Lecture 11: Generalized Lovász Local Lemma. Lovász Local Lemma
|
|
- Britton McBride
- 5 years ago
- Views:
Transcription
1 Lecture 11: Generalized
2 Recall We design an experiment with independent random variables X 1,..., X m We define bad events A 1,..., A n where) the bad event A i depends on the variables (X k1,..., X kni We define Vbl i = {k 1,..., k ni }, the set of all variables that the bad event A i depends on The bad event A i can depend on the bad event A j if Vbl i Vbl j Suppose each bad event A i depends on at most d other bad events Suppose we show that, for each bad event A i, the probability of its occurrence P [A i ] p If ep(d + 1) 1, then ( P [ A 1,... A n ] 1 1 ) n > 0 d + 1
3 Generalized We design an experiment with independent random variables X 1,..., X m We define bad events A 1,..., A n Let D i be the set of indices of bad events that A i depends on Suppose we exhibit the existence of numbers (x 1,..., x n ) such that the following holds. For each i, we have: P [A i ] x i j D i (1 x j ) Then the following holds. P [ A 1,..., A n ] n (1 x i ) > 0 i=1
4 Notes Prove using Generalized Lovász Local Lemma The numbers (x 1,..., x n ) are not probabilities that add up to 1. This is an incorrect intuition Prove the following corollary of the generalized Lovász Local Lemma Corollary If, for all i, we have j D i P [ A j ] < 1/4, then P [ A 1,..., A n ] n (1 2P [A i ]) > 0 i=1 Prove the results in the previous lecture using this corollary albeit with slightly worse parameters
5 Frugal Coloring I Definition (Frugal Coloring) A β-frugal Coloring of a graph satisfies the following two conditions 1 It is a valid coloring, and 2 In the neighborhood N(v) of any vertex v, there are at most β vertices with the same color. For example, a 1-frugal coloring of G is a coloring of G 2
6 Frugal Coloring II We will show the following result Theorem For β N, and a graph G with maximum degree β β there exists a β-frugal coloring using /β colors. Note that a graph with maximum degree can be 1-frugally colored with colors. We will prove the general result using
7 Frugal Coloring III Randomly color the vertices of the graph using C colors. We will consider two types of bad events. A e, where e E(G): If the two vertices at the endpoints of the edge e receive the same color. These will be called type-1 bad events. B {u1,...,u β+1 }, where u 1,..., u β+1 V (G): Suppose there exists a vertex v such that u 1,..., u β+1 are distinct vertices in N(G) with identical color. These will be called type-2 bad events.
8 Frugal Coloring IV Note that one type-1 bad event A e can depends on at most 2 other type-1 bad events A e We are now interested in computing how many type-2 bad events can A e depend on. Consider a type-2 bad event B u1,...,u β+1 such that that are in u 1,..., u β+1 N(v). Suppose e = (a, b). Note that a has at most neighbors. So, there ( are at ) most possible ways of choosing v. Note that we have ways of choosing the remaining vertices β {u 1,..., u β+1 ( } \ ) {a}. Similar case for b as well. So, there are at most 2 type-2 events that A β e can depends on.
9 Frugal Coloring V Similarly, a type-2 event B u1,...,u β+1 can depends on (β + 1) ( ) other type-1 bad events and (β + 1) β other type-2 bad events Note that P [A e ] 1 C ] P [B u1,...,u β+1 1 C β So, to prove that a β-frugal coloring exists, it suffices to prove that ( ) (β + 1) (β + 1) C β C β < 1 4
10 Frugal Coloring VI ( ) n We can use the upper bound k the expression (β + 1) C + ( ) en k k to upper-bound ( ) (β + 1) C β β This is left as an exercise
11 Moser-Tardos Algorithm Now we are interested in computing the solution that is guaranteed by. function Seq_LLL(X = {X 1,..., X m }, A = {A 1,..., A n }) X Random Evaluation while i s.t. A i is satisfied do Pick arbitrary A i that is satisfied Re-sample all X j such that j Vbl i end while Output X end function
12 Performance of Moser-Tardos Theorem Suppose there exists (x 1,..., x n ) such that, for all i [n], we have P [A i ] x i j D i (1 x j ). Then the expected number of times sequential Moser-Tardos samples the event A i is at most x i /(1 x i ) and, hence, the expected number of execution of the inner loop is at most i [n] x i/(1 x i ).
13 Parallel Moser-Tardos Algorithm function Parallel_LLL(X, A) X Random Evaluation while i s.t. A i is satisfied do Let S be a maximal independent set in the dependency graph restricted to satisfied A i s X k S Vbl k end while Output X end function Random Evaluation
14 Performance of Parallel Moder-Tardos Theorem Suppose there exists an ε > 0 and (x 1,..., x n ) such that P [A i ] (1 ε)x i j D i (1 x j ). The expected number of inner loops before all bad events are avoided is at most O ( 1 ε i [n] x i 1 x i ).
Lecture 24: April 12
CS271 Randomness & Computation Spring 2018 Instructor: Alistair Sinclair Lecture 24: April 12 Disclaimer: These notes have not been subjected to the usual scrutiny accorded to formal publications. They
More informationAn Algorithmic Proof of the Lopsided Lovász Local Lemma (simplified and condensed into lecture notes)
An Algorithmic Proof of the Lopsided Lovász Local Lemma (simplified and condensed into lecture notes) Nicholas J. A. Harvey University of British Columbia Vancouver, Canada nickhar@cs.ubc.ca Jan Vondrák
More informationThe Probabilistic Method
The Probabilistic Method Janabel Xia and Tejas Gopalakrishna MIT PRIMES Reading Group, mentors Gwen McKinley and Jake Wellens December 7th, 2018 Janabel Xia and Tejas Gopalakrishna Probabilistic Method
More informationLecture Notes CS:5360 Randomized Algorithms Lecture 20 and 21: Nov 6th and 8th, 2018 Scribe: Qianhang Sun
1 Probabilistic Method Lecture Notes CS:5360 Randomized Algorithms Lecture 20 and 21: Nov 6th and 8th, 2018 Scribe: Qianhang Sun Turning the MaxCut proof into an algorithm. { Las Vegas Algorithm Algorithm
More informationThe Lovász Local Lemma : A constructive proof
The Lovász Local Lemma : A constructive proof Andrew Li 19 May 2016 Abstract The Lovász Local Lemma is a tool used to non-constructively prove existence of combinatorial objects meeting a certain conditions.
More informationn Boolean variables: x 1, x 2,...,x n 2 {true,false} conjunctive normal form:
Advanced Algorithms k-sat n Boolean variables: x 1, x 2,...,x n 2 {true,false} conjunctive normal form: k-cnf = C 1 ^C 2 ^ ^C m Is φ satisfiable? m clauses: C 1,C 2,...,C m each clause C i = `i1 _ `i2
More informationLecture 10: Lovász Local Lemma. Lovász Local Lemma
Lecture 10: Introduction Let A 1,..., A n be indicator variables for bad events in an experiment Suppose P [A i ] p We want to avoid all the bad events If P [ A 1 A n ] > 0, then there exists a way to
More informationNowhere 0 mod p dominating sets in multigraphs
Nowhere 0 mod p dominating sets in multigraphs Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel. e-mail: raphy@research.haifa.ac.il Abstract Let G be a graph with
More informationComputing the Independence Polynomial: from the Tree Threshold Down to the Roots
1 / 16 Computing the Independence Polynomial: from the Tree Threshold Down to the Roots Nick Harvey 1 Piyush Srivastava 2 Jan Vondrák 3 1 UBC 2 Tata Institute 3 Stanford SODA 2018 The Lovász Local Lemma
More information12. LOCAL SEARCH. gradient descent Metropolis algorithm Hopfield neural networks maximum cut Nash equilibria
12. LOCAL SEARCH gradient descent Metropolis algorithm Hopfield neural networks maximum cut Nash equilibria Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley h ttp://www.cs.princeton.edu/~wayne/kleinberg-tardos
More informationTree-width and algorithms
Tree-width and algorithms Zdeněk Dvořák September 14, 2015 1 Algorithmic applications of tree-width Many problems that are hard in general become easy on trees. For example, consider the problem of finding
More informationDominating Set. Chapter 7
Chapter 7 Dominating Set In this chapter we present another randomized algorithm that demonstrates the power of randomization to break symmetries. We study the problem of finding a small dominating set
More informationThe Lopsided Lovász Local Lemma
Joint work with Linyuan Lu and László Székely Georgia Southern University April 27, 2013 The lopsided Lovász local lemma can establish the existence of objects satisfying several weakly correlated conditions
More informationAn asymptotically tight bound on the adaptable chromatic number
An asymptotically tight bound on the adaptable chromatic number Michael Molloy and Giovanna Thron University of Toronto Department of Computer Science 0 King s College Road Toronto, ON, Canada, M5S 3G
More informationGreedy Algorithms. CSE 101: Design and Analysis of Algorithms Lecture 10
Greedy Algorithms CSE 101: Design and Analysis of Algorithms Lecture 10 CSE 101: Design and analysis of algorithms Greedy algorithms Reading: Kleinberg and Tardos, sections 4.1, 4.2, and 4.3 Homework 4
More informationApplications of the Lopsided Lovász Local Lemma Regarding Hypergraphs
Regarding Hypergraphs Ph.D. Dissertation Defense April 15, 2013 Overview The Local Lemmata 2-Coloring Hypergraphs with the Original Local Lemma Counting Derangements with the Lopsided Local Lemma Lopsided
More informationLecture 28: April 26
CS271 Randomness & Computation Spring 2018 Instructor: Alistair Sinclair Lecture 28: April 26 Disclaimer: These notes have not been subjected to the usual scrutiny accorded to formal publications. They
More informationRandomized Algorithms III Min Cut
Chapter 11 Randomized Algorithms III Min Cut CS 57: Algorithms, Fall 01 October 1, 01 11.1 Min Cut 11.1.1 Problem Definition 11. Min cut 11..0.1 Min cut G = V, E): undirected graph, n vertices, m edges.
More informationCS5314 Randomized Algorithms. Lecture 18: Probabilistic Method (De-randomization, Sample-and-Modify)
CS5314 Randomized Algorithms Lecture 18: Probabilistic Method (De-randomization, Sample-and-Modify) 1 Introduce two topics: De-randomize by conditional expectation provides a deterministic way to construct
More informationNotice that lemma 4 has nothing to do with 3-colorability. To obtain a better result for 3-colorable graphs, we need the following observation.
COMPSCI 632: Approximation Algorithms November 1, 2017 Lecturer: Debmalya Panigrahi Lecture 18 Scribe: Feng Gui 1 Overview In this lecture, we examine graph coloring algorithms. We first briefly discuss
More informationPRAMs. M 1 M 2 M p. globaler Speicher
PRAMs A PRAM (parallel random access machine) consists of p many identical processors M,..., M p (RAMs). Processors can read from/write to a shared (global) memory. Processors work synchronously. M M 2
More informationDominating Set. Chapter 26
Chapter 26 Dominating Set In this chapter we present another randomized algorithm that demonstrates the power of randomization to break symmetries. We study the problem of finding a small dominating set
More informationDominating Set. Chapter Sequential Greedy Algorithm 294 CHAPTER 26. DOMINATING SET
294 CHAPTER 26. DOMINATING SET 26.1 Sequential Greedy Algorithm Chapter 26 Dominating Set Intuitively, to end up with a small dominating set S, nodes in S need to cover as many neighbors as possible. It
More informationDynamic Programming on Trees. Example: Independent Set on T = (V, E) rooted at r V.
Dynamic Programming on Trees Example: Independent Set on T = (V, E) rooted at r V. For v V let T v denote the subtree rooted at v. Let f + (v) be the size of a maximum independent set for T v that contains
More informationProbabilistic Methods in Combinatorics Lecture 6
Probabilistic Methods in Combinatorics Lecture 6 Linyuan Lu University of South Carolina Mathematical Sciences Center at Tsinghua University November 16, 2011 December 30, 2011 Balance graphs H has v vertices
More information25 Minimum bandwidth: Approximation via volume respecting embeddings
25 Minimum bandwidth: Approximation via volume respecting embeddings We continue the study of Volume respecting embeddings. In the last lecture, we motivated the use of volume respecting embeddings by
More information12. LOCAL SEARCH. gradient descent Metropolis algorithm Hopfield neural networks maximum cut Nash equilibria
Coping With NP-hardness Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you re unlikely to find poly-time algorithm. Must sacrifice one of three desired features. Solve
More informationThe Lovász Local Lemma: constructive aspects, stronger variants and the hard core model
The Lovász Local Lemma: constructive aspects, stronger variants and the hard core model Jan Vondrák 1 1 Dept. of Mathematics Stanford University joint work with Nick Harvey (UBC) The Lovász Local Lemma
More informationCS 781 Lecture 9 March 10, 2011 Topics: Local Search and Optimization Metropolis Algorithm Greedy Optimization Hopfield Networks Max Cut Problem Nash
CS 781 Lecture 9 March 10, 2011 Topics: Local Search and Optimization Metropolis Algorithm Greedy Optimization Hopfield Networks Max Cut Problem Nash Equilibrium Price of Stability Coping With NP-Hardness
More informationIntroduction to Algorithms
Introduction to Algorithms 6.046J/18.401J/SMA5503 Lecture 18 Prof. Erik Demaine Negative-weight cycles Recall: If a graph G = (V, E) contains a negativeweight cycle, then some shortest paths may not exist.
More informationLecture 18: Zero-Knowledge Proofs
COM S 6810 Theory of Computing March 26, 2009 Lecture 18: Zero-Knowledge Proofs Instructor: Rafael Pass Scribe: Igor Gorodezky 1 The formal definition We intuitively defined an interactive proof to be
More informationThe Lopsided Lovász Local Lemma
Department of Mathematics Nebraska Wesleyan University With Linyuan Lu and László Székely, University of South Carolina Note on Probability Spaces For this talk, every a probability space Ω is assumed
More informationAn extension of the Moser-Tardos algorithmic local lemma
An extension of the Moser-Tardos algorithmic local lemma Wesley Pegden January 26, 2013 Abstract A recent theorem of Bissacot, et al. proved using results about the cluster expansion in statistical mechanics
More informationGood Triangulations. Jean-Daniel Boissonnat DataShape, INRIA
Good Triangulations Jean-Daniel Boissonnat DataShape, INRIA http://www-sop.inria.fr/geometrica Algorithmic Geometry Good Triangulations J-D. Boissonnat 1 / 29 Definition and existence of nets Definition
More informationAditya Bhaskara CS 5968/6968, Lecture 1: Introduction and Review 12 January 2016
Lecture 1: Introduction and Review We begin with a short introduction to the course, and logistics. We then survey some basics about approximation algorithms and probability. We also introduce some of
More informationProbabilistic Constructions of Computable Objects and a Computable Version of Lovász Local Lemma.
Probabilistic Constructions of Computable Objects and a Computable Version of Lovász Local Lemma. Andrei Rumyantsev, Alexander Shen * January 17, 2013 Abstract A nonconstructive proof can be used to prove
More information3. Branching Algorithms
3. Branching Algorithms COMP6741: Parameterized and Exact Computation Serge Gaspers Semester 2, 2015 Contents 1 Introduction 1 2 Maximum Independent Set 3 2.1 Simple Analysis................................................
More informationLecture 6: September 22
CS294 Markov Chain Monte Carlo: Foundations & Applications Fall 2009 Lecture 6: September 22 Lecturer: Prof. Alistair Sinclair Scribes: Alistair Sinclair Disclaimer: These notes have not been subjected
More informationLecture Semidefinite Programming and Graph Partitioning
Approximation Algorithms and Hardness of Approximation April 16, 013 Lecture 14 Lecturer: Alantha Newman Scribes: Marwa El Halabi 1 Semidefinite Programming and Graph Partitioning In previous lectures,
More informationLecture 10 Algorithmic version of the local lemma
Lecture 10 Algorithmic version of the local lemma Uriel Feige Department of Computer Science and Applied Mathematics The Weizman Institute Rehovot 76100, Israel uriel.feige@weizmann.ac.il June 9, 2014
More informationHomology groups of neighborhood complexes of graphs (Introduction to topological combinatorics)
Homology groups of neighborhood complexes of graphs (Introduction to topological combinatorics) October 21, 2017 Y. Hara (Osaka Univ.) Homology groups of neighborhood complexes October 21, 2017 1 / 18
More information3. Branching Algorithms COMP6741: Parameterized and Exact Computation
3. Branching Algorithms COMP6741: Parameterized and Exact Computation Serge Gaspers 12 1 School of Computer Science and Engineering, UNSW Australia 2 Optimisation Resarch Group, NICTA Semester 2, 2015
More informationSubdivisions of a large clique in C 6 -free graphs
Subdivisions of a large clique in C 6 -free graphs József Balogh Hong Liu Maryam Sharifzadeh October 8, 2014 Abstract Mader conjectured that every C 4 -free graph has a subdivision of a clique of order
More informationMATH 409 LECTURES THE KNAPSACK PROBLEM
MATH 409 LECTURES 19-21 THE KNAPSACK PROBLEM REKHA THOMAS We now leave the world of discrete optimization problems that can be solved in polynomial time and look at the easiest case of an integer program,
More informationU.C. Berkeley Better-than-Worst-Case Analysis Handout 3 Luca Trevisan May 24, 2018
U.C. Berkeley Better-than-Worst-Case Analysis Handout 3 Luca Trevisan May 24, 2018 Lecture 3 In which we show how to find a planted clique in a random graph. 1 Finding a Planted Clique We will analyze
More informationRandomized Algorithms
Randomized Algorithms Prof. Tapio Elomaa tapio.elomaa@tut.fi Course Basics A new 4 credit unit course Part of Theoretical Computer Science courses at the Department of Mathematics There will be 4 hours
More informationCographs; chordal graphs and tree decompositions
Cographs; chordal graphs and tree decompositions Zdeněk Dvořák September 14, 2015 Let us now proceed with some more interesting graph classes closed on induced subgraphs. 1 Cographs The class of cographs
More informationExtremal H-colorings of graphs with fixed minimum degree
Extremal H-colorings of graphs with fixed minimum degree John Engbers July 18, 2014 Abstract For graphs G and H, a homomorphism from G to H, or H-coloring of G, is a map from the vertices of G to the vertices
More informationAn Improved Approximation Algorithm for Maximum Edge 2-Coloring in Simple Graphs
An Improved Approximation Algorithm for Maximum Edge 2-Coloring in Simple Graphs Zhi-Zhong Chen Ruka Tanahashi Lusheng Wang Abstract We present a polynomial-time approximation algorithm for legally coloring
More informationLecture 5: January 30
CS71 Randomness & Computation Spring 018 Instructor: Alistair Sinclair Lecture 5: January 30 Disclaimer: These notes have not been subjected to the usual scrutiny accorded to formal publications. They
More informationLecture 31: Miller Rabin Test. Miller Rabin Test
Lecture 31: Recall In the previous lecture we considered an efficient randomized algorithm to generate prime numbers that need n-bits in their binary representation This algorithm sampled a random element
More information20.1 2SAT. CS125 Lecture 20 Fall 2016
CS125 Lecture 20 Fall 2016 20.1 2SAT We show yet another possible way to solve the 2SAT problem. Recall that the input to 2SAT is a logical expression that is the conunction (AND) of a set of clauses,
More informationHW Graph Theory SOLUTIONS (hbovik) - Q
1, Diestel 3.5: Deduce the k = 2 case of Menger s theorem (3.3.1) from Proposition 3.1.1. Let G be 2-connected, and let A and B be 2-sets. We handle some special cases (thus later in the induction if these
More informationarxiv: v1 [cs.dm] 26 Apr 2010
A Simple Polynomial Algorithm for the Longest Path Problem on Cocomparability Graphs George B. Mertzios Derek G. Corneil arxiv:1004.4560v1 [cs.dm] 26 Apr 2010 Abstract Given a graph G, the longest path
More informationExtremal H-colorings of trees and 2-connected graphs
Extremal H-colorings of trees and 2-connected graphs John Engbers David Galvin June 17, 2015 Abstract For graphs G and H, an H-coloring of G is an adjacency preserving map from the vertices of G to the
More informationLecture 17 November 8, 2012
6.841: Advanced Complexity Theory Fall 2012 Prof. Dana Moshkovitz Lecture 17 November 8, 2012 Scribe: Mark Bun 1 Overview In the previous lecture, we saw an overview of probabilistically checkable proofs,
More informationThe max flow problem. Ford-Fulkerson method. A cut. Lemma Corollary Max Flow Min Cut Theorem. Max Flow Min Cut Theorem
The max flow problem Ford-Fulkerson method 7 11 Ford-Fulkerson(G) f = 0 while( simple path p from s to t in G f ) 10-2 2 1 f := f + f p output f 4 9 1 2 A cut Lemma 26. + Corollary 26.6 Let f be a flow
More informationThe Intermediate Value Theorem If a function f (x) is continuous in the closed interval [ a,b] then [ ]
Lecture 2 5B Evaluating Limits Limits x ---> a The Intermediate Value Theorem If a function f (x) is continuous in the closed interval [ a,b] then [ ] the y values f (x) must take on every value on the
More informationObservation 4.1 G has a proper separation of order 0 if and only if G is disconnected.
4 Connectivity 2-connectivity Separation: A separation of G of order k is a pair of subgraphs (H, K) with H K = G and E(H K) = and V (H) V (K) = k. Such a separation is proper if V (H) \ V (K) and V (K)
More informationCoupling. 2/3/2010 and 2/5/2010
Coupling 2/3/2010 and 2/5/2010 1 Introduction Consider the move to middle shuffle where a card from the top is placed uniformly at random at a position in the deck. It is easy to see that this Markov Chain
More informationTopics in Theoretical Computer Science April 08, Lecture 8
Topics in Theoretical Computer Science April 08, 204 Lecture 8 Lecturer: Ola Svensson Scribes: David Leydier and Samuel Grütter Introduction In this lecture we will introduce Linear Programming. It was
More informationUniversity of Chicago Autumn 2003 CS Markov Chain Monte Carlo Methods
University of Chicago Autumn 2003 CS37101-1 Markov Chain Monte Carlo Methods Lecture 4: October 21, 2003 Bounding the mixing time via coupling Eric Vigoda 4.1 Introduction In this lecture we ll use the
More informationAdvanced Combinatorial Optimization September 22, Lecture 4
8.48 Advanced Combinatorial Optimization September 22, 2009 Lecturer: Michel X. Goemans Lecture 4 Scribe: Yufei Zhao In this lecture, we discuss some results on edge coloring and also introduce the notion
More informationObservation 4.1 G has a proper separation of order 0 if and only if G is disconnected.
4 Connectivity 2-connectivity Separation: A separation of G of order k is a pair of subgraphs (H 1, H 2 ) so that H 1 H 2 = G E(H 1 ) E(H 2 ) = V (H 1 ) V (H 2 ) = k Such a separation is proper if V (H
More informationMaximizing the number of independent sets of a fixed size
Maximizing the number of independent sets of a fixed size Wenying Gan Po-Shen Loh Benny Sudakov Abstract Let i t (G be the number of independent sets of size t in a graph G. Engbers and Galvin asked how
More informationSpectral Graph Theory Lecture 2. The Laplacian. Daniel A. Spielman September 4, x T M x. ψ i = arg min
Spectral Graph Theory Lecture 2 The Laplacian Daniel A. Spielman September 4, 2015 Disclaimer These notes are not necessarily an accurate representation of what happened in class. The notes written before
More informationAlgebraic Methods in Combinatorics
Algebraic Methods in Combinatorics Po-Shen Loh 27 June 2008 1 Warm-up 1. (A result of Bourbaki on finite geometries, from Răzvan) Let X be a finite set, and let F be a family of distinct proper subsets
More informationdirected weighted graphs as flow networks the Ford-Fulkerson algorithm termination and running time
Network Flow 1 The Maximum-Flow Problem directed weighted graphs as flow networks the Ford-Fulkerson algorithm termination and running time 2 Maximum Flows and Minimum Cuts flows and cuts max flow equals
More informationAn Improved Algorithm for Parameterized Edge Dominating Set Problem
An Improved Algorithm for Parameterized Edge Dominating Set Problem Ken Iwaide and Hiroshi Nagamochi Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Japan,
More informationKernelization by matroids: Odd Cycle Transversal
Lecture 8 (10.05.2013) Scribe: Tomasz Kociumaka Lecturer: Marek Cygan Kernelization by matroids: Odd Cycle Transversal 1 Introduction The main aim of this lecture is to give a polynomial kernel for the
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Handout 2 Luca Trevisan August 29, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analysis Handout Luca Trevisan August 9, 07 Scribe: Mahshid Montazer Lecture In this lecture, we study the Max Cut problem in random graphs. We compute the probable
More informationPacking and Covering Dense Graphs
Packing and Covering Dense Graphs Noga Alon Yair Caro Raphael Yuster Abstract Let d be a positive integer. A graph G is called d-divisible if d divides the degree of each vertex of G. G is called nowhere
More informationTheorem (Special Case of Ramsey s Theorem) R(k, l) is finite. Furthermore, it satisfies,
Math 16A Notes, Wee 6 Scribe: Jesse Benavides Disclaimer: These notes are not nearly as polished (and quite possibly not nearly as correct) as a published paper. Please use them at your own ris. 1. Ramsey
More informationGraphs with Large Variance
Graphs with Large Variance Yair Caro Raphael Yuster Abstract For a graph G, let V ar(g) denote the variance of the degree sequence of G, let sq(g) denote the sum of the squares of the degrees of G, and
More informationThe Intermediate Value Theorem If a function f (x) is continuous in the closed interval [ a,b] then [ ]
Lecture 4-6B1 Evaluating Limits Limits x ---> a The Intermediate Value Theorem If a function f (x) is continuous in the closed interval [ a,b] then [ ] the y values f (x) must take on every value on the
More informationAdvanced Algorithms 南京大学 尹一通
Advanced Algorithms 南京大学 尹一通 Constraint Satisfaction Problem variables: (CSP) X = {x 1, x2,..., xn} each variable ranges over a finite domain Ω an assignment σ ΩX assigns each variable a value in Ω constraints:
More informationL11.P1 Lecture 11. Quantum statistical mechanics: summary
Lecture 11 Page 1 L11.P1 Lecture 11 Quantum statistical mechanics: summary At absolute zero temperature, a physical system occupies the lowest possible energy configuration. When the temperature increases,
More informationThe Turán number of sparse spanning graphs
The Turán number of sparse spanning graphs Noga Alon Raphael Yuster Abstract For a graph H, the extremal number ex(n, H) is the maximum number of edges in a graph of order n not containing a subgraph isomorphic
More informationProbability Models of Information Exchange on Networks Lecture 6
Probability Models of Information Exchange on Networks Lecture 6 UC Berkeley Many Other Models There are many models of information exchange on networks. Q: Which model to chose? My answer good features
More informationA constructive algorithm for the Lovász Local Lemma on permutations
A constructive algorithm for the Lovász Local Lemma on permutations David G. Harris Aravind Srinivasan Abstract While there has been significant progress on algorithmic aspects of the Lovász Local Lemma
More informationCHAPTER 7. Connectedness
CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set
More informationA Lower Bound for the Distributed Lovász Local Lemma
A Lower Bound for the Distributed Lovász Local Lemma Sebastian Brandt, Orr Fischer, Juho Hirvonen, Barbara Keller, Tuomo Lempiäinen, Joel Rybicki, Jukka Suomela, Jara Uitto Aalto University, Comerge AG,
More informationCS261: A Second Course in Algorithms Lecture #9: Linear Programming Duality (Part 2)
CS261: A Second Course in Algorithms Lecture #9: Linear Programming Duality (Part 2) Tim Roughgarden February 2, 2016 1 Recap This is our third lecture on linear programming, and the second on linear programming
More informationOn the Turán number of forests
On the Turán number of forests Bernard Lidický Hong Liu Cory Palmer April 13, 01 Abstract The Turán number of a graph H, ex(n, H, is the maximum number of edges in a graph on n vertices which does not
More information1 Randomized reduction: a first start
6.S078 Fine-Grained Algorithms and Complexity MIT Lectures 3 and 4 February 14 and 21, 2018 Today: In the previous two lectures, we mentioned ETH and SETH, about the time complexity of SAT. Today we want
More informationMath 5707: Graph Theory, Spring 2017 Midterm 3
University of Minnesota Math 5707: Graph Theory, Spring 2017 Midterm 3 Nicholas Rancourt (edited by Darij Grinberg) December 25, 2017 1 Exercise 1 1.1 Problem Let G be a connected multigraph. Let x, y,
More informationCMSC 451: Lecture 7 Greedy Algorithms for Scheduling Tuesday, Sep 19, 2017
CMSC CMSC : Lecture Greedy Algorithms for Scheduling Tuesday, Sep 9, 0 Reading: Sects.. and. of KT. (Not covered in DPV.) Interval Scheduling: We continue our discussion of greedy algorithms with a number
More informationRobin Moser makes Lovász Local Lemma Algorithmic! Notes of Joel Spencer
Robin Moser makes Lovász Local Lemma Algorithmic! Notes of Joel Spencer 1 Preliminaries The idea in these notes is to explain a new approach of Robin Moser 1 to give an algorithm for the Lovász Local Lemma.
More information1 Primals and Duals: Zero Sum Games
CS 124 Section #11 Zero Sum Games; NP Completeness 4/15/17 1 Primals and Duals: Zero Sum Games We can represent various situations of conflict in life in terms of matrix games. For example, the game shown
More informationNowhere zero flow. Definition: A flow on a graph G = (V, E) is a pair (D, f) such that. 1. D is an orientation of G. 2. f is a function on E.
Nowhere zero flow Definition: A flow on a graph G = (V, E) is a pair (D, f) such that 1. D is an orientation of G. 2. f is a function on E. 3. u N + D (v) f(uv) = w ND f(vw) for every (v) v V. Example:
More informationLecture 15: Expanders
CS 710: Complexity Theory 10/7/011 Lecture 15: Expanders Instructor: Dieter van Melkebeek Scribe: Li-Hsiang Kuo In the last lecture we introduced randomized computation in terms of machines that have access
More informationStability of the path-path Ramsey number
Worcester Polytechnic Institute Digital WPI Computer Science Faculty Publications Department of Computer Science 9-12-2008 Stability of the path-path Ramsey number András Gyárfás Computer and Automation
More informationStrongly chordal and chordal bipartite graphs are sandwich monotone
Strongly chordal and chordal bipartite graphs are sandwich monotone Pinar Heggernes Federico Mancini Charis Papadopoulos R. Sritharan Abstract A graph class is sandwich monotone if, for every pair of its
More information1 Maximum Budgeted Allocation
CS 369P: Polyhedral techniques in combinatorial optimization Instructor: Jan Vondrák Lecture date: November 4, 2010 Scribe: David Tobin 1 Maximum Budgeted Allocation Agents Items Given: n agents and m
More informationA Different Perspective For Approximating Max Set Packing
Weizmann Institute of Science Thesis for the degree Master of Science Submitted to the Scientific Council of the Weizmann Institute of Science Rehovot, Israel A Different Perspective For Approximating
More informationFlow Network. The following figure shows an example of a flow network:
Maximum Flow 1 Flow Network The following figure shows an example of a flow network: 16 V 1 12 V 3 20 s 10 4 9 7 t 13 4 V 2 V 4 14 A flow network G = (V,E) is a directed graph. Each edge (u, v) E has a
More informationLebesgue Measure. Dung Le 1
Lebesgue Measure Dung Le 1 1 Introduction How do we measure the size of a set in IR? Let s start with the simplest ones: intervals. Obviously, the natural candidate for a measure of an interval is its
More informationBetter bounds for k-partitions of graphs
Better bounds for -partitions of graphs Baogang Xu School of Mathematics, Nanjing Normal University 1 Wenyuan Road, Yadong New District, Nanjing, 1006, China Email: baogxu@njnu.edu.cn Xingxing Yu School
More informationLarge topological cliques in graphs without a 4-cycle
Large topological cliques in graphs without a 4-cycle Daniela Kühn Deryk Osthus Abstract Mader asked whether every C 4 -free graph G contains a subdivision of a complete graph whose order is at least linear
More informationCS6999 Probabilistic Methods in Integer Programming Randomized Rounding Andrew D. Smith April 2003
CS6999 Probabilistic Methods in Integer Programming Randomized Rounding April 2003 Overview 2 Background Randomized Rounding Handling Feasibility Derandomization Advanced Techniques Integer Programming
More information