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

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Combinatorics, Probability and Computing (2005) 00, c 2005 Cambridge University Press DOI: /S Printed in the United Kingdom Uniform Multicommodity Flow through the Complete Graph with Random Edge-capacities D A V I D J. A L D O U S Department of Statistics, University of California, Berkeley, CA , USA Give random capacities C to the edges of the complete n-vertex graph. Consider the maximum flow Φ n that can be simultaneously routed between each source-destination pair. We prove that Φ n φ where the limit constant φ depends on the distribution of C in a simple way, and that asymptotically one need use only one- and twostep routes. The proof uses a reduction to the following random graph problem. Make a random graph in which each edge is blue with probability p b or scarlet with probability p s < p b /2 or neither. Then we can find edge-disjoint triangles, with exactly one scarlet edge and exactly two blue edges, which exhaust almost all the scarlet edges. We show this by analyzing a simple greedy algorithm for which one expects a differential equation approximation. The approximation can be justified in a direct way using the local weak convergence methodology, which may more generally be useful in formalizing differential equation approximations in randomized graph algorithm contexts. 1. Introduction This paper is part of a project studying optimal flows through random networks, where a network has both a graph structure and extra structure such as capacities and costs on edges, and where we are in the multicommodity flow setting with simultaneous flows between each source-destination pair. We plan a survey elsewhere. Possible models span a broad spectrum from realistic to mathematically tractable, and at the latter end are models based on the complete graph. Including study of such models within a project is natural both for mathematical completeness and for comparison purposes. Consider first the setting of an arbitrary finite connected undirected graph G. Let φ > 0. A flow of volume φ/2 between vertex v and vertex w has net out-flow = φ/2 at v, net out-flow = φ/2 and w, and zero net out-flow at other vertices. For such a flow write f v,w (e) 0 for the absolute value of the flow volume across an undirected edge e. Suppose we have such a flow simultaneously for each ordered pair (v, w) with w v; call this collection a uniform flow of volume φ and write f(e) := (v,w) f v,w(e) Research supported by N.S.F Grant DMS

2 2 David J. Aldous for the combined volume of flow across the undirected edge e. Suppose now we are given capacities C(e) for edges e. Then the maximum uniform flow volume (MUFV) is defined to be the largest φ such that there exists a uniform flow of volume φ which satisfies the capacity constraints f(e) C(e) e. (1.1) One modeling paradigm, seeking to combine the spatial inhomogeneity of real networks with mathematical tractability, is to consider some standard family G n of n-vertex graphs, and to assume the edge-capacities C(e) are random (specifically, are i.i.d. copies of a reference r.v. C). Now the MUFV is a r.v. Φ n, and one can seek to study its n behavior. Apparently, and somewhat surprisingly, such questions have not been studied before. There is literature [4, 3, 2, 5] on flows with a single source-destination pair and on flows from top to bottom of a square, but these fall into the one-commodity setting of the max-flow min-cut theorem, rather than our multicommodity setting. In this paper we consider the complete graph, and in [1] we consider a similar problem on the m m square grid. An interesting observation is that in both these models the limit constants for Φ n depend on the distribution of C (not just on its expectation EC), but for rather different reasons in the two models. An intermediate model is the cube {0, 1} d, and here we conjecture that the limit constant does depend only on EC when C is bounded away from zero Statement of results Consider the complete n-vertex graph whose edges e have independent random capacities C(e) whose common distribution satisfies P (C c 0 ) = 1; some c 0 > 0. (1.2) Note that the function EC <. (1.3) φ 2E max(φ C, 0) E max(c φ, 0) in continuous and strictly increasing from EC to as φ increases from 0 to, and so we can define a constant 0 < φ < as the unique solution of E max(c φ, 0) = 2E max(φ C, 0). (1.4) Theorem 1.1. Under assumptions (1.2,1.3) the MUFV Φ n satisfies Φ n φ in probability as n. The intuition is very simple. Suppose we wish to route a uniform flow of volume φ. First route as much flow as possible across the direct edge, that is route volume min(φ, C v,w ) across edge {v, w}. This leaves an unsatisfied demand for volume max(φ C v,w, 0) of flow. Now the mean surplus capacity per edge is E max(c φ, 0). We try to route the unsatisfied demand via 2-step paths with surplus capacity; for this to work it seems

3 Flow through complete graphs 3 evidently necessary that E max(c φ, 0) 2E max(φ C v,w, 0). Conversely, this should be sufficient because the set of edges with surplus capacity forms a dense random graph which should be sufficiently well-connected to permit the desired 2-step paths. The necessary part is indeed easy to formalize (Lemma 2.1). For the converse, we start by proving (section 2) a reduction to the following random graph result. To motivate this reduction, consider the case where the edge-capacity C takes only values {0, 1, 2} and where we seek to route a uniform flow of volume 1. Then traffic across capacity-0 edges (colored scarlet, say) needs to be routed through two capacity-2 edges (colored blue, say). Colors are mnemonics for smaller and bigger capacity. Proposition 1.2. Fix 0 < p s < p b /2 with p s + p b 1. Randomly color the edges of the complete n-vertex graph as blue (probability p b ) or scarlet (probability p s ) or neither (probability 1 p b p s ). Then there exists a collection of edge-disjoint triangles, each triangle having one scarlet edge and two blue edges, such that the number N n (v) of scarlet edges incident at v which are not is some triangle satisfies n 1 max N n (v) 0 in probability. (1.5) v Proposition 1.2 is proved in section xxx by analysing the natural greedy algorithm. The particular method of justifying the natural differential equation approximation is perhaps of wider methodological interest, so will be discussed at xxx. The assumption (1.2) is used only in the proof of Lemma 2.3, and we conjecture that it can be eliminated. xxx mention old paper by Frank Kelly (neat, but not diretly relevant). 2. The reduction argument 2.1. The upper bound The upper bound in Theorem 1.1 is provided by Lemma 2.1. lim n P (Φ n φ) = 0, φ > φ. Proof. Fix a realization of the edge-capacities. Suppose a uniform flow of volume ρ exists. For an edge (v, w) (f v,w (e) + f w,v (e)) ρ if C(v, w) ρ e C(v, w) + 2(ρ C(v, w)) if C(v, w) ρ because in the latter case volume of at least ρ C(v, w) must use at least a 2-step route. Combining the two cases, (f v,w (e) + f w,v (e)) min(ρ, C(v, w)) + 2 max(ρ C(v, w), 0). e

4 4 David J. Aldous Summing over edges e = (v, w) and using the capacity constraint (1.1), C(e) (min(ρ, C(e )) + 2 max(ρ C(e ), 0)). e e Dividing by ( n 2) and recalling we supposed that the uniform flow exists, we have shown Q n := ( 1 n ) C(e ) ( 1 n 2 e 2) (min(ρ, C(e )) + 2 max(ρ C(e ), 0)) 0 on {Φ n φ}. e But as n the quantity Q n converges in probability to q := EC (E min(ρ, C) + 2E max(ρ C, 0)) = E max(c ρ, 0) 2E max(ρ C, 0). If ρ > ρ then q < 0 and hence we must have lim n P (Φ n φ) = The reduction In this section we assume the truth of Proposition 1.2 and show how to deduce the lower bound in Theorem 1.1, stated as Lemma 2.3. Note that the condition ρ < ρ is equivalent to E max(ρ C, 0) r := E max(c ρ, 0) < 1 2. (2.1) Lemma 2.2. Suppose C is integer-valued and bounded, and suppose ρ is an integer satisfying (2.1). Then, for (v, w) outside some random subset B n of edges, we can construct flows of volume ρ between v and w such that the capacity constraint (1.1) holds and such that n 1 max {e B n : e incident at v} 0 in probability. v Proof. We first construct a finite set M of marks µ and a joint distribution (C; ξ 1, ξ 2,..., ξ C ρ ) which associates C ρ distinct random marks ξ j with a realization of C, such that the following two properties hold. (i) For each mark µ M there exists exactly one value i = i s (µ) {0, 1,..., ρ 1} such that P (C = i, some ξ j = µ) > 0 and there exists exactly one value i = i b (µ) {ρ + 1, ρ + 2,...} such that P (C = i, some ξ j = µ) > 0. (ii) For each mark µ M P (C = i s (µ), some ξ j = µ) P (C = i b (µ), some ξ j = µ) = r for r at (2.1). The construction is illustrated in Figure 2.2. Set p i = P (C = i). Partition a line of length E max(ρ C, 0) = ρ 1 i=0 (ρ i)p i into (ρ i) consecutive intervals of lengths p i (for i = 0, 1,..., ρ 1). Partition a line of equal length E max(ρ C, 0) = re max(c ρ, 0) = i>ρ r(i ρ)p i into (i ρ) consecutive intervals of lengths rp i (for i = ρ + 1, ρ 2,...). Now identify the two lines, and let M be the set of intervals µ arising as the intersection of some interval in the first partition with some interval in the second partition. Given C = i, choose random points uniformly (with respect to length) from each of the i ρ

5 Flow through complete graphs 5 p 1 p 1 p 1 p 2 p 2 p 3 rp 5 rp 6 rp 6 rp 7 rp 7 rp 7 rp 8 rp 8 rp 8 rp 8 Figure 1. Construction of marks. Here ρ = 4. The middle line shows the intervals which are the marks µ. On the left is a typical set of 3 marks associated with a value C = 1. On the right is a typical set of 4 marks associated with a value C = 8. intervals of length p i or rp i ; each such point is in some intersection-interval µ, so let the marks ξ 1,..., ξ i ρ be these intersection-intervals. Properties (i) and (ii) are immediate from the construction. To prove the lemma, associate each original edge of the complete graph with a realization of (C; ξ 1, ξ 2,..., ξ C ρ ) and replace the original edge by C ρ edges marked (ξ 1, ξ 2,..., ξ C ρ ) and colored scarlet (if C < ρ) or blue (if C > ρ). Now fix a mark µ and consider all edges with mark µ. By (ii) the hypothesis of Proposition 1.2 is satisfied, so we can find edge-disjoint triangles with two scarlet edges and one blue edges, all with mark µ, satisfying (1.5). Repeat for all marks. For each scarlet edge in one of the triangles, route unit flow between its end-vertices by using the two blue edges in the triangle. Then (1.5) and the finiteness of number of marks establish the lemma. Lemma 2.3. Assume (1.2,1.3) and let φ < φ. Then lim n P (Φ n φ) = 1. Proof. Define C k = min(2 k C2 k 1, k) for k sufficiently large that 2 k < c 0 for c 0 at (1.2). So 0 C k C 2 k. Define ρ k as the largest multiple of 2 k for which E max(c k ρ k, 0) > 2E max(ρ k C k, 0). It is easy to check that ρ k ρ as k. Thus it is sufficient to show that, for each fixed large k, P (uniform flow of volume ρ k exists) 0 as n. (2.2) By applying Lemma 2.2 to the integer-valued quantities 2 k C k and 2 k ρ k, then rescaling, we find that for (v, w) outside some random subset B n of edges such that q n := max v q n(v) n 2 0 in probability q n (v) := {e B n : e incident at v} we can construct flows of volume ρ k between v and w such that the total flow volume f(e) satisfies the capacity constraints f(e) C k (e) e. Because C k (e) C(e) 2 k,

6 6 David J. Aldous each edge has unused capacity of at least 2 k. So between each pair (v, w) B n route a flow of volume ρ k by spreading the flow uniformly over all 2-step paths. The volume of flow across an edge (v, w ) created in this way is at most ( ρ qn(v ) k n 2 + qn(w ) n 2 ) 2ρ q n. So if 2ρ q n < 2 k there exists a feasible flow of volume ρ k, establishing (2.2).

7 Flow through complete graphs 7 References [1] D.J. Aldous. Optimal flow through the disordered lattice. In Preparation, [2] J. T. Chayes and L. Chayes. Bulk transport properties and exponent inequalities for random resistor and flow networks. Comm. Math. Phys., 105(1): , [3] G. Grimmett and H. Kesten. First-passage percolation, network flows and electrical resistances. Z. Wahrsch. Verw. Gebiete, 66(3): , [4] H. Kesten. Surfaces with minimal random weights and maximal flows: a higher-dimensional version of first-passage percolation. Illinois J. Math., 31(1):99 166, [5] Y. Zhang. Critical behavior for maximal flows on the cubic lattice. J. Statist. Phys., 98(3-4): , 2000.

### 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

### 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

### 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,

### Lecture 4: BK inequality 27th August and 6th September, 2007

CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound

### A Stronger Form of the Van den Berg-Kesten Inequality

A Stronger Form of the Van den Berg-Kesten Inequality Peter Winkler March 4, 2010 Abstract Let Q n := {0,1} n be the discrete hypercube with uniform probability distribution, and let A and B be two up-events

### 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

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### 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

### Collinear Points in Permutations

Collinear Points in Permutations Joshua N. Cooper Courant Institute of Mathematics New York University, New York, NY József Solymosi Department of Mathematics University of British Columbia, Vancouver,

### 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

### Week 5 Integral Polyhedra

Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory

### 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

### CMSC 858T: Randomized Algorithms Spring 2003 Handout 8: The Local Lemma

CMSC 858T: Randomized Algorithms Spring 2003 Handout 8: The Local Lemma Please Note: The references at the end are given for extra reading if you are interested in exploring these ideas further. You are

### Planar Tree Transformation: Results and Counterexample

Planar Tree Transformation: Results and Counterexample Selim G Akl, Kamrul Islam, and Henk Meijer School of Computing, Queen s University Kingston, Ontario, Canada K7L 3N6 Abstract We consider the problem

### Spectral Measure of Large Random Toeplitz Matrices

Spectral Measure of Large Random Toeplitz Matrices Yongwhan Lim June 5, 2012 Definition (Toepliz Matrix) The symmetric Toeplitz matrix is defined to be [X i j ] where 1 i, j n; that is, X 0 X 1 X 2 X n

### 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

### 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

### 5. Convergence of sequences of random variables

5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,

### 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

### Math 55: Discrete Mathematics

Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What

### A Note on the Ruin Probability in the Delayed Renewal Risk Model

Southeast Asian Bulletin of Mathematics 2004 28: 1 5 Southeast Asian Bulletin of Mathematics c SEAMS. 2004 A Note on the Ruin Probability in the Delayed Renewal Risk Model Chun Su Department of Statistics

### 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

### 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

### CSE 20: Discrete Mathematics for Computer Science. Prof. Miles Jones. Today s Topics: Graphs. The Internet graph

Today s Topics: CSE 0: Discrete Mathematics for Computer Science Prof. Miles Jones. Graphs. Some theorems on graphs. Eulerian graphs Graphs! Model relations between pairs of objects The Internet graph!

### Section 3 Sequences and Limits, Continued.

Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular

### Good luck, veel succes!

Final exam Advanced Linear Programming, May 7, 13.00-16.00 Switch off your mobile phone, PDA and any other mobile device and put it far away. No books or other reading materials are allowed. This exam

### PROBLEM SET 7: PIGEON HOLE PRINCIPLE

PROBLEM SET 7: PIGEON HOLE PRINCIPLE The pigeonhole principle is the following observation: Theorem. Suppose that > kn marbles are distributed over n jars, then one jar will contain at least k + marbles.

### 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

### diffy boxes (iterations of the ducci four number game) 1

diffy boxes (iterations of the ducci four number game) 1 Peter Trapa September 27, 2006 Begin at the beginning and go on till you come to the end: then stop. Lewis Carroll Consider the following game.

### 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

### Midterm 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

### A Non-Linear Schema Theorem for Genetic Algorithms

A Non-Linear Schema Theorem for Genetic Algorithms William A Greene Computer Science Department University of New Orleans New Orleans, LA 70148 bill@csunoedu 504-280-6755 Abstract We generalize Holland

### Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

### Practice Problems Solutions

Practice Problems Solutions The problems below have been carefully selected to illustrate common situations and the techniques and tricks to deal with these. Try to master them all; it is well worth it!

### So here s the next version of Homework Help!!!

HOMEWORK HELP FOR MATH 52 So here s the next version of Homework Help!!! I am going to assume that no one had any great difficulties with the problems assigned this quarter from 4.3 and 4.4. However, if

### Math 317 HW #7 Solutions

Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence

### Fixed Point Theorems

Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation

### A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

### Generalizing the Ramsey Problem through Diameter

Generalizing the Ramsey Problem through Diameter Dhruv Mubayi Submitted: January 8, 001; Accepted: November 13, 001. MR Subject Classifications: 05C1, 05C15, 05C35, 05C55 Abstract Given a graph G and positive

### Graphical degree sequences and realizations

swap Graphical and realizations Péter L. Erdös Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences MAPCON 12 MPIPKS - Dresden, May 15, 2012 swap Graphical and realizations Péter L. Erdös

### MATH 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

### Stationary random graphs on Z with prescribed iid degrees and finite mean connections

Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the non-negative

### 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

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

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

### Lecture 3. Mathematical Induction

Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

### Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

### 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.

### MIX-DECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3. University of Split, Croatia

GLASNIK MATEMATIČKI Vol. 38(58)(2003), 2 232 MIX-DECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3 Damir Vukičević University of Split, Croatia

### CONVERGENCE OF SEQUENCES OF ITERATES OF RANDOM-VALUED VECTOR FUNCTIONS

C O L L O Q U I U M M A T H E M A T I C U M VOL. 97 2003 NO. 1 CONVERGENCE OF SEQUENCES OF ITERATES OF RANDOM-VALUED VECTOR FUNCTIONS BY RAFAŁ KAPICA (Katowice) Abstract. Given a probability space (Ω,

### SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov

Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### 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

### Random Geometric Graph Diameter in the Unit Disk with l p Metric (Extended Abstract)

Random Geometric Graph Diameter in the Unit Disk with l p Metric (Extended Abstract) Robert B. Ellis 1,, Jeremy L. Martin 2,, and Catherine Yan 1, 1 Department of Mathematics, Texas A&M University, College

### Gambling Systems and Multiplication-Invariant Measures

Gambling Systems and Multiplication-Invariant Measures by Jeffrey S. Rosenthal* and Peter O. Schwartz** (May 28, 997.. Introduction. This short paper describes a surprising connection between two previously

### 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

### Best Monotone Degree Bounds for Various Graph Parameters

Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer

### A Practical Scheme for Wireless Network Operation

A Practical Scheme for Wireless Network Operation Radhika Gowaikar, Amir F. Dana, Babak Hassibi, Michelle Effros June 21, 2004 Abstract In many problems in wireline networks, it is known that achieving

### 3.1 Solving Systems Using Tables and Graphs

Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system

### Intersection Dimension and Maximum Degree

Intersection Dimension and Maximum Degree N.R. Aravind and C.R. Subramanian The Institute of Mathematical Sciences, Taramani, Chennai - 600 113, India. email: {nraravind,crs}@imsc.res.in Abstract We show

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### HOLOMORPHIC FRAMES FOR WEAKLY CONVERGING HOLOMORPHIC VECTOR BUNDLES

HOLOMORPHIC FRAMES FOR WEAKLY CONVERGING HOLOMORPHIC VECTOR BUNDLES GEORGIOS D. DASKALOPOULOS AND RICHARD A. WENTWORTH Perhaps the most useful analytic tool in gauge theory is the Uhlenbeck compactness

### 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

### The multi-integer set cover and the facility terminal cover problem

The multi-integer set cover and the facility teral cover problem Dorit S. Hochbaum Asaf Levin December 5, 2007 Abstract The facility teral cover problem is a generalization of the vertex cover problem.

### 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.,

### Basic Notions on Graphs. Planar Graphs and Vertex Colourings. Joe Ryan. Presented by

Basic Notions on Graphs Planar Graphs and Vertex Colourings Presented by Joe Ryan School of Electrical Engineering and Computer Science University of Newcastle, Australia Planar graphs Graphs may be drawn

### Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization

### The degree, size and chromatic index of a uniform hypergraph

The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the

### Algorithm 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;

### 1 The Brownian bridge construction

The Brownian bridge construction The Brownian bridge construction is a way to build a Brownian motion path by successively adding finer scale detail. This construction leads to a relatively easy proof

### 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,

### On an anti-ramsey type result

On an anti-ramsey type result Noga Alon, Hanno Lefmann and Vojtĕch Rödl Abstract We consider anti-ramsey type results. For a given coloring of the k-element subsets of an n-element set X, where two k-element

### Single machine parallel batch scheduling with unbounded capacity

Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University

### 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

### Ri and. i=1. S i N. and. R R i

The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

### Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing

### Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction.

Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction. February 21, 2006 1 Proof by Induction Definition 1.1. A subset S of the natural numbers is said to be inductive if n S we have

### 1 Limiting distribution for a Markov chain

Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easy-to-check

### Course 421: Algebraic Topology Section 1: Topological Spaces

Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............

### UNIVERSAL JUGGLING CYCLES. Fan Chung 1 Department of Mathematics, University of California, San Diego, La Jolla, CA Ron Graham 2.

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7(2) (2007), #A08 UNIVERSAL JUGGLING CYCLES Fan Chung 1 Department of Mathematics, University of California, San Diego, La Jolla, CA 92093 Ron

### 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

### CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

### arxiv: 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

### 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

### 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

### Lecture 7 - Linear Programming

COMPSCI 530: Design and Analysis of Algorithms DATE: 09/17/2013 Lecturer: Debmalya Panigrahi Lecture 7 - Linear Programming Scribe: Hieu Bui 1 Overview In this lecture, we cover basics of linear programming,

### The 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 }

### arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

### Guessing Game: NP-Complete?

Guessing Game: NP-Complete? 1. LONGEST-PATH: Given a graph G = (V, E), does there exists a simple path of length at least k edges? YES 2. SHORTEST-PATH: Given a graph G = (V, E), does there exists a simple

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

### A Network Flow Approach in Cloud Computing

1 A Network Flow Approach in Cloud Computing Soheil Feizi, Amy Zhang, Muriel Médard RLE at MIT Abstract In this paper, by using network flow principles, we propose algorithms to address various challenges

### Complexity and Completeness of Finding Another Solution and Its Application to Puzzles

yato@is.s.u-tokyo.ac.jp seta@is.s.u-tokyo.ac.jp Π (ASP) Π x s x s ASP Ueda Nagao n n-asp parsimonious ASP ASP NP Complexity and Completeness of Finding Another Solution and Its Application to Puzzles Takayuki

### Section 3 Sequences and Limits

Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the n-th term of the sequence.

### Small 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

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### It is time to prove some theorems. There are various strategies for doing

CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it

### 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

### 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