Lecture 1 : Probabilistic Method
|
|
- Brian Dixon
- 6 years ago
- Views:
Transcription
1 IITM-CS6845: Theory Jan 04, 01 Lecturer: N.S.Narayanaswamy Lecture 1 : Probabilistic Method Scribe: R.Krithika The probabilistic method is a technique to deal with combinatorial problems by introducing randomness. Though the method relies on probability theory, it can be used to make deterministic statements. The probabilistic method toolkit typically includes, but is not limited to, Markov s Inequality, Linearity of Expectation, Lovász Local Lemma, Concentration Inequalities and Subadditivity of Probabilities (the union bound), to name a few. This lecture describes examples that illustrate the application of some of these varied tools. 1 Max Cut A cut is a partition of the vertices of a graph into two disjoint sets. An edge is a crossing edge if its end points are in different sets of the partition. The cut-set of a cut is the set of crossing edges. The cut of a graph can sometimes refer to its cut-set instead of the partition. Max Cut Given a graph G, partition V (G) into V 0 and V 1 such that the number of edges crossing the cut is maximum. Max Cut is known to be NP-hard by a reduction from Max -SAT [GJ79]. A simple polynomial-time randomized algorithm achieves an approximation factor of. For each vertex v, flip an unbiased coin to decide which set of the partition v belongs to. Consider this random partition. Linearity of expectation, Independence of events Let X denote the cardinality of the associated cut. For each edge e E(G), define indicator random variable, { 1 if e is a crossing edge X e = 0 otherwise X = X e 1-1
2 By linearity of expectation, E(X) = E[ X e ] = E[X e ] = P r(x e = 1) Note that the coin tosses corresponding to the choices of the sets to the vertices are independent. The probability that an edge e = {u, v} is an crossing edge is given by, Thus, P r(x e = 1) = P r(u V 0 v V 1 ) + P r(v V 0 u V 1 ) = 1 E(X) = 1 = m Since the size of a maximum cardinality cut is at most m, the number of edges in G, the algorithm described is a -approximation for Max Cut. Note that this algorithm can be derandomized with the method of conditional probabilities to obtain a deterministic polynomial-time -approximation algorithm. Further Study The output of the polynomial-time randomized algorithm A described in this section can be viewed as a function of n random bits. The i th bit b i in this n-bit string b {0, 1} n takes the value j {0, 1} denoting that vertex v i is assigned to set V j in the random partition considered. A can be derandomized as follows: Run A for each choice of n-bit string and output the largest value of the cut. By our expectation argument, it follows that, A(b 1 b n ) b 1 b n n = m However, the algorithm is no longer polynomial as there are n choices of the n-bit string to consider. Interestingly, it can be shown that there exists a set S {0, 1} n such that S = O(n ) and, A(b 1 b n ) (b 1 b n) S S Such existential combinatorics and corresponding explicit constructions are interesting to consider. = m Randomized Quick Sort In the classical average case-analysis of quick sort, we rely on the assumption that all permutations of the input array are equally likely. However, in engineering applications, the 1-
3 input distribution is rarely known a priori. instead of assuming a distribution of inputs, we impose a distribution by introducing randomness to the algorithm. We associate the notion of rank with each element in A. That is, r i is the i th smallest element among the elements of A. For simplicity, we assume A has distinct elements. Choose an index i from {1,, n}, where n is the number of elements in A. This choice is made uniformly at random. Now, A[i] is designated as the pivot and the classical quick sort algorithm proceeds. At every choice of the pivot, randomness is introduced. The parameter of interest for the analysis is the expected number of comparisons X made in a run of quick sort on A. Linearity of expectation Define indicator random variable, { 1 if ri and r X ij = j are compared 0 otherwise X = n i=i j>i Here are crucial invariants used in the analysis of randomized quick sort. X ij 1. Once an element x has been selected as a pivot in a call to partition, x does not participate in any comparison in any other calls to partition.. In any call to partition, any comparison involves the pivot of that call. 3. Any pair of elements are compared at most once. 4. Two elements r i and r j are compared if and only if the first element to be chosen as pivot from r i, r i+1,..., r j is either r i or r j. By linearity of expectation, E(X) = n E[X ij ] = i=i j>i n P r(x ij = 1) = i=i j>i n j i + 1 i=i j>i n i=i n i+1 k=1 k = nh n where H n denotes the n th Harmonic number. Thus, E(X) = O(n log n) 1-3
4 Further Study The next inevitable question is to determine if there exists a family S of permutations of n elements such that the average number of comparisons over runs of quick sort on this set is O(n log n). That is, we would typically like to come up with a set S S n, where S n is the set of all permutations of n elements, such that, σ S QuickSort(σ) = O(n log n) S If S = O(n O(1) ), we could use derandomization to obtain a polynomial-time algorithm achieving the desired average-case behaviour. Though, this technique does not yield a sorting algorithm faster than O(n log n), the mathematical artifacts involved are of independent interest. 3 Min Cut Consider a connected, undirected multigraph G on n vertices. By the property of a cut, deleting the edges of a cut disconnects G. A min cut is a cut of minimum cardinality. Min Cut Given a graph G, find a minimum set of edges whose deletion disconnects G. By the max-flow min-cut theorem, there are polynomial-time algorithms to solve Min Cut, notably the Edmonds-Karp algorithm. Non-flow based algorithms are also known for the same. However, they are quite complicated. In this section, we study a simple randomized non-flow based algorithm for finding a min cut. Given G, pick an edge uniformly at random and contract it. The resultant graph is denoted as G e. Observe that no edge contraction reduces the size of a min cut. Further, every cut in G e is a cut in G too. Repeat this contraction step till G has only vertices, say x and y. Note that each contraction reduces the number of vertices of G by one. Eliminate self-loops created as a result of a contraction. Output the cut as X and Y, where X and Y correspond to the sets of vertices merged into x and y, respectively. We will now analyse the probability with which this algorithm outputs a min cut. We denote the graph obtained after i contractions by G i. Method of conditional probabilities Consider a min cut C of G. Let C = k. Since the size of a min cut is at most the min degree δ of G, E(G) nk. We will now estimate the probability with which C survives 1-4
5 after the sequence of contractions during the execution of the algorithm. P r(e C is contracted) k nk = n Also, since contractions are independent, Further, P r(c survives after first contraction) 1 n P r(c survives after contractions) (1 n )P r(c survives in G 1) P r(c survives after the execution of the algorithm ) (1 n )(1 n 1 ) (1 n (n 3) ) n P r(c is output by the algorithm ) (1 n i + 1 ) = n(n 1) Thus, the probability of the algorithm discovering a min cut is at least. The correctness n guarantee can be increased by running the algorithm multiple times and the min cardinality set is output. By n runs of the algorithm, the probability that the a min cut is not output is at most (1 ) n n < 1 e. Further executions of the algorithm makes the failure probability arbitrarily small at the cost of increasing the run-time. i=1 Interesting Observations and Further Study In Max Cut and Randomized Quick Sort, the random variable of interest was split into sub random variables taking significantly smaller range of values as compared to the original random variable. Thus analysis is localised by linearity of expectation and independence of events. However, in the analysis of the algorithm for Min Cut, there is an extensive dependence among events. In such situations, the method of conditional probabilities prove to be an asset to the algorithm analysis. Further, note that Quick Sort runs in O(n log n) time with high probability, where as in Min Cut, the optimum solution is obtained with high probability. The sorting algorithm always produces the correct solution. However, the randomness is in the run-time while in the Min Cut algorithm, the correctness of the solution is random. These randomized algorithms exemplify two different types of randomized algorithms, namely, Las Vegas and Monte Carlo algorithms. 1-5
6 4 Ramsey Numbers Ramsey theory deals with finding order amongst apparent chaos. Given a setting where a mathematical structure may appear, Ramsey theory strives to identify conditions on this setting under which this mathematical structure of interest must appear. In other words, it is an attempt to ascertain that complete disorder is an impossibility and any large structure will necessarily contain an orderly substructure. Ramsey Number R(k, l) R(k, l) is the least integer n such that any edge coloring of K n using colors (say, red and blue) has either a red K k or a blue K l. Equivalently, R(k, l) is the least integer n such that any graph on at least n vertices has a clique of size k or an independent set of size l. To prove bounds on diagonal Ramsey numbers {R(k, k) k = 1, }, we identify possible relations between n and k. Union bound, Independence of events Lemma 1. If k and n are positive integers satisfying ( n k) 1 ( k ) < 1, then R(k, k) > n. Proof. Consider a clique G on n vertices. Color the edges of G uniformly at random with colors, red and blue. Consider a set K of k vertices in G. by the union bound, P r(g[k] is monochromatic) = P r( K V (G), K = k such that G[K] is monochromatic) ( ) n 1 (k ) k Since ( n k) 1 ( k ) < 1, it follows that there is a -edge coloring of G producing neither a red K k nor a blue K k. Equivalently, there are at least graphs on n vertices having neither a K k nor an independent set of size k. These two graphs G 1 and G are the spanning subgraphs of G such that E(G 1 ) = {e E(G) e is red} and E(G ) = {e E(G) e is blue}, respectively. Observe that G 1 and G are complementary graphs of each other. R(k, k) > n. Lemma. For any k 3, R(k, k) > k
7 Proof. Consider a clique G on n vertices where n k 1. Color the edges of G uniformly at random with colors, red and blue. Consider a set K of k vertices in G. P r(k is monochromatic) = by the union bound, P r( K V (G), K = k such that K is monochromatic) ( ) n k Also, ( ) n k nk k( k 1) = k k+1 = (k ) k +1 Since k 3, k > 1. Thus, P r( K V (G), K = k such that K is monochromatic) < 1. It follows that there exists a coloring producing no monochromatic K k in G. Hence, R(k, k) > k 1, for k 3. Further Study Better bounds for R(k, k) and constructive Ramsey numbers are natural directions for a deeper study. As a generalization, multicolour Ramsey numbers are other interesting objects worth searching for. 5 Tournaments with Property S k A tournament T (V, E) is a complete directed graph. That is, for any v i, v j V (T ), exactly one of (v i, v j ) and (v j, v i ) is in E(T ). T is said to have property S k if for every set X of k vertices, there exists a vertex v l V (T ) such that X N(v l ). We refer to such a set X as being dominated by v l. Lemma 3. If ( ) n k (1 1 ) n k < 1, then there exists a tournament on n vertices with property k S k. Proof. Consider a random tournament T on n vertices. By a random tournament, we mean that for each 1 i < j n, with equal probability one of (v i, v j ) and (v j, v i ) is in E(T ). Union bound, Independence of events 1-7
8 Pick a set X V (T ) of size k. P r(x is dominated by v i ) = 1 k and P r(x is not dominated by v i) = 1 1 k by the union bound, P r(x is not dominated) = (1 1 k )n k P r( X V (G), X = k such that X is dominated) ( ) n (1 1 k k )n k As ( ) n k (1 1 ) n k < 1, it follows that there exists a tournament on n vertices with property k S k. Further Study Having proved the existence of a tournament T satisfying S k under appropriate conditions, it is interesting to consider the construction of T. Exciting non-trivial constructions of such tournaments are known [AS9] and are natural directions for deeper study. 6 Hypergraph Coloring A k-uniform hypergraph H is a pair (V, E) where V is the set of vertices and E ( V k) is the set of hyperedges. H is -colorable if its vertices can be colored with colors such that no edge is monochromatic. Lemma 4. For any k-uniform hypergraph H on n vertices, if E(H) < k 1, then H is -colorable. Proof. Consider a random coloring of vertices of an arbitrary k-uniform hypergraph H. Union bound, Independence of events Consider an edge e E(H). P r(e is monochromatic) = k 1-8
9 by the union bound and the fact that E(H) < k 1, P r( e E(H), such that e is monochromatic) < k k 1 = 1 Thus, the probability that all edges in H are non-monochromatic is non-zero. there exists a -coloring for H. Further Study Better bounds, in particular lower bounds are the natural next areas of study. References [AS9] Alon, N. and Spencer, J.H. The probabilistic method. Wiley, New York, 199. [GJ79] Garey, M.R. and Johnson, D.S. Computers and intractability: A guide to the theory of NP-completeness. W.H.Freeman and Company,
Lecture 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 informationMatchings in hypergraphs of large minimum degree
Matchings in hypergraphs of large minimum degree Daniela Kühn Deryk Osthus Abstract It is well known that every bipartite graph with vertex classes of size n whose minimum degree is at least n/2 contains
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 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 informationCSE 548: Analysis of Algorithms. Lectures 18, 19, 20 & 21 ( Randomized Algorithms & High Probability Bounds )
CSE 548: Analysis of Algorithms Lectures 18, 19, 20 & 21 ( Randomized Algorithms & High Probability Bounds ) Rezaul A. Chowdhury Department of Computer Science SUNY Stony Brook Fall 2012 Markov s Inequality
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 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 informationProbabilistic Proofs of Existence of Rare Events. Noga Alon
Probabilistic Proofs of Existence of Rare Events Noga Alon Department of Mathematics Sackler Faculty of Exact Sciences Tel Aviv University Ramat-Aviv, Tel Aviv 69978 ISRAEL 1. The Local Lemma In a typical
More informationRandomness and Computation
Randomness and Computation or, Randomized Algorithms Mary Cryan School of Informatics University of Edinburgh RC (2018/19) Lecture 11 slide 1 The Probabilistic Method The Probabilistic Method is a nonconstructive
More informationThe concentration of the chromatic number of random graphs
The concentration of the chromatic number of random graphs Noga Alon Michael Krivelevich Abstract We prove that for every constant δ > 0 the chromatic number of the random graph G(n, p) with p = n 1/2
More informationIntroduction to Randomized Algorithms: Quick Sort and Quick Selection
Chapter 14 Introduction to Randomized Algorithms: Quick Sort and Quick Selection CS 473: Fundamental Algorithms, Spring 2011 March 10, 2011 14.1 Introduction to Randomized Algorithms 14.2 Introduction
More informationTHE METHOD OF CONDITIONAL PROBABILITIES: DERANDOMIZING THE PROBABILISTIC METHOD
THE METHOD OF CONDITIONAL PROBABILITIES: DERANDOMIZING THE PROBABILISTIC METHOD JAMES ZHOU Abstract. We describe the probabilistic method as a nonconstructive way of proving the existence of combinatorial
More informationTree Decomposition of Graphs
Tree Decomposition of Graphs Raphael Yuster Department of Mathematics University of Haifa-ORANIM Tivon 36006, Israel. e-mail: raphy@math.tau.ac.il Abstract Let H be a tree on h 2 vertices. It is shown
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 informationGraph coloring, perfect graphs
Lecture 5 (05.04.2013) Graph coloring, perfect graphs Scribe: Tomasz Kociumaka Lecturer: Marcin Pilipczuk 1 Introduction to graph coloring Definition 1. Let G be a simple undirected graph and k a positive
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 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 informationarxiv: v1 [math.co] 2 Dec 2013
What is Ramsey-equivalent to a clique? Jacob Fox Andrey Grinshpun Anita Liebenau Yury Person Tibor Szabó arxiv:1312.0299v1 [math.co] 2 Dec 2013 November 4, 2018 Abstract A graph G is Ramsey for H if every
More informationContainment restrictions
Containment restrictions Tibor Szabó Extremal Combinatorics, FU Berlin, WiSe 207 8 In this chapter we switch from studying constraints on the set operation intersection, to constraints on the set relation
More informationSize and degree anti-ramsey numbers
Size and degree anti-ramsey numbers Noga Alon Abstract A copy of a graph H in an edge colored graph G is called rainbow if all edges of H have distinct colors. The size anti-ramsey number of H, denoted
More informationVertex colorings of graphs without short odd cycles
Vertex colorings of graphs without short odd cycles Andrzej Dudek and Reshma Ramadurai Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 1513, USA {adudek,rramadur}@andrew.cmu.edu
More informationA note on network reliability
A note on network reliability Noga Alon Institute for Advanced Study, Princeton, NJ 08540 and Department of Mathematics Tel Aviv University, Tel Aviv, Israel Let G = (V, E) be a loopless undirected multigraph,
More informationThe Probabilistic Method
The Probabilistic Method In Graph Theory Ehssan Khanmohammadi Department of Mathematics The Pennsylvania State University February 25, 2010 What do we mean by the probabilistic method? Why use this method?
More informationNotes 6 : First and second moment methods
Notes 6 : First and second moment methods Math 733-734: Theory of Probability Lecturer: Sebastien Roch References: [Roc, Sections 2.1-2.3]. Recall: THM 6.1 (Markov s inequality) Let X be a non-negative
More informationThe minimum G c cut problem
The minimum G c cut problem Abstract In this paper we define and study the G c -cut problem. Given a complete undirected graph G = (V ; E) with V = n, edge weighted by w(v i, v j ) 0 and an undirected
More informationAcyclic subgraphs with high chromatic number
Acyclic subgraphs with high chromatic number Safwat Nassar Raphael Yuster Abstract For an oriented graph G, let f(g) denote the maximum chromatic number of an acyclic subgraph of G. Let f(n) be the smallest
More informationAlgebraic Methods in Combinatorics
Algebraic Methods in Combinatorics Po-Shen Loh June 2009 1 Linear independence These problems both appeared in a course of Benny Sudakov at Princeton, but the links to Olympiad problems are due to Yufei
More informationIndependence numbers of locally sparse graphs and a Ramsey type problem
Independence numbers of locally sparse graphs and a Ramsey type problem Noga Alon Abstract Let G = (V, E) be a graph on n vertices with average degree t 1 in which for every vertex v V the induced subgraph
More informationOut-colourings of Digraphs
Out-colourings of Digraphs N. Alon J. Bang-Jensen S. Bessy July 13, 2017 Abstract We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an out-colouring.
More informationACO Comprehensive Exam October 14 and 15, 2013
1. Computability, Complexity and Algorithms (a) Let G be the complete graph on n vertices, and let c : V (G) V (G) [0, ) be a symmetric cost function. Consider the following closest point heuristic for
More informationOn shredders and vertex connectivity augmentation
On shredders and vertex connectivity augmentation Gilad Liberman The Open University of Israel giladliberman@gmail.com Zeev Nutov The Open University of Israel nutov@openu.ac.il Abstract We consider the
More informationDecomposing oriented graphs into transitive tournaments
Decomposing oriented graphs into transitive tournaments Raphael Yuster Department of Mathematics University of Haifa Haifa 39105, Israel Abstract For an oriented graph G with n vertices, let f(g) denote
More informationPacking and decomposition of graphs with trees
Packing and decomposition of graphs with trees Raphael Yuster Department of Mathematics University of Haifa-ORANIM Tivon 36006, Israel. e-mail: raphy@math.tau.ac.il Abstract Let H be a tree on h 2 vertices.
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 informationLower bounds for Ramsey numbers for complete bipartite and 3-uniform tripartite subgraphs
Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 7, no. 6, pp. 67 688 (03 DOI: 0.755/jgaa.003 Lower bounds for Ramsey numbers for complete bipartite and 3-uniform tripartite subgraphs
More informationPigeonhole Principle and Ramsey Theory
Pigeonhole Principle and Ramsey Theory The Pigeonhole Principle (PP) has often been termed as one of the most fundamental principles in combinatorics. The familiar statement is that if we have n pigeonholes
More informationInduced subgraphs with many repeated degrees
Induced subgraphs with many repeated degrees Yair Caro Raphael Yuster arxiv:1811.071v1 [math.co] 17 Nov 018 Abstract Erdős, Fajtlowicz and Staton asked for the least integer f(k such that every graph with
More informationLecture 5: Efficient PAC Learning. 1 Consistent Learning: a Bound on Sample Complexity
Universität zu Lübeck Institut für Theoretische Informatik Lecture notes on Knowledge-Based and Learning Systems by Maciej Liśkiewicz Lecture 5: Efficient PAC Learning 1 Consistent Learning: a Bound on
More information1 Introduction and Results
Discussiones Mathematicae Graph Theory 20 (2000 ) 7 79 SOME NEWS ABOUT THE INDEPENDENCE NUMBER OF A GRAPH Jochen Harant Department of Mathematics, Technical University of Ilmenau D-98684 Ilmenau, Germany
More informationQuasi-randomness is determined by the distribution of copies of a fixed graph in equicardinal large sets
Quasi-randomness is determined by the distribution of copies of a fixed graph in equicardinal large sets Raphael Yuster 1 Department of Mathematics, University of Haifa, Haifa, Israel raphy@math.haifa.ac.il
More information< k 2n. 2 1 (n 2). + (1 p) s) N (n < 1
List of Problems jacques@ucsd.edu Those question with a star next to them are considered slightly more challenging. Problems 9, 11, and 19 from the book The probabilistic method, by Alon and Spencer. Question
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 informationOn decomposing graphs of large minimum degree into locally irregular subgraphs
On decomposing graphs of large minimum degree into locally irregular subgraphs Jakub Przyby lo AGH University of Science and Technology al. A. Mickiewicza 0 0-059 Krakow, Poland jakubprz@agh.edu.pl Submitted:
More informationReachability-based matroid-restricted packing of arborescences
Egerváry Research Group on Combinatorial Optimization Technical reports TR-2016-19. Published by the Egerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.
More informationRainbow Hamilton cycles in uniform hypergraphs
Rainbow Hamilton cycles in uniform hypergraphs Andrzej Dude Department of Mathematics Western Michigan University Kalamazoo, MI andrzej.dude@wmich.edu Alan Frieze Department of Mathematical Sciences Carnegie
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 informationMath 261A Probabilistic Combinatorics Instructor: Sam Buss Fall 2015 Homework assignments
Math 261A Probabilistic Combinatorics Instructor: Sam Buss Fall 2015 Homework assignments Problems are mostly taken from the text, The Probabilistic Method (3rd edition) by N Alon and JH Spencer Please
More informationMore on NP and Reductions
Indian Institute of Information Technology Design and Manufacturing, Kancheepuram Chennai 600 127, India An Autonomous Institute under MHRD, Govt of India http://www.iiitdm.ac.in COM 501 Advanced Data
More informationTwo-coloring random hypergraphs
Two-coloring random hypergraphs Dimitris Achlioptas Jeong Han Kim Michael Krivelevich Prasad Tetali December 17, 1999 Technical Report MSR-TR-99-99 Microsoft Research Microsoft Corporation One Microsoft
More informationConstructions in Ramsey theory
Constructions in Ramsey theory Dhruv Mubayi Andrew Suk Abstract We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4-uniform
More informationRainbow Hamilton cycles in uniform hypergraphs
Rainbow Hamilton cycles in uniform hypergraphs Andrzej Dude Department of Mathematics Western Michigan University Kalamazoo, MI andrzej.dude@wmich.edu Alan Frieze Department of Mathematical Sciences Carnegie
More informationHARDNESS AND ALGORITHMS FOR RAINBOW CONNECTIVITY
HARDNESS AND ALGORITHMS FOR RAINBOW CONNECTIVITY SOURAV CHAKRABORTY 1 AND ELDAR FISCHER 1 AND ARIE MATSLIAH 2 AND RAPHAEL YUSTER 3 1 Department of Computer Science, Technion, Haifa 32000, Israel. E-mail
More informationTheorem 1.7 [Bayes' Law]: Assume that,,, are mutually disjoint events in the sample space s.t.. Then Pr( )
Theorem 1.7 [Bayes' Law]: Assume that,,, are mutually disjoint events in the sample space s.t.. Then Pr Pr = Pr Pr Pr() Pr Pr. We are given three coins and are told that two of the coins are fair and the
More informationNotes for Lecture 2. Statement of the PCP Theorem and Constraint Satisfaction
U.C. Berkeley Handout N2 CS294: PCP and Hardness of Approximation January 23, 2006 Professor Luca Trevisan Scribe: Luca Trevisan Notes for Lecture 2 These notes are based on my survey paper [5]. L.T. Statement
More informationMaximum union-free subfamilies
Maximum union-free subfamilies Jacob Fox Choongbum Lee Benny Sudakov Abstract An old problem of Moser asks: how large of a union-free subfamily does every family of m sets have? A family of sets is called
More informationPARTITIONING PROBLEMS IN DENSE HYPERGRAPHS
PARTITIONING PROBLEMS IN DENSE HYPERGRAPHS A. CZYGRINOW Abstract. We study the general partitioning problem and the discrepancy problem in dense hypergraphs. Using the regularity lemma [16] and its algorithmic
More informationDominating a family of graphs with small connected subgraphs
Dominating a family of graphs with small connected subgraphs Yair Caro Raphael Yuster Abstract Let F = {G 1,..., G t } be a family of n-vertex graphs defined on the same vertex-set V, and let k be a positive
More informationRamsey-type problem for an almost monochromatic K 4
Ramsey-type problem for an almost monochromatic K 4 Jacob Fox Benny Sudakov Abstract In this short note we prove that there is a constant c such that every k-edge-coloring of the complete graph K n with
More informationChapter 34: NP-Completeness
Graph Algorithms - Spring 2011 Set 17. Lecturer: Huilan Chang Reference: Cormen, Leiserson, Rivest, and Stein, Introduction to Algorithms, 2nd Edition, The MIT Press. Chapter 34: NP-Completeness 2. Polynomial-time
More informationComputational complexity theory
Computational complexity theory Introduction to computational complexity theory Complexity (computability) theory deals with two aspects: Algorithm s complexity. Problem s complexity. References S. Cook,
More informationConstructive bounds for a Ramsey-type problem
Constructive bounds for a Ramsey-type problem Noga Alon Michael Krivelevich Abstract For every fixed integers r, s satisfying r < s there exists some ɛ = ɛ(r, s > 0 for which we construct explicitly an
More informationProbabilistic Method. Benny Sudakov. Princeton University
Probabilistic Method Benny Sudakov Princeton University Rough outline The basic Probabilistic method can be described as follows: In order to prove the existence of a combinatorial structure with certain
More informationA An Overview of Complexity Theory for the Algorithm Designer
A An Overview of Complexity Theory for the Algorithm Designer A.1 Certificates and the class NP A decision problem is one whose answer is either yes or no. Two examples are: SAT: Given a Boolean formula
More informationHardness and Algorithms for Rainbow Connection
Hardness and Algorithms for Rainbow Connection Sourav Chakraborty Eldar Fischer Arie Matsliah Raphael Yuster Abstract An edge-colored graph G is rainbow connected if any two vertices are connected by a
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 informationProperly colored Hamilton cycles in edge colored complete graphs
Properly colored Hamilton cycles in edge colored complete graphs N. Alon G. Gutin Dedicated to the memory of Paul Erdős Abstract It is shown that for every ɛ > 0 and n > n 0 (ɛ), any complete graph K on
More informationThe Complexity of Optimization Problems
The Complexity of Optimization Problems Summary Lecture 1 - Complexity of algorithms and problems - Complexity classes: P and NP - Reducibility - Karp reducibility - Turing reducibility Uniform and logarithmic
More informationQuick Sort Notes , Spring 2010
Quick Sort Notes 18.310, Spring 2010 0.1 Randomized Median Finding In a previous lecture, we discussed the problem of finding the median of a list of m elements, or more generally the element of rank m.
More informationNordhaus-Gaddum Theorems for k-decompositions
Nordhaus-Gaddum Theorems for k-decompositions Western Michigan University October 12, 2011 A Motivating Problem Consider the following problem. An international round-robin sports tournament is held between
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 informationA = A U. U [n] P(A U ). n 1. 2 k(n k). k. k=1
Lecture I jacques@ucsd.edu Notation: Throughout, P denotes probability and E denotes expectation. Denote (X) (r) = X(X 1)... (X r + 1) and let G n,p denote the Erdős-Rényi model of random graphs. 10 Random
More information1.1 P, NP, and NP-complete
CSC5160: Combinatorial Optimization and Approximation Algorithms Topic: Introduction to NP-complete Problems Date: 11/01/2008 Lecturer: Lap Chi Lau Scribe: Jerry Jilin Le This lecture gives a general introduction
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 informationPreliminaries. Graphs. E : set of edges (arcs) (Undirected) Graph : (i, j) = (j, i) (edges) V = {1, 2, 3, 4, 5}, E = {(1, 3), (3, 2), (2, 4)}
Preliminaries Graphs G = (V, E), V : set of vertices E : set of edges (arcs) (Undirected) Graph : (i, j) = (j, i) (edges) 1 2 3 5 4 V = {1, 2, 3, 4, 5}, E = {(1, 3), (3, 2), (2, 4)} 1 Directed Graph (Digraph)
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 informationA necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees
A necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees Yoshimi Egawa Department of Mathematical Information Science, Tokyo University of
More informationLecture 5: Probabilistic tools and Applications II
T-79.7003: Graphs and Networks Fall 2013 Lecture 5: Probabilistic tools and Applications II Lecturer: Charalampos E. Tsourakakis Oct. 11, 2013 5.1 Overview In the first part of today s lecture we will
More informationInduced subgraphs of prescribed size
Induced subgraphs of prescribed size Noga Alon Michael Krivelevich Benny Sudakov Abstract A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let q(g denote the maximum
More informationan efficient procedure for the decision problem. We illustrate this phenomenon for the Satisfiability problem.
1 More on NP In this set of lecture notes, we examine the class NP in more detail. We give a characterization of NP which justifies the guess and verify paradigm, and study the complexity of solving search
More informationAn Introduction to Randomized algorithms
An Introduction to Randomized algorithms C.R. Subramanian The Institute of Mathematical Sciences, Chennai. Expository talk presented at the Research Promotion Workshop on Introduction to Geometric and
More informationThe number of Euler tours of random directed graphs
The number of Euler tours of random directed graphs Páidí Creed School of Mathematical Sciences Queen Mary, University of London United Kingdom P.Creed@qmul.ac.uk Mary Cryan School of Informatics University
More information1 Agenda. 2 History. 3 Probabilistically Checkable Proofs (PCPs). Lecture Notes Definitions. PCPs. Approximation Algorithms.
CS 221: Computational Complexity Prof. Salil Vadhan Lecture Notes 20 April 12, 2010 Scribe: Jonathan Pines 1 Agenda. PCPs. Approximation Algorithms. PCPs = Inapproximability. 2 History. First, some history
More informationAhlswede Khachatrian Theorems: Weighted, Infinite, and Hamming
Ahlswede Khachatrian Theorems: Weighted, Infinite, and Hamming Yuval Filmus April 4, 2017 Abstract The seminal complete intersection theorem of Ahlswede and Khachatrian gives the maximum cardinality of
More informationDecomposition of random graphs into complete bipartite graphs
Decomposition of random graphs into complete bipartite graphs Fan Chung Xing Peng Abstract We consider the problem of partitioning the edge set of a graph G into the minimum number τg) of edge-disjoint
More informationApproximate Hypergraph Coloring. 1 Introduction. Noga Alon 1 Pierre Kelsen 2 Sanjeev Mahajan 3 Hariharan Ramesh 4
Approximate Hypergraph Coloring Noga Alon 1 Pierre Kelsen 2 Sanjeev Mahajan Hariharan Ramesh 4 Abstract A coloring of a hypergraph is a mapping of vertices to colors such that no hyperedge is monochromatic.
More informationComputational complexity theory
Computational complexity theory Introduction to computational complexity theory Complexity (computability) theory deals with two aspects: Algorithm s complexity. Problem s complexity. References S. Cook,
More informationarxiv: v2 [math.co] 19 Jun 2018
arxiv:1705.06268v2 [math.co] 19 Jun 2018 On the Nonexistence of Some Generalized Folkman Numbers Xiaodong Xu Guangxi Academy of Sciences Nanning 530007, P.R. China xxdmaths@sina.com Meilian Liang School
More informationOn a Conjecture of Thomassen
On a Conjecture of Thomassen Michelle Delcourt Department of Mathematics University of Illinois Urbana, Illinois 61801, U.S.A. delcour2@illinois.edu Asaf Ferber Department of Mathematics Yale University,
More informationBlock Sensitivity of Minterm-Transitive Functions
Block Sensitivity of Minterm-Transitive Functions Andrew Drucker Abstract Boolean functions with a high degree of symmetry are interesting from a complexity theory perspective: extensive research has shown
More informationDiscrete mathematics , Fall Instructor: prof. János Pach
Discrete mathematics 016-017, Fall Instructor: prof. János Pach - covered material - Lecture 1. Counting problems To read: [Lov]: 1.. Sets, 1.3. Number of subsets, 1.5. Sequences, 1.6. Permutations, 1.7.
More informationCS Homework Chapter 6 ( 6.14 )
CS50 - Homework Chapter 6 ( 6. Dan Li, Xiaohui Kong, Hammad Ibqal and Ihsan A. Qazi Department of Computer Science, University of Pittsburgh, Pittsburgh, PA 560 Intelligent Systems Program, University
More informationRandom Variable. Pr(X = a) = Pr(s)
Random Variable Definition A random variable X on a sample space Ω is a real-valued function on Ω; that is, X : Ω R. A discrete random variable is a random variable that takes on only a finite or countably
More informationChromatic Ramsey number of acyclic hypergraphs
Chromatic Ramsey number of acyclic hypergraphs András Gyárfás Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Budapest, P.O. Box 127 Budapest, Hungary, H-1364 gyarfas@renyi.hu Alexander
More informationHypergraph Ramsey numbers
Hypergraph Ramsey numbers David Conlon Jacob Fox Benny Sudakov Abstract The Ramsey number r k (s, n is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains a red
More informationIndependent Transversals in r-partite Graphs
Independent Transversals in r-partite Graphs Raphael Yuster Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel Abstract Let G(r, n) denote
More informationOn a hypergraph matching problem
On a hypergraph matching problem Noga Alon Raphael Yuster Abstract Let H = (V, E) be an r-uniform hypergraph and let F 2 V. A matching M of H is (α, F)- perfect if for each F F, at least α F vertices of
More informationMinimum spanning tree
Minimum spanning tree Jean Cousty MorphoGraph and Imagery 2011 J. Cousty : MorphoGraph and Imagery 1/17 Outline of the lecture 1 Minimum spanning tree 2 Cut theorem for MST 3 Kruskal algorithm J. Cousty
More informationEdge-disjoint induced subgraphs with given minimum degree
Edge-disjoint induced subgraphs with given minimum degree Raphael Yuster Department of Mathematics University of Haifa Haifa 31905, Israel raphy@math.haifa.ac.il Submitted: Nov 9, 01; Accepted: Feb 5,
More information(x 1 +x 2 )(x 1 x 2 )+(x 2 +x 3 )(x 2 x 3 )+(x 3 +x 1 )(x 3 x 1 ).
CMPSCI611: Verifying Polynomial Identities Lecture 13 Here is a problem that has a polynomial-time randomized solution, but so far no poly-time deterministic solution. Let F be any field and let Q(x 1,...,
More information18.5 Crossings and incidences
18.5 Crossings and incidences 257 The celebrated theorem due to P. Turán (1941) states: if a graph G has n vertices and has no k-clique then it has at most (1 1/(k 1)) n 2 /2 edges (see Theorem 4.8). Its
More information