Fixed-Parameter Algorithms for (k,r)-Center in Planar Graphs and Map Graphs Erik D. Demaine, Fedor V. Fomin, MohammadTaghi Hajiaghayi, and Dimitrios M.

Slides:



Advertisements
Similar presentations
Bart Jansen 1.  Problem definition  Instance: Connected graph G, positive integer k  Question: Is there a spanning tree for G with at least k leaves?
Advertisements

Minimum Vertex Cover in Rectangle Graphs
Graph Isomorphism Algorithms and networks. Graph Isomorphism 2 Today Graph isomorphism: definition Complexity: isomorphism completeness The refinement.
Presented by Yuval Shimron Course
Bart Jansen, Utrecht University. 2  Max Leaf  Instance: Connected graph G, positive integer k  Question: Is there a spanning tree for G with at least.
Bart Jansen, Utrecht University. 2  Max Leaf  Instance: Connected graph G, positive integer k  Question: Is there a spanning tree for G with at least.
Combinatorial Algorithms
CS774. Markov Random Field : Theory and Application Lecture 17 Kyomin Jung KAIST Nov
Welcome to the TACO Project Finding tree decompositions Hans L. Bodlaender Institute of Information and Computing Sciences Utrecht University.
CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch.
What is the next line of the proof? a). Let G be a graph with k vertices. b). Assume the theorem holds for all graphs with k+1 vertices. c). Let G be a.
Greedy Algorithms Reading Material: Chapter 8 (Except Section 8.5)
“IBM Research Report A faster Exponential-Time Algorithm for Max 2-Sat, Max Cut, and Max k- Cut”, Alexander D. Scott, Gregory B. Sorkin, IBM Research Division.
Utrecht, february 22, 2002 Applications of Tree Decompositions Stan van Hoesel KE-FdEWB Universiteit Maastricht
Greedy Algorithms Like dynamic programming algorithms, greedy algorithms are usually designed to solve optimization problems Unlike dynamic programming.
New Algorithm DOM for Graph Coloring by Domination Covering
Steiner trees Algorithms and Networks. Steiner Trees2 Today Steiner trees: what and why? NP-completeness Approximation algorithms Preprocessing.
What is the next line of the proof? a). Assume the theorem holds for all graphs with k edges. b). Let G be a graph with k edges. c). Assume the theorem.
K-Coloring k-coloring: A k-coloring of a graph G is a labeling f: V(G)  S, where |S|=k. The labels are colors; the vertices of one color form a color.
1 Refined Search Tree Technique for Dominating Set on Planar Graphs Jochen Alber, Hongbing Fan, Michael R. Fellows, Henning Fernau, Rolf Niedermeier, Fran.
K-Coloring k-coloring: A k-coloring of a graph G is a labeling f: V(G)  S, where |S|=k. The labels are colors; the vertices of one color form a color.
A linear time algorithm for recognizing a K 5 -minor Bruce Reed Zhentao Li.
Fixed Parameter Complexity Algorithms and Networks.
Simple and Improved Parameterized Algorithms for Multiterminal Cuts Mingyu Xiao The Chinese University of Hong Kong Hong Kong SAR, CHINA CSR 2008 Presentation,
Graph Coalition Structure Generation Maria Polukarov University of Southampton Joint work with Tom Voice and Nick Jennings HUJI, 25 th September 2011.
Kernel Bounds for Structural Parameterizations of Pathwidth Bart M. P. Jansen Joint work with Hans L. Bodlaender & Stefan Kratsch July 6th 2012, SWAT 2012,
Approximating Minimum Bounded Degree Spanning Tree (MBDST) Mohit Singh and Lap Chi Lau “Approximating Minimum Bounded DegreeApproximating Minimum Bounded.
Trees and Distance. 2.1 Basic properties Acyclic : a graph with no cycle Forest : acyclic graph Tree : connected acyclic graph Leaf : a vertex of degree.
1 Bart Jansen Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter STACS 2011, Dortmund March 10 th, 2011 Joint work with.
Edge-disjoint induced subgraphs with given minimum degree Raphael Yuster 2012.
NP-complete Problems SAT 3SAT Independent Set Hamiltonian Cycle
Graph Colouring L09: Oct 10. This Lecture Graph coloring is another important problem in graph theory. It also has many applications, including the famous.
The length of vertex pursuit games Anthony Bonato Ryerson University CCC 2013.
NP-Complete Problems. Running Time v.s. Input Size Concern with problems whose complexity may be described by exponential functions. Tractable problems.
The Dominating Set and its Parametric Dual  the Dominated Set  Lan Lin prepared for theory group meeting on June 11, 2003.
Graph.
The Vertex Arboricity of Integer Distance Graph with a Special Distance Set Juan Liu* and Qinglin Yu Center for Combinatorics, LPMC Nankai University,
1 IM.CJCU Hsin-Hung Chou The Node-Searching Problem on Special Graphs 周信宏 長榮大學 資訊管理學系
Computing Branchwidth via Efficient Triangulations and Blocks Authors: F.V. Fomin, F. Mazoit, I. Todinca Presented by: Elif Kolotoglu, ISE, Texas A&M University.
CSE 421 Algorithms Richard Anderson Winter 2009 Lecture 5.
Introduction to Graph Theory
Algorithms for hard problems Parameterized complexity – definitions, sample algorithms Juris Viksna, 2015.
NP-completeness NP-complete problems. Homework Vertex Cover Instance. A graph G and an integer k. Question. Is there a vertex cover of cardinality k?
NPC.
Indian Institute of Technology Kharagpur PALLAB DASGUPTA Graph Theory: Trees Pallab Dasgupta, Professor, Dept. of Computer Sc. and Engineering, IIT
1 Assignment #3 is posted: Due Thursday Nov. 15 at the beginning of class. Make sure you are also working on your projects. Come see me if you are unsure.
Algorithms for hard problems Parameterized complexity Bounded tree width approaches Juris Viksna, 2015.
CSE 421 Algorithms Richard Anderson Autumn 2015 Lecture 5.
Approximating graph coloring of minor-closed graphs Joint Work with Erik Demaine, Mohammad Hajiaghayi, Bojan Mohar, Robin Thomas Partially joint Work with.
Branchwidth via Integer Programming
Algorithms for Finding Distance-Edge-Colorings of Graphs
What is the next line of the proof?
Exact Algorithms via Monotone Local Search
Algorithms and networks
Dissertation for the degree of Philosophiae Doctor (PhD)
Autumn 2016 Lecture 11 Minimum Spanning Trees (Part II)
Planarity Testing.
Autumn 2015 Lecture 11 Minimum Spanning Trees (Part II)
Trees.
REDUCESEARCH Polynomial Kernels for Hitting Forbidden Minors under Structural Parameterizations Bart M. P. Jansen Astrid Pieterse ESA 2018 August.
Problem Solving 4.
Richard Anderson Autumn 2016 Lecture 5
Three-coloring triangle-free planar graphs in linear time (SODA 09’)
Distance Approximating Trees: Complexity and Algorithms
Richard Anderson Winter 2009 Lecture 6
Clustering.
Winter 2019 Lecture 11 Minimum Spanning Trees (Part II)
Richard Anderson Lecture 5 Graph Theory
Treewidth meets Planarity
Autumn 2019 Lecture 11 Minimum Spanning Trees (Part II)
Presentation transcript:

Fixed-Parameter Algorithms for (k,r)-Center in Planar Graphs and Map Graphs Erik D. Demaine, Fedor V. Fomin, MohammadTaghi Hajiaghayi, and Dimitrios M. Thilikos ACM Transactions on Algorithms, volume 1, number 1, July 2005, pages Jeryann Huang

(k,r)-center problem Given an unweighted graph G and asks whether G has ≤ k vertices (centers) such that every vertex of G is within distance ≤ r from some center. Results:  For planar graphs, the running time is  For map graphs, the running time is where n is the number of vertices

Previous Results Alber et al. [2002]: Kanj and Perkovi´c [2002]: Fomin and Thilikos [2003]: Most of these problems have reductions to the dominating set problem.

Minor A graph G has a minor H if H can be formed by removing and contracting edges of G Otherwise, G is H-minor-free For example, planar graphs are both K 3,3 -minor-free and K 5 -minor-free delete contract

Map Graphs THEOREM 2.1 [CHEN ET AL. 2002]. A graph G M is a map graph if and only if it is the half-square of some planar bipartite graph H.  Graph H is a witness for G M.  Finding a (k, r )-center in a map graph G M is equivalent to finding in a witness H of G M a set S ⊆ V(G M ) of size k such that every vertex in V(G M ) − S has distance ≤ 2r in H from some vertex of S

Branchwidth Branch decomposition: A pair (T, v), where T is a tree with vertices of degree 1 or 3 and v is a bijection from E(G) to the set of leaves of T order function ω : E(T ) → 2 V(G) of a branch decomposition maps every edge e of T to a subset of vertices ω(e) ⊆ V(G) G e (T, )

Branchwidth (cont.) The width of (T, v) is equal to max e ∈ E(T ) |ω(e)| and the branchwidth of G, bw(G), is the minimum width over all branch decompositions of G. G a b d e f g c b g d c a f e width = 4

Combinatorial Bounds LEMMA 3.1. Let p, k, r ≥ 1 be integers and G be a planar graph having a (k, r )-center and with a (p × p)-grid as a minor. Then k ≥ ( p−2r / 2r+1 ) 2. THEOREM 2.2 [ROBERTSON ET AL. 1994] G contains a (p ×p)-grid as a minor where p = (2r + 1)√k + 2r + c/4 for some c, 0 < c <= 4 p p

Combinatorial Bounds (cont.) THEOREM 3.2. For any planar graph G having a (k, r )-center, bw(G) ≤ 4(2r+ 1)√k + 8r + 1 THEOREM 3.3. For any map graph G M having a (k, r )-center and its witness H, bw(H) ≤ 4(4r + 3)√k + 16r + 9

Algorithms Step1: Check whether the branchwidth of G is at most 4(2r +1)√k+8r +1  If the answer is negative, report no and stop.  Running time: O((|V(G)| + |E(G)|) 2 ) Step2: Compute an optimal branch- decomposition of graph G  Running time: O((|V(G)| + |E(G)|) 4 ) Step 3: Compute, if is exists, a (k,r)-center of G using dynamic-programming algorithm

Results Dynamic program are defined by a coloring of the vertices in ω( f ) for every edge f of T. Each vertex will be assigned one of 2r + 1 colors {0, ↑1, ↑2,..., ↑r, ↓1, ↓2,..., ↓r }  0 means that the vertex v is a chosen center.  ↓i means that vertex v has distance exactly i to the closest center c with a neighbor that has distance i-1  ↑i means that vertex v has distance exactly i to the closest center c with no neighbor

Results (cont.) For a graph G on m edges and with a given branch decomposition of width ≤ L, and integers k, r, the existence of a (k, r )-center in G can be checked in O((2r + 1) 3/2*L ·m) time and, in case of a positive answer, constructs a (k, r )- center of G in the same time. For planar graph, L= 4(2r+ 1)√k + 8r + 1 Running time = For map graph, L = 4(4r + 3)√k + 16r + 9 Running time =