DOCSLIB.ORG

Graph Minors, Bidimensionality and Algorithms

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Graph Minors, Bidimensionality and Algorithms . FOMIN FEDOR V Graph Minors, Bidimensionality and Algorithms PART I Warsaw, 2010 Outline of the lectures I Part I: Graph Minors WQO Kraskul's theorem Robertson-Seymour Graph Minors Theorem Obstructions Meta-algorithmic consequences Outline of the lectures I Part II: Bidimensionality Branch-width Dynamic programming Excluding Planar grid Parameterized Algorithms Outline of the lectures I Part III: Bidimensionality Sphere-cut decompositions and Catalan structure EPTAS Kernelization (if time allows) PART I Wagner's Conjecture and Graph Minors series I Relation \ " is quasi-ordering on X if it is reflexive and ≤ transitive I X is well-quasi-ordered by if for every infinite sequence ≤ x = x ; x ;::: in X, i < j, s.t. x x . 0 1 9 i ≤ j I (xixj): a good pair of x . Sequence x is a good sequence. W.Q.O. definition I X be a collection of mathematical objects I X is well-quasi-ordered by if for every infinite sequence ≤ x = x ; x ;::: in X, i < j, s.t. x x . 0 1 9 i ≤ j I (xixj): a good pair of x . Sequence x is a good sequence. W.Q.O. definition I X be a collection of mathematical objects I Relation \ " is quasi-ordering on X if it is reflexive and ≤ transitive I (xixj): a good pair of x . Sequence x is a good sequence. W.Q.O. definition I X be a collection of mathematical objects I Relation \ " is quasi-ordering on X if it is reflexive and ≤ transitive I X is well-quasi-ordered by if for every infinite sequence ≤ x = x ; x ;::: in X, i < j, s.t. x x . 0 1 9 i ≤ j W.Q.O. definition I X be a collection of mathematical objects I Relation \ " is quasi-ordering on X if it is reflexive and ≤ transitive I X is well-quasi-ordered by if for every infinite sequence ≤ x = x ; x ;::: in X, i < j, s.t. x x . 0 1 9 i ≤ j I (xixj): a good pair of x . Sequence x is a good sequence. Proposition A quasi-ordering X is a well-quasi-ordering if and only if X contains I neither infinite antichain I nor strictly decreasing sequence x0 > x1 > : : : . PROOF: Ramsey arguments v 1. nv: vertex removal: e 2. ne: edge removal: v 3. =v: deg-2 vertex dissolution: e 4. =e: edge contraction Let L ⊆ fnv; ne; =v; =eg H ≤L G if H can be obtained from G after a sequence of operations in L We define4 local operations: e 2. ne: edge removal: v 3. =v: deg-2 vertex dissolution: e 4. =e: edge contraction Let L ⊆ fnv; ne; =v; =eg H ≤L G if H can be obtained from G after a sequence of operations in L We define4 local operations: v 1. nv: vertex removal: v 3. =v: deg-2 vertex dissolution: e 4. =e: edge contraction Let L ⊆ fnv; ne; =v; =eg H ≤L G if H can be obtained from G after a sequence of operations in L We define4 local operations: v 1. nv: vertex removal: e 2. ne: edge removal: e 4. =e: edge contraction Let L ⊆ fnv; ne; =v; =eg H ≤L G if H can be obtained from G after a sequence of operations in L We define4 local operations: v 1. nv: vertex removal: e 2. ne: edge removal: v 3. =v: deg-2 vertex dissolution: Let L ⊆ fnv; ne; =v; =eg H ≤L G if H can be obtained from G after a sequence of operations in L We define4 local operations: v 1. nv: vertex removal: e 2. ne: edge removal: v 3. =v: deg-2 vertex dissolution: e 4. =e: edge contraction H ≤L G if H can be obtained from G after a sequence of operations in L We define4 local operations: v 1. nv: vertex removal: e 2. ne: edge removal: v 3. =v: deg-2 vertex dissolution: e 4. =e: edge contraction Let L ⊆ fnv; ne; =v; =eg We define4 local operations: v 1. nv: vertex removal: e 2. ne: edge removal: v 3. =v: deg-2 vertex dissolution: e 4. =e: edge contraction Let L ⊆ fnv; ne; =v; =eg H ≤L G if H can be obtained from G after a sequence of operations in L Relation Notation nv ne =v =e WQO induced subgraph (H ⊆in G) • NO subgraph (H ⊆sb G) • • NO spanning subgraph (H ⊆sp G) • NO induced topological minor (H ≤it G) • • NO topological minor (H ≤tp G) • • • NO induced minor (H ≤in G) • • NO contraction (H ≤cn G) • NO minor (H ≤mn G) • • • YES The fact that graphs are WQO by the minor relation was known as The Wagner's Conjecture formulated by Klaus Wagner in the 1930s (?). The Graph Minors Series The conjecture was proven by Neil Robertson and Paul Seymour in their Graph Minor series of papers. Now it is known as the Robertson & Seymour Theorem. Width of the proof: < 10 cm (23 papers) Kruskal's theorem Theorem (Kruskal, 1960) Trees are W.Q.O. by the topological minor relation. (Formerly also known as the V´azsonyiconjecture) Proof of Kruskal's theorem We prove the following stronger statement: I Rooted trees are W.Q.O. by the topological minor containment Trees T and T 0 with roots r and r0. T T if isomorphism ' from some subdivision of T to a subtree ≤ 0 9 of T 0 preserving the tree-order on V (T ) associated with r and T . Proof of Kruskal's theorem We prove the following stronger statement: I Rooted trees are W.Q.O. by the topological minor containment r φ(r) Proof of Kruskal's theorem We prove the following stronger statement: I Rooted trees are W.Q.O. by the topological minor containment r φ(r) I Choose Tn to be the minimum order rooted tree such that there is a bad sequence starting with T0;T1;T2;:::;Tn I (Tn)n 0 is a bad sequence ≥ Proof sketch: Suppose that there is a bad sequence of rooted trees. For n 0 ≥ we select inductively the following bad sequence: I Assume we constructed a sequence T0;T1;T2;:::;Tn 1 s.t. − there is a bad sequence starting with T0;T1;T2;:::;Tn 1 − I (Tn)n 0 is a bad sequence ≥ Proof sketch: Suppose that there is a bad sequence of rooted trees. For n 0 ≥ we select inductively the following bad sequence: I Assume we constructed a sequence T0;T1;T2;:::;Tn 1 s.t. − there is a bad sequence starting with T0;T1;T2;:::;Tn 1 − I Choose Tn to be the minimum order rooted tree such that there is a bad sequence starting with T0;T1;T2;:::;Tn Proof sketch: Suppose that there is a bad sequence of rooted trees. For n 0 ≥ we select inductively the following bad sequence: I Assume we constructed a sequence T0;T1;T2;:::;Tn 1 s.t. − there is a bad sequence starting with T0;T1;T2;:::;Tn 1 − I Choose Tn to be the minimum order rooted tree such that there is a bad sequence starting with T0;T1;T2;:::;Tn I (Tn)n 0 is a bad sequence ≥ Let A1;A2;::: be a sequence of rooted trees in A k k Let A be such that the tree Ti where A comes from has the smallest possible index. k k+1 Then the sequence T0;:::;Ti−1;A ;A ;:::; is good (by (*)). k0 k00 00 0 Then a comparable pair is of the form A tp A for k > k k. ≤ ≥ I For (Tn)n≥0 = T0;T1;T2;::: I An the set of components Tn rn. − S I = An A n≥0 r0 r1 r2 r3 r4 A We first prove that is WQO. I A Choose k with the smallest n(k). k k+1 Then the sequence T0;:::;Tn(k) 1;T ;T ;::: is good (by − minimality of T and because T k T ). n(k) ⊂ n(k) Proof that is WQO A r0 r1 r2 r3 r4 Tk A k 1 2 Let (T )k 0 = T ;T ;::: be a sequence of rooted trees in ≥ A For k define n(k) to be the minimum n such that T k A . 2 n k k+1 Then the sequence T0;:::;Tn(k) 1;T ;T ;::: is good (by − minimality of T and because T k T ). n(k) ⊂ n(k) Proof that is WQO A r0 r1 r2 r3 r4 Tk A k 1 2 Let (T )k 0 = T ;T ;::: be a sequence of rooted trees in ≥ A For k define n(k) to be the minimum n such that T k A . 2 n Choose k with the smallest n(k). Proof that is WQO A r0 r1 r2 r3 r4 Tk A k 1 2 Let (T )k 0 = T ;T ;::: be a sequence of rooted trees in ≥ A For k define n(k) to be the minimum n such that T k A . 2 n Choose k with the smallest n(k). k k+1 Then the sequence T0;:::;Tn(k) 1;T ;T ;::: is good (by − minimality of T and because T k T ). n(k) ⊂ n(k) Thus it has a good pair (T;T 0) Proof that is WQO A r0 r1 r2 r3 r4 Tk A k k+1 The sequence T0;:::;Tn(k) 1;T ;T ;::: is good − Proof that is WQO A r0 r1 r2 r3 r4 Tk A k k+1 The sequence T0;:::;Tn(k) 1;T ;T ;::: is good − Thus it has a good pair (T;T 0) Proof that is WQO: T T0;:::;Tn(k) 1 ProofProof that thatis WQOis WQOA 62 − A A ProofProof that thatis WQOis WQO A A r0 r1 r2 r3 r4 r0 r1 Tk r2 r3 r4 r0 r1 r2 r3 r4 r r r r r Tk 0 1 Tk 2 3 4 A Tk A A A k k+1 The sequence T0,...,Tn(k) 1, T ,T ,..
Load more

Recommended publications
  • Nowhere-Zero Flows in Regular Matroids and Hadwiger's Conjecture
    NOWHERE-ZERO FLOWS IN REGULAR MATROIDS AND HADWIGER'S CONJECTURE LUIS A. GODDYN AND WINFRIED HOCHSTATTLER¨ To the memory of Reinhard B¨orger Abstract. We present a tool that shows, that the existence of a k-nowhere-zero-flow is compatible with 1-,2- and 3-sums in regular matroids. As application we present a conjecture for regular matroids that is equivalent to Hadwiger's conjecture for graphs and Tuttes's 4- and 5-flow conjectures. Keywords: nowhere zero flow, regular matroid, chromatic number, flow number, total unimodularity 1. Introduction A (real) matrix is totally unimodular (TUM) if each subdeterminant belongs to f0; ±1g. Totally unimodular matrices enjoy several nice properties which give them a fundamental role in combinatorial optimization and matroid theory. In this note we prove that the TUM possesses an attractive property. Let S ⊆ R, and let A be a real matrix. A column vector f is a S-flow of A if Af = 0 and every entry of f is a member of ±S. For any additive abelian group Γ use the notation Γ∗ = Γ n f0g. For a TUM A and a column vector f with entries in Γ, the product Af is a well defined column vector with entries in Γ, by interpreting (−1)γ to be the additive inverse of γ. It is convenient to use the language of matroids. A regular oriented matroid M is an oriented matroid that is representable M = M[A] by a TUM matrix A. Here the elements E(M) of M label the columns of A. Each (signed) cocircuit D = (D+;D−) of M corresponds to a f0; ±1g-valued vector in the row space of A and having minimal support.
    [Show full text]
  • Bidimensionality Theory and Algorithmic Graph Minor Theory Lecture Notes for Mohammadtaghi Hajiaghayi’S Tutorial
    Bidimensionality Theory and Algorithmic Graph Minor Theory Lecture Notes for MohammadTaghi Hajiaghayi’s Tutorial Mareike Massow,∗ Jens Schmidt,† Daria Schymura,‡ Siamak Tazari§ MDS Fall School Blankensee October 2007 1 Introduction Dealing with Hard Graph Problems Many graph problems cannot be computed in polynomial time unless P = NP , which most computer scientists and mathematicians doubt. Examples are the Traveling Salesman Problem (TSP), vertex cover, and dominating set. TSP is to find a Hamiltonian cycle with the least weight in a complete weighted graph. A vertex cover of a graph is a subset of the vertices that covers all edges. A dominating set in a graph is a subset of the vertices such that each vertex is contained in the set or has a neighbour in the set. The decision problem is to answer the question whether there is a vertex cover resp. a dominating set with at most k elements. The optimization problem is to find a vertex cover resp. a dominating set of minimal size. How can we solve these problems despite their computational complexity? The four main theoretical approaches to handle NP -hard problems are the following. • Average case: Prove that an algorithm is efficient in the expected case. • Special instances: A problem is solved efficiently for a special graph class. For example, graph isomor- phism is computable in linear time on planar graphs whereas graph isomorphism in general is known to be in NP and suspected not to be in P . • Approximation algorithms: Approximate the optimal solution within a constant factor c, or even within 1 + . An algorithmic scheme that gives for each > 0 a polynomial-time (1 + )-approximation is called a polynomial-time approximation scheme (PTAS).
    [Show full text]
  • Sphere-Cut Decompositions and Dominating Sets in Planar Graphs
    Sphere-cut Decompositions and Dominating Sets in Planar Graphs Michalis Samaris R.N. 201314 Scientific committee: Dimitrios M. Thilikos, Professor, Dep. of Mathematics, National and Kapodistrian University of Athens. Supervisor: Stavros G. Kolliopoulos, Dimitrios M. Thilikos, Associate Professor, Professor, Dep. of Informatics and Dep. of Mathematics, National and Telecommunications, National and Kapodistrian University of Athens. Kapodistrian University of Athens. white Lefteris M. Kirousis, Professor, Dep. of Mathematics, National and Kapodistrian University of Athens. Aposunjèseic sfairik¸n tom¸n kai σύνοla kuriarqÐac se epÐpeda γραφήματa Miχάλης Σάμαρης A.M. 201314 Τριμελής Epiτροπή: Δημήτρioc M. Jhlυκός, Epiblèpwn: Kajhγητής, Tm. Majhmatik¸n, E.K.P.A. Δημήτρioc M. Jhlυκός, Staύρoc G. Kolliόποuloc, Kajhγητής tou Τμήμatoc Anaπληρωτής Kajhγητής, Tm. Plhroforiκής Majhmatik¸n tou PanepisthmÐou kai Thl/ni¸n, E.K.P.A. Ajhn¸n Leutèrhc M. Kuroύσης, white Kajhγητής, Tm. Majhmatik¸n, E.K.P.A. PerÐlhyh 'Ena σημαντικό apotèlesma sth JewrÐa Γραφημάτwn apoteleÐ h apόdeixh thc eikasÐac tou Wagner από touc Neil Robertson kai Paul D. Seymour. sth σειρά ergasi¸n ‘Ελλάσσοna Γραφήματα’ apo to 1983 e¸c to 2011. H eikasÐa αυτή lèei όti sthn κλάση twn γραφημάtwn den υπάρχει άπειρη antialusÐda ¸c proc th sqèsh twn ελλασόnwn γραφημάτwn. H JewrÐa pou αναπτύχθηκε gia thn απόδειξη αυτής thc eikasÐac eÐqe kai èqei ακόμα σημαντικό antÐktupo tόσο sthn δομική όσο kai sthn algoriθμική JewrÐa Γραφημάτwn, άλλα kai se άλλα pedÐa όπως h Παραμετρική Poλυπλοκόthta. Sta πλάιsia thc απόδειξης oi suggrafeÐc eiσήγαγαν kai nèec paramètrouc πλά- touc. Se autèc ήτan h κλαδοαποσύνθεση kai to κλαδοπλάτoc ενός γραφήματoc. H παράμετρος αυτή χρησιμοποιήθηκε idiaÐtera sto σχεδιασμό algorÐjmwn kai sthn χρήση thc τεχνικής ‘διαίρει kai basÐleue’.
    [Show full text]
  • Derandomizing Isolation Lemma for K3,3-Free and K5-Free Bipartite Graphs
    Derandomizing Isolation Lemma for K3,3-free and K5-free Bipartite Graphs Rahul Arora1, Ashu Gupta2, Rohit Gurjar∗3, and Raghunath Tewari4 1 University of Toronto, Canada; and Indian Institute of Technology Kanpur, India [email protected], [email protected] 2 University of Illinois at Urbana-Champaign, USA [email protected] 3 HTW Aalen, Germany [email protected] 4 Indian Institute of Technology Kanpur, India [email protected] Abstract The perfect matching problem has a randomized NC algorithm, using the celebrated Isolation Lemma of Mulmuley, Vazirani and Vazirani. The Isolation Lemma states that giving a random weight assignment to the edges of a graph ensures that it has a unique minimum weight perfect matching, with a good probability. We derandomize this lemma for K3,3-free and K5-free bipart- ite graphs. That is, we give a deterministic log-space construction of such a weight assignment for these graphs. Such a construction was known previously for planar bipartite graphs. Our result implies that the perfect matching problem for K3,3-free and K5-free bipartite graphs is in SPL. It also gives an alternate proof for an already known result – reachability for K3,3-free and K5-free graphs is in UL. 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes, G.2.2 Graph Theory Keywords and phrases bipartite matching, derandomization, isolation lemma, SPL, minor-free graph Digital Object Identifier 10.4230/LIPIcs.STACS.2016.10 1 Introduction The perfect matching problem is one of the most extensively studied problem in combinatorics, algorithms and complexity.
    [Show full text]
  • Bidimensionality and Kernels1,2
    Bidimensionality and Kernels1;2 Fedor V. Fomin3;4;5 Daniel Lokshtanov6 Saket Saurabh3;7;8 Dimitrios M. Thilikos5;9;10 Abstract Bidimensionality Theory was introduced by [E. D. Demaine, F. V. Fomin, M. Hajiaghayi, and D. M. Thilikos. Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs, J. ACM, 52 (2005), pp. 866{893] as a tool to obtain sub-exponential time parameterized algorithms on H-minor-free graphs. In [E. D. Demaine and M. Haji- aghayi, Bidimensionality: new connections between FPT algorithms and PTASs, in Proceed- ings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2005, pp. 590{601] this theory was extended in order to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this work, we establish a third meta-algorithmic direc- tion for bidimensionality theory by relating it to the existence of linear kernels for parameter- ized problems. In particular, we prove that every minor (respectively contraction) bidimensional problem that satisfies a separation property and is expressible in Countable Monadic Second Order Logic (CMSO), admits a linear kernel for classes of graphs that exclude a fixed graph (respectively an apex graph) H as a minor. Our results imply that a multitude of bidimensional problems admit linear kernels on the corresponding graph classes. For most of these problems no polynomial kernels on H-minor-free graphs were known prior to our work. Keywords: Kernelization, Parameterized algorithms, Treewidth, Bidimensionality 1Part of the results of this paper have appeared in [42]. arXiv:1606.05689v3 [cs.DS] 1 Sep 2020 2Emails of authors: [email protected], [email protected], [email protected]/[email protected], [email protected].
    [Show full text]
  • Linearity of Grid Minors in Treewidth with Applications Through Bidimensionality∗
    Linearity of Grid Minors in Treewidth with Applications through Bidimensionality∗ Erik D. Demaine† MohammadTaghi Hajiaghayi† Abstract We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an Ω(w) × Ω(w) grid graph as a minor. Thus grid minors suffice to certify that H-minor-free graphs have large treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and bounded-genus graphs, and is known not to hold for general graphs. The approach of this paper can be viewed more generally as a framework for extending combinatorial results on planar graphs to hold on H-minor-free graphs for any fixed H. Our result has many combinatorial consequences on bidimensionality theory, parameter-treewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial results has several algorithmic consequences including subexponential fixed-parameter algorithms and approximation algorithms. 1 Introduction The r × r grid graph1 is the canonical planar graph of treewidth Θ(r). In particular, an important result of Robertson, Seymour, and Thomas [38] is that every planar graph of treewidth w has an Ω(w) × Ω(w) grid graph as a minor. Thus every planar graph of large treewidth has a grid minor certifying that its treewidth is almost as large (up to constant factors). In their Graph Minor Theory, Robertson and Seymour [36] have generalized this result in some sense to any graph excluding a fixed minor: for every graph H and integer r > 0, there is an integer w > 0 such that every H-minor-free graph with treewidth at least w has an r × r grid graph as a minor.
    [Show full text]
  • The Bidimensionality Theory and Its Algorithmic
    The Bidimensionality Theory and Its Algorithmic Applications by MohammadTaghi Hajiaghayi B.S., Sharif University of Technology, 2000 M.S., University of Waterloo, 2001 Submitted to the Department of Mathematics in partial ful¯llment of the requirements for the degree of DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2005 °c MohammadTaghi Hajiaghayi, 2005. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author.............................................................. Department of Mathematics April 29, 2005 Certi¯ed by. Erik D. Demaine Associate Professor of Electrical Engineering and Computer Science Thesis Supervisor Accepted by . Rodolfo Ruben Rosales Chairman, Applied Mathematics Committee Accepted by . Pavel I. Etingof Chairman, Department Committee on Graduate Students 2 The Bidimensionality Theory and Its Algorithmic Applications by MohammadTaghi Hajiaghayi Submitted to the Department of Mathematics on April 29, 2005, in partial ful¯llment of the requirements for the degree of DOCTOR OF PHILOSOPHY Abstract Our newly developing theory of bidimensional graph problems provides general techniques for designing e±cient ¯xed-parameter algorithms and approximation algorithms for NP- hard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k £ k grid graph (and similar graphs) grows with k, typically as ­(k2), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex- removal parameters, dominating set, edge dominating set, r-dominating set, connected dominating set, connected edge dominating set, connected r-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices).
    [Show full text]
  • Planar Graphs and Wagner's And
    PLANAR GRAPHS AND WAGNER'S AND KURATOWSKI'S THEOREMS SQUID TAMAR-MATTIS Abstract. This is an expository paper in which we rigorously prove Wagner's Theorem and Kuratowski's Theorem, both of which establish necessary and sufficient conditions for a graph to be planar. Contents 1. Introduction and Basic Definitions 1 2. Kuratowski's Theorem 3 3. Wagner's Theorem 12 Acknowledgments 13 References 13 1. Introduction and Basic Definitions The planarity of a graph|its ability to be embedded in a plane|is a deceptively meaningful property which, through various theorems, can tell us many other things about a graph. Among other properties, planar graphs were famously found to be 4-colorable. Of course, to use such theorems to determine whether a graph has these properties, we must first determine whether that graph is planar. Wagner's and Kuratowki's theorems show that there are simple and easily testable charac- terizations of planarity, but proving that they work is much less simple. Before we can do that, we must establish some definitions. Definition 1.1. An undirected graph is an ordered pair G = (V; E), where V is a set of vertices, and E is a set of edges, where each edge is a set of two vertices. We will mostly refer to these simply as graphs, as this paper does not deal with directed graphs. Definition 1.2. A graph H is a subgraph of G if H = (W; F ) for some W ⊆ V and F ⊆ E. Definition 1.3. Two vertices u; v 2 V are adjacent in G if fu; vg 2 E.
    [Show full text]
  • Contraction-Bidimensionality of Geometric Intersection Graphs∗
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Dagstuhl Research Online Publication Server Contraction-Bidimensionality of Geometric Intersection Graphs∗ Julien Baste1 and Dimitrios M. Thilikos2 1 Université de Montpellier, LIRMM, Montpellier, France [email protected] 2 AlGCo project team, CNRS, LIRMM, Montpellier, France and Department of Mathematics, National and Kapodistrian University of Athens, Greece [email protected] Abstract Given a graph G, we define bcg(G) as the minimum k for which G can be contracted to the uniformly triangulated grid Γk. A graph class G has the SQGC property if every graph G ∈ G has treewidth O(bcg(G)c) for some 1 ≤ c < 2. The SQGC property is important for algo- rithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes. These results apply to a wide family of problems, namely problems that are contraction-bidimensional. Our main combinatorial result reveals a general family of graph classes that satisfy the SQGC property and includes bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for several intersection graph classes of 2-dimensional geometrical objects. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.2 Graph Theory Keywords and phrases Grid exlusion theorem, Bidimensionality, Geometric intersection graphs, String Graphs Digital Object Identifier 10.4230/LIPIcs.IPEC.2017.5 1 Introduction Treewidth is one of most well-studied parameters in graph algorithms. It serves as a measure of how close a graph is to the topological structure of a tree (see Section 2 for the formal definition).
    [Show full text]
  • Contraction-Bidimensionality of Geometric Intersection Graphs Julien Baste, Dimitrios M
    Contraction-Bidimensionality of Geometric Intersection Graphs Julien Baste, Dimitrios M. Thilikos To cite this version: Julien Baste, Dimitrios M. Thilikos. Contraction-Bidimensionality of Geometric Intersection Graphs. 12th International Symposium on Parameterized and Exact Computation (IPEC 2017), Sep 2017, Vienne, Austria. pp.5:1–5:13, 10.4230/LIPIcs.IPEC.2017.5. lirmm-01890527 HAL Id: lirmm-01890527 https://hal-lirmm.ccsd.cnrs.fr/lirmm-01890527 Submitted on 8 Oct 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Contraction-Bidimensionality of Geometric Intersection Graphs∗ Julien Baste1 and Dimitrios M. Thilikos2,3 1 Université de Montpellier, LIRMM, Montpellier, France 2 AlGCo project team, CNRS, LIRMM, Montpellier, France 3 Department of Mathematics, National and Kapodistrian University of Athens, Greece Abstract Given a graph G, we define bcg(G) as the minimum k for which G can be contracted to the uniformly triangulated grid Γk. A graph class G has the SQGC property if every graph G ∈ G has treewidth O(bcg(G)c) for some 1 ≤ c < 2. The SQGC property is important for algo- rithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes.
    [Show full text]
  • Fast Algorithms for Hard Graph Problems: Bidimensionality, Minors, and Local Treewidth
    Fast Algorithms for Hard Graph Problems: Bidimensionality, Minors, and Local Treewidth Erik D. Demaine and MohammadTaghi Hajiaghayi MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar Street, Cambridge, MA 02139, USA, {edemaine,hajiagha}@mit.edu Abstract. This paper surveys the theory of bidimensional graph prob- lems. We summarize the known combinatorial and algorithmic results of this theory, the foundational Graph Minor results on which this theory is based, and the remaining open problems. 1 Introduction The newly developing theory of bidimensional graph problems, developed in a series of papers [DHT,DHN+04,DFHT,DH04a,DFHT04b,DH04b,DFHT04a, DHT04,DH05b,DH05a], provides general techniques for designing efficient fixed- parameter algorithms and approximation algorithms for NP-hard graph prob- lems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k × k grid graph (and similar graphs) grows with k, typically as Ω(k2), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex-removal parameters, dominating set, edge dominating set, R-dominating set, connected dominating set, connected edge dominating set, connected R-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). Bidimensional problems have many structural properties; for example, any graph in an appropriate minor-closed class has treewidth bounded above in terms of the problem’s solution value, typically by the square root of that value. These properties lead to efficient—often subexponential—fixed-parameter algorithms, as well as polynomial-time approximation schemes, for many minor-closed graph classes.
    [Show full text]
  • Lecture Beyond Planar Graphs 1 Overview 2 Preliminaries
    Approximation Algorithms Workshop June 17, 2011, Princeton Lecture Beyond Planar Graphs Erik Demaine Scribe: Siamak Tazari 1 Overview In the last lecture we learned about approximation schemes in planar graphs. In this lecture we go beyond planar graphs and consider bounded-genus graphs, and more generally, minor-closed graph classes. In particular, we discuss the framework of bidimensionality that can be used to obtain approximation schemes for many problems in minor-closed graph classes. Afterwards, we consider decomposition techniques that can be applied to other types of problems. The main motivation behind this line of work is that some networks are not planar but conform to some generalization of the concept of planarity, like being embeddable on a surface of bounded genus. Furthermore, by Kuratowski's famous theorem [Kur30], planarity can be characterized be excluding K5 and K3;3 as minors. Hence, it seems natural to study further classes with excluded minors as a generalization of planar graphs. One of the goals of this research is to identify the largest classes of graphs for which we can still ob- tain good and efficient approximations, i.e. to find out how far beyond planarity we can go. Natural candidate classes are graphs of bounded genus, graphs excluding a fixed minor, powers thereof, but also further generalizations, like odd-minor-free graphs, graphs of bounded expansion and nowhere dense classes of graphs. Another objective is to derive general approximation frameworks, two of which we are going to discuss here: bidimensionality and contraction decomposition. The main tools we are going to use come on one hand from graph structure theory and on the other hand from algorithmic results.
    [Show full text]