Computing a partial solution . 76 ix 3.13 Example of a non-basic family of paths. P “ tQiu is not basic. Let T1 “ D1 Y D2 and T2 “ D3. Then Qi covers T1 Y T2 and avoids T zpT1 Y T2q, but 1 2 there is no edge-disjoint subpaths Qi;Qi of Qi satisfying the third condition. 81 3.14 Illustration of the elementary proof . 88 3.15 Example of good non-trivial trees T0, T1, and T2 that are pairwise friends; note that T0 is of the second type while T1 and T2 are of the third type. 92 3.16 The instance of Edge Balancing constructed in the proof of Lemma 3.13.4. The values on the edges indicate the value of χ corresponding to a solu- tion pvi;ji qi“1;:::;2k of the Multicolored Biclique instance (we have Y1 “ 1¤i¤k xji and Y2 “ k`1¤i¤2k xji )........................ 100 3.17 Theř gadget of Lemmař 3.13.6 with two collections of paths satisfying it. 105 3.18 The gadget HX of Lemma 3.13.7 for a set X “ tx1; x2; x3u of three integers. The gray rectangles represent the internal vertices of the three gadgets Hx1 , Hx2 , and Hx3 .................................... 107 4.1 Maintaining a Eulerian graph with bounded degree. 118 4.2 Decomposition of the grid minor . 120 1 4.3 The connected components of GzC in Γ0 .................... 122 4.4 Construction of cq ................................. 125 x Chapter 1: Introduction Wagner's theorem is one of the most well-known results in graph theory.
Load more

Recommended publications
  • Characterizations of Graphs Without Certain Small Minors by J. Zachary
    Characterizations of Graphs Without Certain Small Minors By J. Zachary Gaslowitz Dissertation Submitted to the Faculty of the Graduate School of Vanderbilt University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mathematics May 11, 2018 Nashville, Tennessee Approved: Mark Ellingham, Ph.D. Paul Edelman, Ph.D. Bruce Hughes, Ph.D. Mike Mihalik, Ph.D. Jerry Spinrad, Ph.D. Dong Ye, Ph.D. TABLE OF CONTENTS Page 1 Introduction . 2 2 Previous Work . 5 2.1 Planar Graphs . 5 2.2 Robertson and Seymour's Graph Minor Project . 7 2.2.1 Well-Quasi-Orderings . 7 2.2.2 Tree Decomposition and Treewidth . 8 2.2.3 Grids and Other Graphs with Large Treewidth . 10 2.2.4 The Structure Theorem and Graph Minor Theorem . 11 2.3 Graphs Without K2;t as a Minor . 15 2.3.1 Outerplanar and K2;3-Minor-Free Graphs . 15 2.3.2 Edge-Density for K2;t-Minor-Free Graphs . 16 2.3.3 On the Structure of K2;t-Minor-Free Graphs . 17 3 Algorithmic Aspects of Graph Minor Theory . 21 3.1 Theoretical Results . 21 3.2 Practical Graph Minor Containment . 22 4 Characterization and Enumeration of 4-Connected K2;5-Minor-Free Graphs 25 4.1 Preliminary Definitions . 25 4.2 Characterization . 30 4.3 Enumeration . 38 5 Characterization of Planar 4-Connected DW6-minor-free Graphs . 51 6 Future Directions . 91 BIBLIOGRAPHY . 93 1 Chapter 1 Introduction All graphs in this paper are finite and simple. Given a graph G, the vertex set of G is denoted V (G) and the edge set is denoted E(G).
    [Show full text]
  • Topics in Low Dimensional Computational Topology
    THÈSE DE DOCTORAT présentée et soutenue publiquement le 7 juillet 2014 en vue de l’obtention du grade de Docteur de l’École normale supérieure Spécialité : Informatique par ARNAUD DE MESMAY Topics in Low-Dimensional Computational Topology Membres du jury : M. Frédéric CHAZAL (INRIA Saclay – Île de France ) rapporteur M. Éric COLIN DE VERDIÈRE (ENS Paris et CNRS) directeur de thèse M. Jeff ERICKSON (University of Illinois at Urbana-Champaign) rapporteur M. Cyril GAVOILLE (Université de Bordeaux) examinateur M. Pierre PANSU (Université Paris-Sud) examinateur M. Jorge RAMÍREZ-ALFONSÍN (Université Montpellier 2) examinateur Mme Monique TEILLAUD (INRIA Sophia-Antipolis – Méditerranée) examinatrice Autre rapporteur : M. Eric SEDGWICK (DePaul University) Unité mixte de recherche 8548 : Département d’Informatique de l’École normale supérieure École doctorale 386 : Sciences mathématiques de Paris Centre Numéro identifiant de la thèse : 70791 À Monsieur Lagarde, qui m’a donné l’envie d’apprendre. Résumé La topologie, c’est-à-dire l’étude qualitative des formes et des espaces, constitue un domaine classique des mathématiques depuis plus d’un siècle, mais il n’est apparu que récemment que pour de nombreuses applications, il est important de pouvoir calculer in- formatiquement les propriétés topologiques d’un objet. Ce point de vue est la base de la topologie algorithmique, un domaine très actif à l’interface des mathématiques et de l’in- formatique auquel ce travail se rattache. Les trois contributions de cette thèse concernent le développement et l’étude d’algorithmes topologiques pour calculer des décompositions et des déformations d’objets de basse dimension, comme des graphes, des surfaces ou des 3-variétés.
    [Show full text]
  • Rooted Graph Minors and Reducibility of Graph Polynomials Arxiv
    Rooted Graph Minors and Reducibility of Graph Polynomials by Benjamin Richard Moore B.A., Thompson Rivers University, 2015 Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Masters of Science in the Department of Mathematics and Statistics Faculty of Science c Benjamin Richard Moore 2017 SIMON FRASER UNIVERSITY Spring 2017 arXiv:1704.04701v1 [math.CO] 15 Apr 2017 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced without authorization under the conditions for “Fair Dealing.” Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately. Approval Name: Benjamin Richard Moore Degree: Masters of Science (Mathematics) Title: Rooted Graph Minors and Reducibility of Graph Poly- nomials Examining Committee: Dr. Ladislav Stacho (chair) Associate Professor Dr. Karen Yeats Senior Supervisor Associate Professor Dr. Luis Goddyn Supervisor Professor Dr. Bojan Mohar Internal Examiner Professor Date Defended: April 6, 2017 ii Abstract In 2009, Brown gave a set of conditions which when satisfied imply that a Feynman integral evaluates to a multiple zeta value. One of these conditions is called reducibility, which loosely says there is an order of integration for the Feynman integral for which Brown’s techniques will succeed. Reducibility can be abstracted away from the Feynman integral to just being a condition on two polynomials, the first and second Symanzik polynomials. The first Symanzik polynomial is defined from the spanning trees of a graph, and the second Symanzik polynomial is defined from both spanning forests of a graph and some edge and vertex weights, called external momenta and masses.
    [Show full text]
  • Minor Excluded Network Families Admit Fast Distributed Algorithms∗
    Minor Excluded Network Families Admit Fast Distributed Algorithms∗ Bernhard Haeupler,y Jason Li,y Goran Zuzicy January 22, 2018 Abstract Distributed network optimization algorithms, such as minimum spanning tree, minimum cut, and shortest path, are an active research area in distributed computing. This paper presents a fast distributed algorithm for such problems in the CONGEST model, on networks that exclude a fixed minor. pOn general graphs, many optimization problems, including the ones mentioned above, require Ω(~ n) rounds of communication in the CONGEST model, even if the network graph has a much smaller diameter. Naturally, the next step in algorithm design is to design efficient algorithms which bypass this lower bound on a restricted class of graphs. Currently, the only known method of doing so uses the low-congestion shortcut framework of Ghaffari and Haeupler [SODA'16]. Building off of their work, this paper proves that excluded minor graphs admit high-quality shortcuts, leading to an O~(D2) round algorithm for the aforementioned problems, where D is the diameter of the network graph. To work with excluded minor graph families, we utilize the Graph Structure Theorem of Robertson and Seymour. To the best of our knowledge, this is the first time the Graph Structure Theorem has been used for an algorithmic result in the distributed setting. Even though the proof is involved, merely showing the existence of good shortcuts is sufficient to obtain simple, efficient distributed algorithms. In particular, the shortcut framework can effi- ciently construct near-optimal shortcuts and then use them to solve the optimization problems. This, combined with the very general family of excluded minor graphs, which includes most other important graph classes, makes this result of significant interest.
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach Graph Theory
    Mathematisches Forschungsinstitut Oberwolfach Report No. 2/2016 DOI: 10.4171/OWR/2016/2 Graph Theory Organised by Reinhard Diestel, Hamburg Daniel Kr´al’, Warwick Paul Seymour, Princeton 10 January – 16 January 2016 Abstract. This workshop focused on recent developments in graph theory. These included in particular recent breakthroughs on nowhere-zero flows in graphs, width parameters, applications of graph sparsity in algorithms, and matroid structure results. Mathematics Subject Classification (2010): 05C. Introduction by the Organisers The aim of the workshop was to offer a forum to communicate recent develop- ments in graph theory and discuss directions for further research in the area. The atmosphere of the workshop was extremely lively and collaborative. On the first day of the workshop, each participant introduced her/himself and briefly presented her/his research interests, which helped to establish a working atmosphere right from the very beginning of the workshop. The workshop program consisted of 9 long more general talks and 21 short more focused talks. The talks were comple- mented by six evening workshops to discuss particular topics at a larger depth; these were held in ad hoc formed groups in the evenings. The workshop focused on four interlinked topics in graph theory, which have recently seen new exciting developments. These topics were nowhere-zero flows and the dual notion of graph colorings, • sparsity of graphs and its algorithmic applications, • width parameters and graph decompositions, and • results following the proof of Rota’s conjecture in matroid theory. • 52 Oberwolfach Report 2/2016 The four main themes of the workshop were reflected in the selection of the work- shop participants and in the choice of the talks for the program.
    [Show full text]
  • A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface∗
    A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface∗ Vincent Cohen-Addady Arnaud de Mesmayz Abstract Given a graph G cellularly embedded on a surface Σ of genus g, a cut graph is a subgraph of G such that cutting Σ along G yields a topological disk. We provide a fixed parameter tractable approximation scheme for the problem of computing the shortest cut graph, that is, for any " > 0, we show how to compute a (1 + ") approximation of the shortest cut graph in time f("; g)n3. Our techniques first rely on the computation of a spanner for the problem using the technique of brick decompositions, to reduce the problem to the case of bounded tree- width. Then, to solve the bounded tree-width case, we introduce a variant of the surface-cut decomposition of Ru´e,Sau and Thilikos, which may be of independent interest. 1 Introduction Embedded graphs are commonly used to model a wide array of discrete structures, and in many cases it is necessary to consider embeddings into surfaces instead of the plane or the sphere. For example, many instances of network design actually feature some crossings, coming from tunnels or overpasses, which are appropriately modeled by a surface of small genus. In other settings, such as in computer graphics or computer-aided design, we are looking for a discrete model for objects which inherently display a non-trivial topology (e.g., holes), and graphs embedded on surfaces are the natural tool for that. From a more theoretical point of view, the graph structure theorem of Robertson and Seymour showcases a very strong connection between graphs embedded on surfaces and minor-closed families of graphs.
    [Show full text]
  • Lipics-STACS-2021-2.Pdf (0.7
    First-Order Transductions of Graphs Patrice Ossona de Mendez ! Ï Centre d’Analyse et de Mathématique Sociales CNRS UMR 8557, Paris, France Charles University, Prague, Czech Republic Abstract This paper is an extended abstract of my STACS 2021 talk “First-order transductions of graphs”. 2012 ACM Subject Classification Theory of computation → Finite Model Theory; Mathematics of computing → Graph theory Keywords and phrases Finite model theory, structural graph theory Digital Object Identifier 10.4230/LIPIcs.STACS.2021.2 Category Invited Talk Funding This paper is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 810115 – Dynasnet). Acknowledgements I would like to thank Jarik Nešetřil and Sebastian Siebertz for their most valuable help in preparing this presentation. 1 Introduction Logical methods in Computer Science have a long history, as witnessed e.g. by the relative longevity of SQL in relational database management. More recently, Courcelle’s theorem, which combines second-order logic and tree decompositions of graphs, showed that a many NP-complete algorithmic problems in graph theory can be solved in polynomial time on graphs with bounded tree-width (and even on graphs with bounded clique-width). At the heart of the latter result is the notion of monadic second-order transductions, which are a way to encode a graph within a structure using coloring and monadic second-order logic formulas. In this presentation we consider first-order transductions, for which the formulas have to be first-order formulas. As a counterpart for this strong restriction, many algorithmic problems become fixed parameter tractable when restricted to nowhere dense classes, which include classes excluding a topological minor thus, in particular, classes of planar graphs and classes of graphs with bounded degrees.
    [Show full text]
  • Algorithmic Graph Structure Theory
    Algorithmic graph structure theory Dániel Marx1 1Computer and Automation Research Institute, Hungarian Academy of Sciences (MTA SZTAKI) Budapest, Hungary STACS 2013 Tutorial February 27, 2013 Kiel, Germany 1 Classes of graphs Classes of graphs can be described by 1 what they do not have, (excluded structures) 2 how they look like (constructions and decompositions). In general, the second description is more useful for algorithmic purposes. 2 Classes of graphs Example: Trees 1 Do not contain cycles (and connected) 2 Have a tree structure. Example: Bipartite graphs 1 Do not contain odd cycles, 2 Edges going only between two classes. Example: Chordal graphs 1 Do not contain induced cycles, 2 Clique-tree decomposition and simplicial ordering. 3 Graph Structure Theory “Graph structure theory” usually refers to the theory developed by Robertson and Seymour on graphs excluding minors. Definition Graph H is a minor of G (H ≤ G) if H can be obtained from G by deleting edges, deleting vertices, and contracting edges. uv deleting uv contracting uv vu w 4 How do graphs excluding H (or H1, ::: , Hk ) look like? What other classes can be defined this way? The work of Robertson and Seymour gives some kind of combinatorial answer to that and provides tools for the related algorithmic questions. Excluding minors Theorem [Wagner 1937] A graph is a planar if and only if it excludes K5 and K3;3 as a minor. K5 K3;3 5 Excluding minors Theorem [Wagner 1937] A graph is a planar if and only if it excludes K5 and K3;3 as a minor. K5 K3;3 How do graphs excluding H (or H1, ::: , Hk ) look like? What other classes can be defined this way? The work of Robertson and Seymour gives some kind of combinatorial answer to that and provides tools for the related algorithmic questions.
    [Show full text]
  • Structure Theorem and Isomorphism Test for Graphs with Excluded Topological Subgraphs
    Structure Theorem and Isomorphism Test for Graphs with Excluded Topological Subgraphs Martin Grohe Dániel Marx∗ Institut für Informatik Computer and Automation Research Institute, Humboldt-Universität zu Berlin Hungarian Academy of Sciences (MTA SZTAKI) Berlin, Germany Budapest, Hungary [email protected] [email protected] ABSTRACT Keywords We generalize the structure theorem of Robertson and Sey- topological minors, fixed-parameter tractability, graph iso- mour for graphs excluding a fixed graph H as a minor to morphism graphs excluding H as a topological subgraph. We prove that for a fixed H, every graph excluding H as a topological subgraph has a tree decomposition where each part is either 1. INTRODUCTION \almost embeddable" to a fixed surface or has bounded de- We say that a graph H is a minor of G if H can be ob- gree with the exception of a bounded number of vertices. tained from G by deleting vertices, deleting edges, and con- Furthermore, such a decomposition is computable by an al- tracting edges. A graph G is H-minor free if H is not a minor gorithm that is fixed-parameter tractable with parameter of G. Robertson and Seymour [25] proved a structure theo- SHS. rem for the class of H-minor graphs: roughly speaking, every We present two algorithmic applications of our structure H-minor free graph can be decomposed in a way such that theorem. To illustrate the mechanics of a \typical" appli- each part is \almost embeddable" into a fixed surface. This cation of the structure theorem, we show that on graphs structure theorem has important algorithmic consequences: excluding H as a topological subgraph, Partial Dominat- many natural computational problems become easier when ing Set (find k vertices whose closed neighborhood has restricted to H-minor free graphs [4, 13, 6, 15, 14, 5, 10].
    [Show full text]
  • Cyclability: Combinatorial Properties, Algorithms and Complexity
    National and Kapodistrian University of Athens School of Science, Department of Mathematics Cyclability: Combinatorial Properties, Algorithms and Complexity Spyridon Maniatis PhD Thesis Supervised by Prof. Dimitrios M. Thilikos June, 2018 ΠΕΡΙΛΗΨΗ Ένα γράφημα G καλείται k-κυκλώσιμο, αν για κάθε k από τις κορυφές του υπάρχει ένας κύκλος στο G που τις περιέχει. Η κυκλωσιμότητα ενός γραφήματος G είναι ο μέγιστος ακέραιος k για τον οποίο το G είναι k-κυκλώσιμο και είναι μία παράμετρος που σχετίζεται με τη συνεκτικότητα. Σε αυτή τη διδακτορική διατριβή μελετάμε, κυρίως από τη σκοπιά της Παραμετρικής Πολυ- πλοκότητας, το πρόβλημα ΚΥΚΛΩΣΙΜΟΤΗΤΑ: Δεδομένου ενός γραφήματος G = (V; E) και ενός μη αρνητικού ακεραίου k (η παράμετρος), να αποφασιστεί αν η κυκλωσιμότητα του G είναι ίση με k. Το πρώτο μας αποτέλεσμα είναι αρνητικό και δείχνει ότι η ύπαρξη ενός FPT-αλγορίθμου για την επίλυση του προβλήματος ΚΥΚΛΩΣΙΜΟΤΗΤΑ είναι απίθανη (εκτός αν FPT = co-W[1], το οποίο θεωρείται απίθανο). Πιο συγκεκριμέ- να, αποδεικνύουμε ότι το πρόβλημα ΚΥΚΛΩΣΙΜΟΤΗΤΑ είναι co-W[1]-δύσκολο, ακόμα και αν περιορίσουμε την είσοδο στο να είναι χωριζόμενο γράφημα. Από την άλλη, δίνουμε έναν FPT-αλγόριθμο για το ίδιο πρόβλημα περι- ορισμένο στην κλάση των επίπεδων γραφημάτων. Για να το πετύχουμε αυτό αποδεικνύουμε μια σειρά από συνδυαστικά αποτελέσματα σχετικά με την κυκλωσιμότητα και εφαρμόζουμε μια εκδοχή δύο βημάτων της περίφημης τεχνι- κής της άσχετης κορυφής, που εισήχθη από τους Robertson και Seymour στη σειρά εργασιών τους για Ελλάσονα Γραφήματα, ως ένα κρίσιμο συστατικό του αλγορίθμου τους για την επίλυση του προβλήματος των ΔΙΑΚΕΚΡΙΜΕΝΩΝ ΜΟΝΟΠΑΤΙΩΝ. Για να αποδείξουμε την ορθότητα του αλγορίθμου μας εισάγου- με έννοιες, όπως αυτή των ζωτικών κυκλικών συνδέσμων, και αποδεικνύουμε αποτελέσματα με ανεξάρτητου γραφοθεωρητικού ενδιαφέροντος.
    [Show full text]
  • Minor Excluded Network Families Admit Fast Distributed Algorithms∗
    Session 3D: Graphs and Population PODC’18, July 23-27, 2018, Egham, United Kingdom Minor Excluded Network Families Admit Fast Distributed Algorithms∗ Bernhard Haeupler Jason Li Goran Zuzic Carnegie Mellon University Carnegie Mellon University Carnegie Mellon University Pittsburgh, PA Pittsburgh, PA Pittsburgh, PA [email protected] [email protected] [email protected] ABSTRACT KEYWORDS Distributed network optimization problems, such as minimum span- Distributed Computing; CONGEST; Graph Structure Theorem; Mi- ning tree, minimum cut, and shortest path, are an active research nor Excluded Graphs; Shortcuts; Low-congestion Shortcuts area in distributed computing. This paper presents a fast distributed ACM Reference Format: algorithm for such problems in the CONGEST model, on networks Bernhard Haeupler, Jason Li, and Goran Zuzic. 2018. Minor Excluded Net- that exclude a fixed minor. work Families Admit Fast Distributed Algorithms. In PODC ’18: ACM Sym- On general graphs, many optimization problems, including the posium on Principles of Distributed Computing, July 23ś27, 2018, Egham, ones mentioned above, require Ω(˜ √n) rounds of communication United Kingdom. ACM, New York, NY, USA, 10 pages. https://doi.org/10. in the CONGEST model, even if the network graph has a much 1145/3212734.3212776 smaller diameter. Naturally, the next step in algorithm design is to design efficient algorithms which bypass this lower bound on 1 INTRODUCTION a restricted class of graphs. Currently, the only known method of doing so uses the low-congestion shortcut framework of Ghaffari Network optimization problems in the CONGEST model, such as and Haeupler [SODA’16]. Building off of their work, this paper Minimum Spanning Tree, Min-Cut or Shortest Path are an active proves that excluded minor graphs admit high-quality shortcuts, research area in theoretical distributed computing [8, 11, 22, 27].
    [Show full text]
  • Arxiv:1307.2187V3 [Cs.DS] 25 Aug 2013
    Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask) D´anielMarx ∗ Micha lPilipczuk y Abstract Given two graphs H and G, the Subgraph Isomorphism problem asks if H is iso- morphic to a subgraph of G. While NP-hard in general, algorithms exist for various pa- rameterized versions of the problem: for example, the problem can be solved (1) in time 2O(jV (H)j) · nO(tw(H)) using the color-coding technique of Alon, Yuster, and Zwick [1]; (2) in time f(jV (H)j; tw(G)) · n using Courcelle's Theorem; (3) in time f(jV (H)j; genus(G)) · n using a result on first-order model checking by Frick and Grohe [28]; or (4) in time f(∆(H))· nO(tw(G)) for connected H using the algorithm of Matouˇsekand Thomas [50]. Already this small sample of results shows that the way an algorithm can depend on the parameters is highly nontrivial and subtle. However, the literature contains very few negative results ruling out that certain combination of parameters cannot be exploited algorithmically. Our goal is to systematically investigate the possible parameterized algorithms that can exist for Subgraph Isomorphism. We develop a framework involving 10 relevant parameters for each of H and G (such as treewidth, pathwidth, genus, maximum degree, number of vertices, number of components, etc.), and ask if an algorithm with running time f2(p`+1;:::;pk) f1(p1; p2; : : : ; p`) · n exists, where each of p1; : : : ; pk is one of the 10 parameters depending only on H or G.
    [Show full text]