Small independent edge dominating sets in graphs of maximum degree three


 Paula James
 2 years ago
 Views:
Transcription
1 Small independent edge dominating sets in graphs of maimum degree three Grażna Zwoźniak Institute of Computer Science Wrocław Universit Small independent edge dominating sets in graphs of maimum degree three p. 1/34
2 Problem definition Given a connected undirected graph G = (V, E) Find a minimum set of edges E E(G) which fulfils the following conditions: for ever edge (u, w) E(G) either (u, w) E or one of (, u), (w, ) E(G) belongs to E no two edges of E share a common endpoint. Small independent edge dominating sets in graphs of maimum degree three p. 2/34
3 Arbitrar graphs results Minimum independent edge dominating set problem is NPcomplete even when restricted to planar or bipartite graphs of maimum degree three M. Yannakakis, F. Gavril, Journal on Appl. Math The problem remains NPcomplete for planar bipartite graphs, line graphs, total graphs, planar cubic graphs J. D. Horton, K. Kilakos, Journal on Disc. Math There is a polnomial time algorithm for trees M. Yannakakis, F. Gavril, Journal on Appl. Math Small independent edge dominating sets in graphs of maimum degree three p. 3/34
4 Arbitrar graphs approimation It is NPhard to approimate the problem within an factor smaller than 7/6 in general graphs and smaller than 1+1/487 in cubic graphs M. Chlebík, J. Chlebíková, Algorithms and Computations 2003 Constructing an maimal matching gives approimation ratio of 2 for the problem Small independent edge dominating sets in graphs of maimum degree three p. 4/34
5 Cubic graphs results The minimum independent edge dominating set of an nverte cubic graph is at most 9n/20+O(1) W. Duckworth, N. C. Wormald, 2000 There are families of cubic graphs for which the size of the minimum independent edge dominating set is at least 3n/8 W. Duckworth, N. C. Wormald, 2000 The lower bound for the size of the minimum independent edge dominating set is 3n/10 Small independent edge dominating sets in graphs of maimum degree three p. 5/34
6 Our results An approimation algorithm which finds the independent edge dominating set of size at most 4n/9+1/3 in an nverte graph of maimum degree three. As a collorar we obtain an approimation ratio of 40/27 for the problem in cubic graphs. Small independent edge dominating sets in graphs of maimum degree three p. 6/34
7 Rules (a) (b) (c) (d) (e) (f) T i T i T i T i T i T i v v v v v v u w u u u 1 u 1 1 u 1 w 1 w 2 w w u s u 2 w u s w Small independent edge dominating sets in graphs of maimum degree three p. 7/34
8 Construction of the forest F 1. F 2. V (T 0 ) {r 0, v 1, v 2, v 3 }, where r 0 is an degree three verte of G and v 1, v 2, v 3 Γ G (r 0 ); E(T 0 ) {(r 0, v 1 ), (r 0, v 2 ), (r 0, v 3 )}; let r 0 be a root of T 0 ; i 0 3. if it is possible find the leftmost leaf in T i that can be epanded b the Rule 1 and epand it; go to the step 3; else go to the step 4; 4. if it is possible find the leftmost leaf in T i that can be epanded b the Rule 2 and epand it; go to the step 3 else F F T i and go to the step 5 5. if it is possible find the leftmost leaf v in T i such that Rule 3 can be applied to v and appl this rule to v; i i + 1; let T i be a new tree created in this step; go to the step 3 with T i else go to the step 5 with the father of T i Small independent edge dominating sets in graphs of maimum degree three p. 8/34
9 Properties of F Fact 1 Let v V (T i ), T i F. 1. If deg Ti (v) = 2 and w Γ Ti (v) then deg Ti (w) = If deg Ti (v) = 2 and w Γ G (v) then w V (T i ). 3. If v is adjacent to the verte w EX R, u Γ G (v) and u w then u V (T i ). Small independent edge dominating sets in graphs of maimum degree three p. 9/34
10 The edges e E(G)\E(F) (a) (b) (c) (d) (e) (f) T i T i T i T i T j T j T j T j T j T j (g) (h) (i) (j) T i T i T i T i T j T j T j T j Small independent edge dominating sets in graphs of maimum degree three p. 10/34
11 Notation G current graph F current forest EX = V (G )\V (F ) S the subtree of T F rooted at some verte V (T ) EDS(G) independent edge dominating set contructed b our algorithm Small independent edge dominating sets in graphs of maimum degree three p. 11/34
12 DOMINATE i DOMINATE 1 (S S ) dominates the edges of the subtrees S, S, where deg T () = 2 and is the endpoint of the edge (, ) E(G)\E(F) together with edges e E(G )\E(F ) incident to V (S ) V (S ). DOMINATE 2 (S ) which dominates the edges of some paths which end at the leaves of S and go through the edges e E(G )\E(F ). DOMINATE 3 (S ) dominates E(S ) and all the edges incident to V (S ). Small independent edge dominating sets in graphs of maimum degree three p. 12/34
13 Main propert Propert 1 The procedure DOMINATE i (), i {1, 2, 3} adds m edges to EDS(G) and removes n vertices from V (G ), where m 4 9 n. The onl eception ma be the case when DOMINATE 3 () is applied to the tree rooted at r 0 or at the son of r 0 ; then m 4 9 n Small independent edge dominating sets in graphs of maimum degree three p. 13/34
14 Required edges (a) (b) (c) w w z u u z w V (F) u v L(S ) v EX R v Small independent edge dominating sets in graphs of maimum degree three p. 14/34
15 DOMINATE 3 (S )  basic (a) (b) (c) (d) (e) (f) (g) Small independent edge dominating sets in graphs of maimum degree three p. 15/34
16 DOMINATE 3 (S ) when h(s ) = 2 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) Small independent edge dominating sets in graphs of maimum degree three p. 16/34
17 S definition Let S be a subgraph of S which fulfils the following condition: if two edges (u 1, u 2 ), (v 1, v 2 ) E(S ) where either the path from u 2 to in T is shorter than the path from v 2 to in T or u 2 is on the left of v 2 in S, are incident to an edge (u 2, v 2 ) REQUIRED(S ), then the edge (u 1, u 2 ) E(S ). Small independent edge dominating sets in graphs of maimum degree three p. 17/34
18 DOMINATE 3 (S ) when h(s ) = 2 and = r (a) (b) (c) r r r u 1 u 1 u 2 w 1 w 1 w 2 Small independent edge dominating sets in graphs of maimum degree three p. 18/34
19 DOMINATE 3 (S ) when h(s ) = 2 (a) (b) (c) z 2 z 1 w z 2 z 1 z 2 z 1 w w z 1 w Small independent edge dominating sets in graphs of maimum degree three p. 19/34
20 Impossible to appl DOMINATE 3 (S ) (a) (b) (c) (d) (e) (f) z z z Small independent edge dominating sets in graphs of maimum degree three p. 20/34
21 DOMINATE 3 (S ) when h(s ) = 3 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) Small independent edge dominating sets in graphs of maimum degree three p. 21/34
22 DOMINATE 3 (S ) when h(s ) = 3 (a) (b) (c) (d) (e) w w w w w Small independent edge dominating sets in graphs of maimum degree three p. 22/34
23 DOMINATE(S ) when = r 0 (a) (b) (c) (d) (e) r 0 r 0 r 0 r 0 r 0 (f) (g) (h) (i) r 0 r 0 r 0 r 0 r 0 (j) (k) (l) r 0 r 0 Small independent edge dominating sets in graphs of maimum degree three p. 23/34
24 DOMINATE(S ) when is the last son of r 0 (a) (b) (c) (d) (e) (f) (g) (h) (i) Small independent edge dominating sets in graphs of maimum degree three p. 24/34
25 DOMINATE 1 (S S ) (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) Small independent edge dominating sets in graphs of maimum degree three p. 25/34
26 DOMINATE 2 (S ) (a) (b) (c) (d) S S S S v 4 v 6 u 1 w 1 u 1 w 1 w 2 v 3 v 2 v 1 v 5 v 4 v 3 v 2 v 1 v 1 v 2 v 3 v 1 v 2 v 3 v 4 v 5 (e) (f) (g) T T T w 2 S u 2 S u 2 S w 1 w 1 w 2 u 1 u 2 w 1 w 1 u 1 u 1 u 2 u 1 w 1 v 3 v 2 v 1 v 1 v 2 v 3 v 1 v 2 v 3 Small independent edge dominating sets in graphs of maimum degree three p. 26/34
27 EDS_T(T ) 1. appl procedure EDS_T() to the sons T i F of T (successivel from the leftmost to the rightmost); 2. EDS(T ). Small independent edge dominating sets in graphs of maimum degree three p. 27/34
28 EDS(S ) Let W = {u V (S ) : deg F (u) = 2 and deg G (u) = 3}. 1. while W (a) appl inorder search and find the first verte u W ; let w be such a verte that (u, w) E(G)\E(F); (b) EDS(S u); (c) if w V (F ): EDS(S w); (d) if deg G (u) = 3 or deg G (w) = 3: MARK_REQUIRED(S u S w); DOMINATE_PATHS(S u S w); DOMINATE 1 (S u S w); 2. MARK(S ). Small independent edge dominating sets in graphs of maimum degree three p. 28/34
29 MARK_REQUIRED(S ) Let W = {(u, w) E(G )\E(F ) : u L(S )}. 1. REQUIRED(S ) := 2. while there is a verte u L(S ) and the edges (u, v), (u, w) W, v L(S ), w L(S ): find the first u, add (u, w) to REQUIRED(S ) and remove the edges incident to u, w from W. 3. while there is a verte u L(S ) and the edge (u, w) W, w EX R: find the first u, add (u, w) to REQUIRED(S ) and remove the edges incident to u, w from W. 4. while there is a verte u L(S ) and the edge (u, w) W : find the first u, add (u, w) to REQUIRED(S ) and remove the edges incident to u, w from W. Small independent edge dominating sets in graphs of maimum degree three p. 29/34
30 DOMINATE_PATHS(S ) 1. if TAILS 1 (S ) or EARS 1 (S ) : DOMINATE 2 (S ); 2. MARK_REQUIRED(S ); 3. if TAILS 2 (S ) or EARS 2 (S ) or EARS 3 (S ) : DOMINATE 2 (S ); goto 2. Small independent edge dominating sets in graphs of maimum degree three p. 30/34
31 MARK(S ) 1. if h(s ) > 2: appl procedure MARK() to the trees rooted at the sons of (successivel from the leftmost to the rightmost); 2. MARK_REQUIRED(S ); 3. DOMINATE_PATHS(S ); 4. if has at most two sons u i, 1 i 2: (a) while DOMINATION_POSSIBLE(S u i ): DOMINATE 3 (S u i ); (b) if DOMINATION_POSSIBLE(S ): DOMINATE 3 (S ); else: return; 5. if has three sons u i, 1 i 3 (in this case = r 0 ): (a) while DOMINATION_POSSIBLE(S u i ): DOMINATE 3 (S u i ); (b) if h(s r 0 ) > 1 and deg G (r 0 ) = 3: let u 3 V ( S r 0 ); i. DOMINATE 3 ( S r 0 ); ii. if u 3 V (G ): MARK_REQUIRED(S u 3 ); DOMINATE 3 (S u 3 ); (c) else if r 0 V (G ): DOMINATE 3 (S rsmall 0 ). independent edge dominating sets in graphs of maimum degree three p. 31/34
32 Properties of F Fact 2 Let deg T (u) = deg T (w) = 2, T F, (u, w) E(G)\E(F), a = LCA T (u, w) and let u be on the left of w in T. Then w is the right son of a, and there are no vertices of degree 2 on the path from u to a in T. Small independent edge dominating sets in graphs of maimum degree three p. 32/34
33 Properties of F Fact 3 Let r be a root of a tree T F. Let u L(T) and let w be the first verte of degree 2 on the path from u to r in T. If there is v V (T) such that deg F (v) = 2 and (u, v) E(G)\E(T) then w is an ancestor of v in T. Small independent edge dominating sets in graphs of maimum degree three p. 33/34
34 Main theorem Theorem 1 Let G be a graph of maimum degree three, where V (G) = n. Our algorithm constructs for G an independent edge dominating set EDS(G) of size at most 4n/9 +1/3 in linear time. Small independent edge dominating sets in graphs of maimum degree three p. 34/34
On the kpath cover problem for cacti
On the kpath cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we
More informationConnectivity and cuts
Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every
More informationIE 680 Special Topics in Production Systems: Networks, Routing and Logistics*
IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* Rakesh Nagi Department of Industrial Engineering University at Buffalo (SUNY) *Lecture notes from Network Flows by Ahuja, Magnanti
More informationA2 1 10Approximation Algorithm for a Generalization of the Weighted EdgeDominating Set Problem
Journal of Combinatorial Optimization, 5, 317 326, 2001 c 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. A2 1 Approximation Algorithm for a Generalization of the Weighted EdgeDominating
More informationDiscrete Applied Mathematics. The firefighter problem with more than one firefighter on trees
Discrete Applied Mathematics 161 (2013) 899 908 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The firefighter problem with
More information2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8]
Code No: R05220502 Set No. 1 1. (a) Describe the performance analysis in detail. (b) Show that f 1 (n)+f 2 (n) = 0(max(g 1 (n), g 2 (n)) where f 1 (n) = 0(g 1 (n)) and f 2 (n) = 0(g 2 (n)). [8+8] 2. (a)
More informationCombinatorial 5/6approximation of Max Cut in graphs of maximum degree 3
Combinatorial 5/6approximation of Max Cut in graphs of maximum degree 3 Cristina Bazgan a and Zsolt Tuza b,c,d a LAMSADE, Université ParisDauphine, Place du Marechal de Lattre de Tassigny, F75775 Paris
More informationAlgorithms Chapter 12 Binary Search Trees
Algorithms Chapter 1 Binary Search Trees Outline Assistant Professor: Ching Chi Lin 林 清 池 助 理 教 授 chingchi.lin@gmail.com Department of Computer Science and Engineering National Taiwan Ocean University
More informationAnalysis of Algorithms, I
Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadthfirst search (BFS) 4 Applications
More informationAnalysis of Algorithms I: Binary Search Trees
Analysis of Algorithms I: Binary Search Trees Xi Chen Columbia University Hash table: A data structure that maintains a subset of keys from a universe set U = {0, 1,..., p 1} and supports all three dictionary
More informationTHE PROBLEM WORMS (1) WORMS (2) THE PROBLEM OF WORM PROPAGATION/PREVENTION THE MINIMUM VERTEX COVER PROBLEM
1 THE PROBLEM OF WORM PROPAGATION/PREVENTION I.E. THE MINIMUM VERTEX COVER PROBLEM Prof. Tiziana Calamoneri Network Algorithms A.y. 2014/15 2 THE PROBLEM WORMS (1)! A computer worm is a standalone malware
More informationA Fast Algorithm For Finding Hamilton Cycles
A Fast Algorithm For Finding Hamilton Cycles by Andrew Chalaturnyk A thesis presented to the University of Manitoba in partial fulfillment of the requirements for the degree of Masters of Science in Computer
More informationInstitut für Informatik Lehrstuhl Theoretische Informatik I / Komplexitätstheorie. An Iterative Compression Algorithm for Vertex Cover
FriedrichSchillerUniversität Jena Institut für Informatik Lehrstuhl Theoretische Informatik I / Komplexitätstheorie Studienarbeit An Iterative Compression Algorithm for Vertex Cover von Thomas Peiselt
More informationA binary search tree is a binary tree with a special property called the BSTproperty, which is given as follows:
Chapter 12: Binary Search Trees A binary search tree is a binary tree with a special property called the BSTproperty, which is given as follows: For all nodes x and y, if y belongs to the left subtree
More informationSocial Media Mining. Graph Essentials
Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures
More informationTotal colorings of planar graphs with small maximum degree
Total colorings of planar graphs with small maximum degree Bing Wang 1,, JianLiang Wu, SiFeng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong
More informationUPPER BOUNDS ON THE L(2, 1)LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE
UPPER BOUNDS ON THE L(2, 1)LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE ANDREW LUM ADVISOR: DAVID GUICHARD ABSTRACT. L(2,1)labeling was first defined by Jerrold Griggs [Gr, 1992] as a way to use graphs
More informationApproximating Minimum Bounded Degree Spanning Trees to within One of Optimal
Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal ABSTACT Mohit Singh Tepper School of Business Carnegie Mellon University Pittsburgh, PA USA mohits@andrew.cmu.edu In the MINIMUM
More informationCIS 700: algorithms for Big Data
CIS 700: algorithms for Big Data Lecture 6: Graph Sketching Slides at http://grigory.us/bigdataclass.html Grigory Yaroslavtsev http://grigory.us Sketching Graphs? We know how to sketch vectors: v Mv
More informationRotated Ellipses. And Their Intersections With Lines. Mark C. Hendricks, Ph.D. Copyright March 8, 2012
Rotated Ellipses And Their Intersections With Lines b Mark C. Hendricks, Ph.D. Copright March 8, 0 Abstract: This paper addresses the mathematical equations for ellipses rotated at an angle and how to
More informationSAMPLE. Polynomial functions
Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through
More informationa > 0 parabola opens a < 0 parabola opens
Objective 8 Quadratic Functions The simplest quadratic function is f() = 2. Objective 8b Quadratic Functions in (h, k) form Appling all of Obj 4 (reflections and translations) to the function. f() = a(
More informationEvery tree contains a large induced subgraph with all degrees odd
Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University
More informationONLINE DEGREEBOUNDED STEINER NETWORK DESIGN. Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015
ONLINE DEGREEBOUNDED STEINER NETWORK DESIGN Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015 ONLINE STEINER FOREST PROBLEM An initially given graph G. s 1 s 2 A sequence of demands (s i, t i ) arriving
More informationProduct irregularity strength of certain graphs
Also available at http://amc.imfm.si ISSN 18553966 (printed edn.), ISSN 18553974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (014) 3 9 Product irregularity strength of certain graphs Marcin Anholcer
More informationWhen I was 3.1 POLYNOMIAL FUNCTIONS
146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we
More informationScheduling. Open shop, job shop, flow shop scheduling. Related problems. Open shop, job shop, flow shop scheduling
Scheduling Basic scheduling problems: open shop, job shop, flow job The disjunctive graph representation Algorithms for solving the job shop problem Computational complexity of the job shop problem Open
More informationPolynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will
More informationPart 2: Community Detection
Chapter 8: Graph Data Part 2: Community Detection Based on Leskovec, Rajaraman, Ullman 2014: Mining of Massive Datasets Big Data Management and Analytics Outline Community Detection  Social networks 
More informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NPCompleteness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
More informationBOUNDARY EDGE DOMINATION IN GRAPHS
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 04874, ISSN (o) 04955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 19704 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA
More informationIdentify a pattern and find the next three numbers in the pattern. 5. 5(2s 2 1) 2 3(s 1 2); s 5 4
Chapter 1 Test Do ou know HOW? Identif a pattern and find the net three numbers in the pattern. 1. 5, 1, 3, 7, c. 6, 3, 16, 8, c Each term is more than the previous Each term is half of the previous term;
More informationSmall Maximal Independent Sets and Faster Exact Graph Coloring
Small Maximal Independent Sets and Faster Exact Graph Coloring David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science The Exact Graph Coloring Problem: Given an undirected
More information1 Definitions. Supplementary Material for: Digraphs. Concept graphs
Supplementary Material for: van Rooij, I., Evans, P., Müller, M., Gedge, J. & Wareham, T. (2008). Identifying Sources of Intractability in Cognitive Models: An Illustration using Analogical Structure Mapping.
More informationPolytope Examples (PolyComp Fukuda) Matching Polytope 1
Polytope Examples (PolyComp Fukuda) Matching Polytope 1 Matching Polytope Let G = (V,E) be a graph. A matching in G is a subset of edges M E such that every vertex meets at most one member of M. A matching
More informationFinding and counting given length cycles
Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected
More informationGlobal secure sets of trees and gridlike graphs. Yiu Yu Ho
Global secure sets of trees and gridlike graphs by Yiu Yu Ho B.S. University of Central Florida, 2006 M.S. University of Central Florida, 2010 A dissertation submitted in partial fulfillment of the requirements
More informationroot node level: internal node edge leaf node CS@VT Data Structures & Algorithms 20002009 McQuain
inary Trees 1 A binary tree is either empty, or it consists of a node called the root together with two binary trees called the left subtree and the right subtree of the root, which are disjoint from each
More informationData Management in Hierarchical Bus Networks
Data Management in Hierarchical Bus Networks F. Meyer auf der Heide Department of Mathematics and Computer Science and Heinz Nixdorf Institute, Paderborn University, Germany fmadh@upb.de H. Räcke Department
More informationHome Page. Data Structures. Title Page. Page 1 of 24. Go Back. Full Screen. Close. Quit
Data Structures Page 1 of 24 A.1. Arrays (Vectors) nelement vector start address + ielementsize 0 +1 +2 +3 +4... +n1 start address continuous memory block static, if size is known at compile time dynamic,
More informationBinary Search Trees > = Binary Search Trees 1. 2004 Goodrich, Tamassia
Binary Search Trees < 6 2 > = 1 4 8 9 Binary Search Trees 1 Binary Search Trees A binary search tree is a binary tree storing keyvalue entries at its internal nodes and satisfying the following property:
More informationAnswers to some of the exercises.
Answers to some of the exercises. Chapter 2. Ex.2.1 (a) There are several ways to do this. Here is one possibility. The idea is to apply the kcenter algorithm first to D and then for each center in D
More informationGRAPH THEORY LECTURE 4: TREES
GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection
More informationApproximating the Minimum Chain Completion problem
Approximating the Minimum Chain Completion problem Tomás Feder Heikki Mannila Evimaria Terzi Abstract A bipartite graph G = (U, V, E) is a chain graph [9] if there is a bijection π : {1,..., U } U such
More informationCounting Falsifying Assignments of a 2CF via Recurrence Equations
Counting Falsifying Assignments of a 2CF via Recurrence Equations Guillermo De Ita Luna, J. Raymundo Marcial Romero, Fernando Zacarias Flores, Meliza Contreras González and Pedro Bello López Abstract
More informationRegular Augmentations of Planar Graphs with Low Degree
Regular Augmentations of Planar Graphs with Low Degree Bachelor Thesis of Huyen Chau Nguyen At the Department of Informatics Institute of Theoretical Computer Science Reviewers: Advisors: Prof. Dr. Dorothea
More informationMATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2
MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we
More informationBargaining Solutions in a Social Network
Bargaining Solutions in a Social Network Tanmoy Chakraborty and Michael Kearns Department of Computer and Information Science University of Pennsylvania Abstract. We study the concept of bargaining solutions,
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. #approximation algorithm.
Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of three
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationV. Adamchik 1. Graph Theory. Victor Adamchik. Fall of 2005
V. Adamchik 1 Graph Theory Victor Adamchik Fall of 2005 Plan 1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs Introduction A.Aho and J.Ulman acknowledge that Fundamentally, computer
More informationApproximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs
Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and HingFung Ting 2 1 College of Mathematics and Computer Science, Hebei University,
More informationA Structural View on Parameterizing Problems: Distance from Triviality
A Structural View on Parameterizing Problems: Distance from Triviality Jiong Guo, Falk Hüffner, and Rolf Niedermeier WilhelmSchickardInstitut für Informatik, Universität Tübingen, Sand 13, D72076 Tübingen,
More informationOptimal Binary Space Partitions in the Plane
Optimal Binary Space Partitions in the Plane Mark de Berg and Amirali Khosravi TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, the Netherlands Abstract. An optimal bsp for a set S of disjoint line segments
More informationNetwork (Tree) Topology Inference Based on Prüfer Sequence
Network (Tree) Topology Inference Based on Prüfer Sequence C. Vanniarajan and Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology Madras Chennai 600036 vanniarajanc@hcl.in,
More informationBinary Search Trees (BST)
Binary Search Trees (BST) 1. Hierarchical data structure with a single reference to node 2. Each node has at most two child nodes (a left and a right child) 3. Nodes are organized by the Binary Search
More informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
More informationAn algorithmic classification of open surfaces
An algorithmic classification of open surfaces Sylvain Maillot January 8, 2013 Abstract We propose a formulation for the homeomorphism problem for open ndimensional manifolds and use the Kerekjarto classification
More information135 Final Review. Determine whether the graph is symmetric with respect to the xaxis, the yaxis, and/or the origin.
13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, 6); P2 = (7, 2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the ais, the ais, and/or the
More informationmost 4 Mirka Miller 1,2, Guillermo PinedaVillavicencio 3, The University of Newcastle Callaghan, NSW 2308, Australia University of West Bohemia
Complete catalogue of graphs of maimum degree 3 and defect at most 4 Mirka Miller 1,2, Guillermo PinedaVillavicencio 3, 1 School of Electrical Engineering and Computer Science The University of Newcastle
More informationNetwork File Storage with Graceful Performance Degradation
Network File Storage with Graceful Performance Degradation ANXIAO (ANDREW) JIANG California Institute of Technology and JEHOSHUA BRUCK California Institute of Technology A file storage scheme is proposed
More informationGENERATING THE FIBONACCI CHAIN IN O(log n) SPACE AND O(n) TIME J. Patera
ˆ ˆŠ Œ ˆ ˆ Œ ƒ Ÿ 2002.. 33.. 7 Š 539.12.01 GENERATING THE FIBONACCI CHAIN IN O(log n) SPACE AND O(n) TIME J. Patera Department of Mathematics, Faculty of Nuclear Science and Physical Engineering, Czech
More informationarxiv:1409.4299v1 [cs.cg] 15 Sep 2014
Planar Embeddings with Small and Uniform Faces Giordano Da Lozzo, Vít Jelínek, Jan Kratochvíl 3, and Ignaz Rutter 3,4 arxiv:409.499v [cs.cg] 5 Sep 04 Department of Engineering, Roma Tre University, Italy
More informationMore Equations and Inequalities
Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities
More information5.1 Bipartite Matching
CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the FordFulkerson
More informationMaking life easier for firefighters
Making life easier for firefighters Fedor V. Fomin, Pinar Heggernes, and Erik Jan van Leeuwen Department of Informatics, University of Bergen, Norway {fedor.fomin, pinar.heggernes, e.j.van.leeuwen}@ii.uib.no
More informationStudents Currently in Algebra 2 Maine East Math Placement Exam Review Problems
Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write
More informationA Turán Type Problem Concerning the Powers of the Degrees of a Graph
A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of HaifaORANIM, Tivon 36006, Israel. AMS Subject Classification:
More informationAutomated Plausibility Analysis of Large Phylogenies
Automated Plausibility Analysis of Large Phylogenies Bachelor Thesis of David Dao At the Department of Informatics Institute of Theoretical Computer Science Reviewers: Advisors: Prof. Dr. Alexandros Stamatakis
More informationSpecial Classes of Divisor Cordial Graphs
International Mathematical Forum, Vol. 7,, no. 35, 737749 Special Classes of Divisor Cordial Graphs R. Varatharajan Department of Mathematics, Sri S.R.N.M. College Sattur  66 3, Tamil Nadu, India varatharajansrnm@gmail.com
More informationComparing of SGML documents
Comparing of SGML documents MohammadTaghi Hajiaghayi mhajiaghayi@uwaterloo.ca Department of computer science University of Waterloo Abstract Documents can be represented as structures with a hierarchial
More informationThe Firefighter Problem: A Structural Analysis
Electronic Colloquium on Computational Complexity, Report No. 162 (2013) The Firefighter Problem: A Structural Analysis Janka Chlebíková a, Morgan Chopin b,1 a University of Portsmouth, School of Computing,
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More informationHigh degree graphs contain largestar factors
High degree graphs contain largestar factors Dedicated to László Lovász, for his 60th birthday Noga Alon Nicholas Wormald Abstract We show that any finite simple graph with minimum degree d contains a
More informationComputer Science Department. Technion  IIT, Haifa, Israel. Itai and Rodeh [IR] have proved that for any 2connected graph G and any vertex s G there
 1  THREE TREEPATHS Avram Zehavi Alon Itai Computer Science Department Technion  IIT, Haifa, Israel Abstract Itai and Rodeh [IR] have proved that for any 2connected graph G and any vertex s G there
More informationData Caching in Networks with Reading, Writing and Storage Costs
Data Caching in Networks with Reading, 1 Writing and Storage Costs Himanshu Gupta Computer Science Department Stony Brook University Stony Brook, NY 11790 Email: hgupta@cs.sunysb.edu Bin Tang Computer
More informationLecture 7: Approximation via Randomized Rounding
Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;
More informationarxiv: v2 [math.co] 30 Nov 2015
PLANAR GRAPH IS ON FIRE PRZEMYSŁAW GORDINOWICZ arxiv:1311.1158v [math.co] 30 Nov 015 Abstract. Let G be any connected graph on n vertices, n. Let k be any positive integer. Suppose that a fire breaks out
More informationy intercept Gradient Facts Lines that have the same gradient are PARALLEL
CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or
More informationBinary Search Trees. A Generic Tree. Binary Trees. Nodes in a binary search tree ( BST) are of the form. P parent. Key. Satellite data L R
Binary Search Trees A Generic Tree Nodes in a binary search tree ( BST) are of the form P parent Key A Satellite data L R B C D E F G H I J The BST has a root node which is the only node whose parent
More informationCSC 373: Algorithm Design and Analysis Lecture 16
CSC 373: Algorithm Design and Analysis Lecture 16 Allan Borodin February 25, 2013 Some materials are from Stephen Cook s IIT talk and Keven Wayne s slides. 1 / 17 Announcements and Outline Announcements
More informationGraphs without proper subgraphs of minimum degree 3 and short cycles
Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract
More informationBrendan D. McKay Computer Science Dept., Vanderbilt University, Nashville, Tennessee 37235
139. Proc. 3rd Car. Conf. Comb. & Comp. pp. 139143 SPANNING TREES IN RANDOM REGULAR GRAPHS Brendan D. McKay Computer Science Dept., Vanderbilt University, Nashville, Tennessee 37235 Let n~ < n2
More informationCMPSCI611: Approximating MAXCUT Lecture 20
CMPSCI611: Approximating MAXCUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NPhard problems. Today we consider MAXCUT, which we proved to
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationDisjoint Compatible Geometric Matchings
Disjoint Compatible Geometric Matchings Mashhood Ishaque Diane L. Souvaine Csaba D. Tóth Abstract We prove that for every even set of n pairwise disjoint line segments in the plane in general position,
More informationNPcomplete? NPhard? Some Foundations of Complexity. Prof. Sven Hartmann Clausthal University of Technology Department of Informatics
NPcomplete? NPhard? Some Foundations of Complexity Prof. Sven Hartmann Clausthal University of Technology Department of Informatics Tractability of Problems Some problems are undecidable: no computer
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. !approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of
More informationThe Goldberg Rao Algorithm for the Maximum Flow Problem
The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }
More informationNotes on NP Completeness
Notes on NP Completeness Rich Schwartz November 10, 2013 1 Overview Here are some notes which I wrote to try to understand what NP completeness means. Most of these notes are taken from Appendix B in Douglas
More informationLocal search for the minimum label spanning tree problem with bounded color classes
Available online at www.sciencedirect.com Operations Research Letters 31 (003) 195 01 Operations Research Letters www.elsevier.com/locate/dsw Local search for the minimum label spanning tree problem with
More information8.1 Min Degree Spanning Tree
CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree
More informationER E P M A S S I CONSTRUCTING A BINARY TREE EFFICIENTLYFROM ITS TRAVERSALS DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF TAMPERE REPORT A19985
S I N S UN I ER E P S I T VER M A TA S CONSTRUCTING A BINARY TREE EFFICIENTLYFROM ITS TRAVERSALS DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF TAMPERE REPORT A19985 UNIVERSITY OF TAMPERE DEPARTMENT OF
More informationMidterm Practice Problems
6.042/8.062J Mathematics for Computer Science October 2, 200 Tom Leighton, Marten van Dijk, and Brooke Cowan Midterm Practice Problems Problem. [0 points] In problem set you showed that the nand operator
More informationFairness in Routing and Load Balancing
Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria
More informationMathematics Placement Packet Colorado College Department of Mathematics and Computer Science
Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking
More informationFINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) 
More information