# CS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010

Size: px
Start display at page:

## Transcription

1 CS 598CSC: Combinatorial Optimization Lecture date: /4/010 Instructor: Chandra Chekuri Scribe: David Morrison Gomory-Hu Trees (The work in this section closely follows [3]) Let G = (V, E) be an undirected graph with non-negative edge capacities defined by c : E R. We would like to be able to compute the global minimum cut on the graph (i.e., the minimum over all min-cuts between pairs of vertices s and t). Clearly, this can be done by computing the minimum cut for all ( n ) pairs of vertices, but this can take a lot of time. Gomory and Hu showed that the number of distinct cuts in the graph is at most n 1, and furthermore that there is an efficient tree structure that can be maintained to compute this set of distinct cuts [1] (note that there is also a very nice randomized algorithm due to Karger and Stein that can compute the global minimum cut in near-linear time with high probability []). An important note is that Gomory-Hu trees work because the cut function is both submodular and symmetric. We will see later that any submodular, symmetric function will induce a Gomory- Hu tree. Definition 1. Given a graph G = (V, E), we define α G (u, v) to be the value of a minimum u, v cut in G. Furthermore, for some set of vertices U, we define δ(u) to be the set of edges with one endpoint in U. Definition. Let G, c, and α G be defined as above. Then, a tree T = (V (G), E T ) is a Gomory- Hu tree if for all st E T, δ(w ) is a minimum s, t cut in G, where W is one component of T st. The natural question is whether such a tree even exists; we will return to this question shortly. However, if we are given such a tree for an arbitrary graph G, we know that this tree obeys some very nice properties. In particular, we can label the edges of the tree with the values of the minimum cuts, as the following theorem shows (an example of this can be seen in figure 1): Theorem 1. Let T be a Gomory-Hu tree for a graph G = (V, E). Then, for all u, v V, let st be the edge on the unique path in T from u to v such that α G (s, t) is minimized. Then, α G (u, v) = α G (s, t) and the cut δ(w ) induced by T st is a u, v minimum cut in G. Thus α G (s, t) = α T (s, t) for each s, t V where the capacity of an edge st in T is equal to α G (s, t). Proof. We first note that α G obeys a triangle inequality. That is, α G (a, b) min(α G (a, c), α G (b, c)) for any undirected graph G and vertices a, b, c (to see this, note that c has to be on one side or the other of any a, b cut). Consider the path from u to v in T. We note that if uv = st, then α G (u, v) = α G (s, t). Otherwise, let w v be the neighbor of u on the u-v path in T. By the triangle inequality mentioned above, α G (u, v) min(α G (u, w), α G (w, v)). If uw = st, then α G (u, v) α G (s, t); otherwise, by induction on the path length, we have that α G (u, v) α G (w, v) α G (s, t). However, by the definition of Gomory-Hu trees, we have that α G (u, v) α G (s, t), since the cut induced by T st is a valid cut for u, v. Thus, we have α G (u, v) = α G (s, t) and the cut induced by T st is a u, v minimum cut in G.

2 10 b 4 c 5 a 3 d 8 f 3 e a b f e c 14 d Figure 1: A graph G with its corresponding Gomory-Hu tree [4]. Remark. Gomory-Hu trees can be (and are often) defined by asking for the property described in Theorem 1. However, the proof shows that the basic requirement in Definition implies the other property. The above theorem shows that we can represent compactly all of the minimum cuts in an undirected graph. Several non-trivial facts about undirected graphs fall out of the definition and the above result. The only remaining question is Does such a tree exist? And if so, how does one compute it efficiently? We will answer both questions by giving a constructive proof of Gomory-Hu trees for any undirected graph G. However, first we must discuss some properties of submodular functions. Definition 3. Given a finite set E, f : E R is submodular if for all A, B E, f(a)+f(b) f(a B) + f(a B). An alternate definition based on the idea of decreasing marginal value is the following: Definition 4. Given E and f as above, f is submodular if f(a + e) f(a) f(b + e) f(b) for all A B and e E. To see the equivalence of these definitions, let f A (e) = f(a + e) f(a), and similarly for f B (e). Take any A, B E and e E such that A B, and let f be submodular according to definition 3. Then f(a + e) + f(b) f((a + e) B) + f((a + e) B) = f(b + e) + f(a). Rearranging shows that f A (e) f B (e). Showing that definition 4 implies definition 3 is slightly more complicated, but can be done (Exercise). There are three types of submodular functions that will be of interest: 1. Arbitrary submodular functions. Non-negative (range is [0, )). Two subclasses of non-negative submodular functions are monotone (f(a) f(b) whenever A B) and non-monotone. 3. Symmetric submodular functions where f(a) = f(e \ A) for all A E. As an example of a submodular function, consider a graph G = (V, E) with capacity function c : E R +. Then f : V tor + defined by f(a) = c(δ(a)) (i.e., the capacity of a cut induced by a set A) is submodular.

3 To see this, notice that f(a) + f(b) = a + b + c + d + e + f, for any arbitrary A and B, and a, b, c, d, e, f are as shown in figure. Here, a (for example) represents the total capacity of edges with one endpoint in A and the other in V \ (A B). Also notice that f(a B) + f(a B) = a + b + c + d + e, and since all values are positive, we see that f(a) + f(b) f(a B) + f(a B), satisfying definition 3. a c b A d e B Figure : Given a graph G and two sets A, B V, this diagram shows all of the possible classes of edges of interest in G. In particular, there could be edges with both endpoints in V \ (A B), A, or B that are not shown here. f Exercise 1. Show that cut function on the vertices of a directed graph is submodular. Another nice property about this function f is that it is posi-modular, meaning that f(a) + f(b) f(a B) + f(b A). In fact, posi-modularity follows for any symmetric submodular function: f(a) + f(b) = f(v A) + f(b) f ((V A) B) + f ((V A) B) = f(b A) + f (V (A B)) = f(b A) + f(a B) We use symmetry in the first and last lines above. In fact, it turns out that the above two properties of the cut function are the only two properties necessary for the proof of existence of Gomory-Hu trees. As mentioned before, this will give us a Gomory-Hu tree for any non-negative symmetric submodular function. We now prove the following lemma, which will be instrumental in constructing Gomory-Hu trees: Key Lemma. Let δ(w ) be an s, t minimum cut in a graph G with respect to a capacity function c. Then for any u, v W, u v, there is a u, v minimum cut δ(x) where X W. Proof. Let δ(x) be any u, v minimum cut that crosses W. Suppose without loss of generality that s W, s X, and u X. If one of these are not the case, we can invert the roles of s and t or X and V \ X. Then there are two cases to consider: Case 1: t X (see figure 3). Then, since c is submodular,

4 X t s u v W Figure 3: δ(w ) is a minimum s, t cut. δ(x) is a minimum u, v cut that crosses W. This diagram shows the situation in Case 1; a similar picture can be drawn for Case c(δ(x)) + c(δ(w )) c(δ(x W )) + c(δ(x W )) (1) But notice that δ(x W ) is a u, v cut, so since δ(x) is a minimum cut, we have c(δ(x W )) c(δ(x)). Also, X W is a s, t cut, so c(δ(x W )) c(δ(w )). Thus, equality holds in equation (1), and X W is a minimum u, v cut. Case : t X. Since c is posi-modular, we have that c(δ(x)) + c(δ(w )) c(δ(w \ X)) + c(δ(x \ W )) () However, δ(w \ X) is a u, v cut, so c(δ(w \ X)) c(δ(x)). Similarly, δ(x \ W ) is an s, t cut, so c(δ(x \ W )) c(δ(w )). Therefore, equality holds in equation (), and W \ X is a u, v minimum cut. The above argument shows that minimum cuts can be uncrossed, a technique that is useful in many settings. In order to construct a Gomory-Hu tree for a graph, we need to consider a slightly generalized definition: Definition 5. Let G = (V, E), R V. Then a Gomory-Hu tree for R in G is a pair consisting of T = (R, E T ) and a partition (C r r R) of V associated with each r R such that 1. For all r R, r C r. For all st E T, T st induces a minimum cut in G between s and t defined by δ(u) = r X where X is the vertex set of a component of T st. Notice that a Gomory-Hu tree for G is simply a generalized Gomory-Hu tree with R = V. C r

5 Algorithm 1 GomoryHuAlg(G, R) if R = 1 then return T = ({r}, ), C r = V else Let r 1, r R, and let δ(w ) be an r 1, r minimum cut Create two subinstances of the problem G 1 = G with V \ W shrunk to a single vertex, v 1 ; R 1 = R W G = G with W shrunk to a single vertex, v ; R = R \ W Now we recurse T 1, (C 1 r r R 1 ) = GomoryHuAlg(G 1, R 1 ) T, (C r r R ) = GomoryHuAlg(G, R ) Note that r, r are not necessarily r 1, r! Let r be the vertex such that v 1 C 1 r Let r be the vertex such that v C r See figure 4 T = (R 1 R, E T1 E T {rr }) (C r r R) = ComputePartitions(R 1, R, C 1 r, C r, r, r ) return T, C r end if Algorithm ComputePartitions(R 1, R, Cr 1, c r, r, r ) We use the returned partitions, except we remove v 1 and v from C r For r R 1, r r, C r = Cr 1 For r R 1, r r, C r = Cr C r = Cr 1 {v 1 }, C r = Cr {v } return (C r r R) and C r, respectively Intuitively, we associate with each vertex v in the tree a bucket that contains all of the vertices that have to appear on the same side as v in some minimum cut. This allows us to define the algorithm GomoryHuAlg. Theorem 3. GomoryHuAlg returns a valid Gomory-Hu tree for a set R. Proof. We need to show that any st E T satisfies the key property of Gomory-Hu trees. That is, we need to show that T st induces a minimum cut in G between s and t. The base case is trivial. Then, suppose that st T 1 or st T. By the Key Lemma, we can ignore all of the vertices outside of T 1 or T, because they have no effect on the minimum cut, and by our induction hypothesis, we know that T 1 and T are correct. Thus, the only edge we need to care about is the edge we added from r to r. First, consider the simple case when α G (r 1, r ) is minimum over all pairs of vertices in R. In this case, we see that in particular, α G (r 1, r ) α G (r, r ), so we are done. However, in general this may not always be the case. Let δ(w ) be a minimum cut between r 1 and r, and suppose that there is a smaller r, r minimum cut δ(x) than what W induces; that is c(δ(x)) < ( δ(w )). Assume without loss of generality that r 1, r W. Notice that if r 1 X, we

6 T 1 T r r 1 C = {v,...} r 1 r r C = {v,...} r 1 Figure 4: T 1 and T have been recursively computed by GomoryHuAlg. Then we find r and r such that v 1 (the shrunken vertex corresponding to V \ W in T 1 ) is in the partition of r, and similarly for r and v. Then, to compute T, we connect r and r, and recompute the partitions for the whole tree according to ComputePartitions. have a smaller r 1, r cut than δ(w ), and similarly if r X. So, it is clear that X separates r 1 and r. By our key lemma, we can then uncross and assume that X W. Now, consider the path from r to r 1 in T 1. There exists an edge uv on this path such that the weight of uv in T 1, w 1 (uv), is at most c(δ(x)). Because T 1 is a Gomory-Hu tree, uv induces an r 1, r cut in G of capacity w 1 (uv) (since v 1 Cr 1 ). But this contradicts the fact that W is a r 1, r minimum cut. Threfore, e can pick r 1 and r arbitrarily from R 1 and R, and GomoryHuAlg is correct. This immediately implies the following corollary: Corollary 4. A Gomory-Hu tree for R V in G can be computed in the time needed to compute R 1 minimum-cuts in graphs of size at most that of G. Finally, we present the following alternative proof of the last step of theorem 3 (that is, showing that we can choose r 1 and r arbitrarily in GomoryHuAlg). As before, let δ(w ) be an r 1, r minimum cut, and assume that r 1 W, r V \ W. Assume for simplicity that r 1 r and r r (the other cases are similar). We claim that α G1 (r 1, r ) = α G (r 1, r ) α G (r 1, r ). To see this, note that if α G1 (r 1, r ) < α G (r 1, r ), there is an edge uv E T1 on the path from r 1 to r that has weight less than α G (r 1, r ), which gives a smaller r 1, r cut in G than W (since v 1 C 1 r ). For similar reasons, we see that α G (r, r ) α G (r 1, r ). Thus, by the triangle inequality we have α G (r, r ) min(α G (r, r 1 ), α G (r, r ), α G (r 1, r ) α G (r 1, r ) which completes the proof. Gomory-Hu trees allow one to easily show some facts that are otherwise hard to prove directly. Some examples are the following.

7 Exercise. For any undirected graph there is a pair of nodes s, t and an s-t minimum cut consisting of a singleton node (either s or t). Such a pair is called a pendant pair. Exercise 3. Let G be a graph such that deg(v) k for all v V. Show that there is some pair s, t such that α G (s, t) k. Notice that the proof of the correctness of the algorithm relied only on the key lemma which in turn used only the symmetry and submodularity of the cut function. One can directly extend the proof to show the following theorem. Theorem 5. Let V be a ground set, and let f : V R + be a symmetric submodular function. Given s, t in V, define the minimum cut between s and t as α f (s, t) = min f(w ) W V, W {s,t} =1 Then, there is a Gomory-Hu tree that represents α f. That is, there is a tree T = (V, E T ) and a capacity function c : E T R + such that α f (s, t) = α T (s, t) for all s, t in V, and moreover, the minimum cut in T induces a minimum cut according to f for each s, t. Exercise 4. Let G = (V, ξ) be a hypergraph. That is, each hyper-edge S ξ is a subset of V. Define f : V R + as f(w ) = δ(w ), where S ξ is in δ(w ) iff S W and S \ W are non-empty. Show that f is a symmetric, submodular function. References [1] R. E. Gomory, T. C. Hu. Multi-terminal network flows. Journal of the Society for Industrial and Applied Mathematics, vol. 9, [] D. R. Karger. Minimum cuts in near-linear time. Journal of the ACM, vol. 47, 000. [3] A. Schrijver. Combinatorial Optimization. Springer-Verlag Berlin Heidelberg, 003. Chapter [4] V. Vazirani. Approximation Algorithms. Springer, 004.

### Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs

MCS-236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set

### 1 Polyhedra and Linear Programming

CS 598CSC: Combinatorial Optimization Lecture date: January 21, 2009 Instructor: Chandra Chekuri Scribe: Sungjin Im 1 Polyhedra and Linear Programming In this lecture, we will cover some basic material

### Lecture 22: November 10

CS271 Randomness & Computation Fall 2011 Lecture 22: November 10 Lecturer: Alistair Sinclair Based on scribe notes by Rafael Frongillo Disclaimer: These notes have not been subjected to the usual scrutiny

### CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992

CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992 Original Lecture #6: 28 January 1991 Topics: Triangulating Simple Polygons

### Math 443/543 Graph Theory Notes 4: Connector Problems

Math 443/543 Graph Theory Notes 4: Connector Problems David Glickenstein September 19, 2012 1 Trees and the Minimal Connector Problem Here is the problem: Suppose we have a collection of cities which we

### Every 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

### Solutions to Exercises 8

Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### P. Jeyanthi and N. Angel Benseera

Opuscula Math. 34, no. 1 (014), 115 1 http://dx.doi.org/10.7494/opmath.014.34.1.115 Opuscula Mathematica A TOTALLY MAGIC CORDIAL LABELING OF ONE-POINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel

### Theorem A graph T is a tree if, and only if, every two distinct vertices of T are joined by a unique path.

Chapter 3 Trees Section 3. Fundamental Properties of Trees Suppose your city is planning to construct a rapid rail system. They want to construct the most economical system possible that will meet the

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

### Divisor graphs have arbitrary order and size

Divisor graphs have arbitrary order and size arxiv:math/0606483v1 [math.co] 20 Jun 2006 Le Anh Vinh School of Mathematics University of New South Wales Sydney 2052 Australia Abstract A divisor graph G

### On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

### Approximating 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

### Tree-representation of set families and applications to combinatorial decompositions

Tree-representation of set families and applications to combinatorial decompositions Binh-Minh Bui-Xuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no

### Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C

### CSL851: Algorithmic Graph Theory Semester I Lecture 1: July 24

CSL851: Algorithmic Graph Theory Semester I 2013-2014 Lecture 1: July 24 Lecturer: Naveen Garg Scribes: Suyash Roongta Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have

### Lecture 9. 1 Introduction. 2 Random Walks in Graphs. 1.1 How To Explore a Graph? CS-621 Theory Gems October 17, 2012

CS-62 Theory Gems October 7, 202 Lecture 9 Lecturer: Aleksander Mądry Scribes: Dorina Thanou, Xiaowen Dong Introduction Over the next couple of lectures, our focus will be on graphs. Graphs are one of

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

### Triangle deletion. Ernie Croot. February 3, 2010

Triangle deletion Ernie Croot February 3, 2010 1 Introduction The purpose of this note is to give an intuitive outline of the triangle deletion theorem of Ruzsa and Szemerédi, which says that if G = (V,

### Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

### Connectivity 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

### Cycles in a Graph Whose Lengths Differ by One or Two

Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

### Lecture Notes on Spanning Trees

Lecture Notes on Spanning Trees 15-122: Principles of Imperative Computation Frank Pfenning Lecture 26 April 26, 2011 1 Introduction In this lecture we introduce graphs. Graphs provide a uniform model

### / Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry

600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry 2.1 Steiner Tree Definition 2.1.1 In the Steiner Tree problem the input

### 1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution.

Name : Instructor: Marius Ionescu Instructions: 1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution. 3. If you need extra pages

### Sample Problems in Discrete Mathematics

Sample Problems in Discrete Mathematics This handout lists some sample problems that you should be able to solve as a pre-requisite to Computer Algorithms Try to solve all of them You should also read

### 1 Basic Definitions and Concepts in Graph Theory

CME 305: Discrete Mathematics and Algorithms 1 Basic Definitions and Concepts in Graph Theory A graph G(V, E) is a set V of vertices and a set E of edges. In an undirected graph, an edge is an unordered

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

### Relations Graphical View

Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binary

### Markov random fields and Gibbs measures

Chapter Markov random fields and Gibbs measures 1. Conditional independence Suppose X i is a random element of (X i, B i ), for i = 1, 2, 3, with all X i defined on the same probability space (.F, P).

### Math 421, Homework #5 Solutions

Math 421, Homework #5 Solutions (1) (8.3.6) Suppose that E R n and C is a subset of E. (a) Prove that if E is closed, then C is relatively closed in E if and only if C is a closed set (as defined in Definition

### x a x 2 (1 + x 2 ) n.

Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

### 136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics

### A CHARACTERIZATION OF TREE TYPE

A CHARACTERIZATION OF TREE TYPE LON H MITCHELL Abstract Let L(G) be the Laplacian matrix of a simple graph G The characteristic valuation associated with the algebraic connectivity a(g) is used in classifying

### CSC2420 Fall 2012: Algorithm Design, Analysis and Theory

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory Allan Borodin November 15, 2012; Lecture 10 1 / 27 Randomized online bipartite matching and the adwords problem. We briefly return to online algorithms

### Graph Theory Notes. Vadim Lozin. Institute of Mathematics University of Warwick

Graph Theory Notes Vadim Lozin Institute of Mathematics University of Warwick 1 Introduction A graph G = (V, E) consists of two sets V and E. The elements of V are called the vertices and the elements

### Combinatorics: The Fine Art of Counting

Combinatorics: The Fine Art of Counting Week 9 Lecture Notes Graph Theory For completeness I have included the definitions from last week s lecture which we will be using in today s lecture along with

### Mean Ramsey-Turán numbers

Mean Ramsey-Turán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρ-mean coloring of a graph is a coloring of the edges such that the average

### 8.1 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

### Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,

### Lecture 17 : Equivalence and Order Relations DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

### On three zero-sum Ramsey-type problems

On three zero-sum Ramsey-type problems Noga Alon Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel and Yair Caro Department of Mathematics

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### 4 Basics of Trees. Petr Hliněný, FI MU Brno 1 FI: MA010: Trees and Forests

4 Basics of Trees Trees, actually acyclic connected simple graphs, are among the simplest graph classes. Despite their simplicity, they still have rich structure and many useful application, such as in

### Why graph clustering is useful?

Graph Clustering Why graph clustering is useful? Distance matrices are graphs as useful as any other clustering Identification of communities in social networks Webpage clustering for better data management

### Graph Theory Problems and Solutions

raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is

### On-line secret sharing

On-line secret sharing László Csirmaz Gábor Tardos Abstract In a perfect secret sharing scheme the dealer distributes shares to participants so that qualified subsets can recover the secret, while unqualified

### COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

### INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### Analysis 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 Breadth-first search (BFS) 4 Applications

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

### arxiv:1203.1525v1 [math.co] 7 Mar 2012

Constructing subset partition graphs with strong adjacency and end-point count properties Nicolai Hähnle haehnle@math.tu-berlin.de arxiv:1203.1525v1 [math.co] 7 Mar 2012 March 8, 2012 Abstract Kim defined

### THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

### A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

### Minimum cost maximum flow, Minimum cost circulation, Cost/Capacity scaling

6.854 Advanced Algorithms Lecture 16: 10/11/2006 Lecturer: David Karger Scribe: Kermin Fleming and Chris Crutchfield, based on notes by Wendy Chu and Tudor Leu Minimum cost maximum flow, Minimum cost circulation,

### Outline 2.1 Graph Isomorphism 2.2 Automorphisms and Symmetry 2.3 Subgraphs, part 1

GRAPH THEORY LECTURE STRUCTURE AND REPRESENTATION PART A Abstract. Chapter focuses on the question of when two graphs are to be regarded as the same, on symmetries, and on subgraphs.. discusses the concept

### SCORE SETS IN ORIENTED GRAPHS

Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in

### Lecture 4: The Chromatic Number

Introduction to Graph Theory Instructor: Padraic Bartlett Lecture 4: The Chromatic Number Week 1 Mathcamp 2011 In our discussion of bipartite graphs, we mentioned that one way to classify bipartite graphs

### Generating models of a matched formula with a polynomial delay

Generating models of a matched formula with a polynomial delay Petr Savicky Institute of Computer Science, Academy of Sciences of Czech Republic, Pod Vodárenskou Věží 2, 182 07 Praha 8, Czech Republic

### 6.042/18.062J Mathematics for Computer Science October 3, 2006 Tom Leighton and Ronitt Rubinfeld. Graph Theory III

6.04/8.06J Mathematics for Computer Science October 3, 006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Graph Theory III Draft: please check back in a couple of days for a modified version of these

### On the independence number of graphs with maximum degree 3

On the independence number of graphs with maximum degree 3 Iyad A. Kanj Fenghui Zhang Abstract Let G be an undirected graph with maximum degree at most 3 such that G does not contain any of the three graphs

### All trees contain a large induced subgraph having all degrees 1 (mod k)

All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New

### Diameter and Treewidth in Minor-Closed Graph Families, Revisited

Algorithmica manuscript No. (will be inserted by the editor) Diameter and Treewidth in Minor-Closed Graph Families, Revisited Erik D. Demaine, MohammadTaghi Hajiaghayi MIT Computer Science and Artificial

### The chromatic spectrum of mixed hypergraphs

The chromatic spectrum of mixed hypergraphs Tao Jiang, Dhruv Mubayi, Zsolt Tuza, Vitaly Voloshin, Douglas B. West March 30, 2003 Abstract A mixed hypergraph is a triple H = (X, C, D), where X is the vertex

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

### Electrical Resistances in Products of Graphs

Electrical Resistances in Products of Graphs By Shelley Welke Under the direction of Dr. John S. Caughman In partial fulfillment of the requirements for the degree of: Masters of Science in Teaching Mathematics

### Lecture 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

### CS5314 Randomized Algorithms. Lecture 16: Balls, Bins, Random Graphs (Random Graphs, Hamiltonian Cycles)

CS5314 Randomized Algorithms Lecture 16: Balls, Bins, Random Graphs (Random Graphs, Hamiltonian Cycles) 1 Objectives Introduce Random Graph Model used to define a probability space for all graphs with

### COLORED GRAPHS AND THEIR PROPERTIES

COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring

### Uniform Multicommodity Flow through the Complete Graph with Random Edge-capacities

Combinatorics, Probability and Computing (2005) 00, 000 000. c 2005 Cambridge University Press DOI: 10.1017/S0000000000000000 Printed in the United Kingdom Uniform Multicommodity Flow through the Complete

### 1 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.

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

### 8. Matchings and Factors

8. Matchings and Factors Consider the formation of an executive council by the parliament committee. Each committee needs to designate one of its members as an official representative to sit on the council,

### A tree can be defined in a variety of ways as is shown in the following theorem: 2. There exists a unique path between every two vertices of G.

7 Basic Properties 24 TREES 7 Basic Properties Definition 7.1: A connected graph G is called a tree if the removal of any of its edges makes G disconnected. A tree can be defined in a variety of ways as

### Even Faster Algorithm for Set Splitting!

Even Faster Algorithm for Set Splitting! Daniel Lokshtanov Saket Saurabh Abstract In the p-set Splitting problem we are given a universe U, a family F of subsets of U and a positive integer k and the objective

### The positive minimum degree game on sparse graphs

The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University

### Tools for parsimonious edge-colouring of graphs with maximum degree three. J.L. Fouquet and J.M. Vanherpe. Rapport n o RR-2010-10

Tools for parsimonious edge-colouring of graphs with maximum degree three J.L. Fouquet and J.M. Vanherpe LIFO, Université d Orléans Rapport n o RR-2010-10 Tools for parsimonious edge-colouring of graphs

### Lecture 1: Course overview, circuits, and formulas

Lecture 1: Course overview, circuits, and formulas Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: John Kim, Ben Lund 1 Course Information Swastik

### Mechanism Design without Money I: House Allocation, Kidney Exchange, Stable Matching

Algorithmic Game Theory Summer 2015, Week 12 Mechanism Design without Money I: House Allocation, Kidney Exchange, Stable Matching ETH Zürich Peter Widmayer, Paul Dütting Looking at the past few lectures

### An inequality for the group chromatic number of a graph

An inequality for the group chromatic number of a graph Hong-Jian Lai 1, Xiangwen Li 2 and Gexin Yu 3 1 Department of Mathematics, West Virginia University Morgantown, WV 26505 USA 2 Department of Mathematics

### On Total Domination in Graphs

University of Houston - Downtown Senior Project - Fall 2012 On Total Domination in Graphs Author: David Amos Advisor: Dr. Ermelinda DeLaViña Senior Project Committee: Dr. Sergiy Koshkin Dr. Ryan Pepper

### Lecture notes from Foundations of Markov chain Monte Carlo methods University of Chicago, Spring 2002 Lecture 1, March 29, 2002

Lecture notes from Foundations of Markov chain Monte Carlo methods University of Chicago, Spring 2002 Lecture 1, March 29, 2002 Eric Vigoda Scribe: Varsha Dani & Tom Hayes 1.1 Introduction The aim of this

### princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

### Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

### each college c i C has a capacity q i - the maximum number of students it will admit

n colleges in a set C, m applicants in a set A, where m is much larger than n. each college c i C has a capacity q i - the maximum number of students it will admit each college c i has a strict order i

### Brendan D. McKay Computer Science Dept., Vanderbilt University, Nashville, Tennessee 37235

139. Proc. 3rd Car. Conf. Comb. & Comp. pp. 139-143 SPANNING TREES IN RANDOM REGULAR GRAPHS Brendan D. McKay Computer Science Dept., Vanderbilt University, Nashville, Tennessee 37235 Let n~ < n2

### Chapter 4. Trees. 4.1 Basics

Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.

### Lemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S.

Definition 51 Let S be a set bijection f : S S 5 Permutation groups A permutation of S is simply a Lemma 52 Let S be a set (1) Let f and g be two permutations of S Then the composition of f and g is a

### Characterizations of Arboricity of Graphs

Characterizations of Arboricity of Graphs Ruth Haas Smith College Northampton, MA USA Abstract The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs

### THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER

THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER JONATHAN P. MCCAMMOND AND GÜNTER ZIEGLER Abstract. In 1933 Karol Borsuk asked whether every compact subset of R d can be decomposed