# Lecture 27: Nov 17, Hardness of approximation of MIN-VERTEX-COVER

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 E0 224 Computational Complexity Theory Indian Institute of Science Fall 2014 Department of Computer Science and Automation Lecture 27: Nov 17, 2014 Lecturer: Chandan Saha Scribe: Bibaswan & Sarath In the previous few lectures, we were introduced to the PCP theorem and were shown an alternative hardness of approximation view of the theorem. The PCP theorem implies the hardness of approximation results for many problems. In the last lecture we saw a strong hardness of approximation result for MAX-INDSET. In this lecture we will see a weaker result for MIN-VERTEX-COVER. We will also start the proof of a weaker PCP theorem Hardness of approximation of MIN-VERTEX-COVER Theorem There is a constant ρ > 1 s.t. ρ -approximation of MIN-VERTEX-COVER is NP-hard. Proof. We already know from PCP theorem that ρ-approximation of MAX-3SAT is NP-hard where ρ is some constant < 1. Now similar to the proof of hardness of approximation for MAX-INDSET given in the previous lecture, given an instance ϕ of MAX-3SAT, we construct a graph G ϕ as follows: Assuming ϕ has m clauses and n variables, where each clause has three literals, for each clause we will create a cluster of 7 vertices, each vertex represents one of the 7 possible satisfying assignments for that clause. Thus we will have a total of t = 7m vertices in G ϕ. Within each cluster, there will be edges between each of the 7 vertices among each other. For any two clusters, there will be an edge between the vertex in one cluster to a vertex in another cluster if and only if the two assignments the vertices represent are conflicting. As already stated in the previous lecture, val(ϕ) is the maximum fraction of clauses that can be satisfied by any assignment for ϕ iff G ϕ has a MAX-INDSET of size val(ϕ)m Now, if G ϕ has a MAX-INDSET of size val(ϕ)m or val(ϕ) t 7 then it has a MIN-VERTEX-COVER of size (t val(ϕ) t 7 ). Consider a YES instance of ϕ. In that case val(ϕ) = 1 and hence, MIN-VERTEX-COVER(G ϕ ) = t t 7 = 6t 7. Consider a NO instance of ϕ. In that case val(ϕ) < ρ and hence, MIN-VERTEX-COVER(G ϕ ) t ρt 7 = (7 ρ)t 7. Now, suppose we have a ρ -approximation algorithm for MIN-VERTEX-COVER. In that case to distinguish between the YES and NO instances of ϕ, we would want ρ 6t 7 (7 ρ)t 7 which implies ρ (7 ρ) 6. Thus we obtain a ρ for which ρ -approximation of MIN-VERTEX-COVER is NP-hard. Due to the unique games conjecture some people believe, the best efficient approximation algorithm for MIN-VERTEX- COVER is a 2-approximation [SKhot08] Proof of PCP theorem (weaker version) Theorem (Exponential size PCP system for NP)[ALM+] NP P CP (poly(n), 1) Proof idea: We already know that language QUADEQ is NP-Complete (given in assignment 1). If we can show that QUADEQ P CP (poly(n), 1) then we are done. To show this we show that there exists a PCP system for QUADEQ. 27-1

2 Lecture 27: Nov 17, The PCP (poly(n), 1)-verifier expects the proof to contain an encoded version of the original proof, which is a satisfying assignment for the input instance of QUADEQ. The verifier checks the such an encoded certificate by simple probabilistic tests. Note: The size of the proof can be exponential in size of the problem size. As long as the verifier makes only a constant number of queries to the proof and uses O(poly(n)) random bits, we are fine Some preliminaries: Definition Tensor Product of two vectors: Let x = (x 1, x 2,..., x n ) F n 2, y = (y 1, y 2,..., y n ) F n 2 be two vectors. Then x y is defined as x 1 y 1 x 1 y 2... x 1 y j... x 1 y n x 2 y 1 x 2 y 2... x 2 y j... x 2 y n x y = x i y 1 x i y 2... x i y j... x i y n x n y 1 x n y 2... x n y j... x n y n x y can be thought of as a vector of size n 2. Definition Walsh-Hadamard code: Let z {0, 1} n. Then W-H code of z is denoted by the 2 n length vector f z f z = (z.r) r {0,1} n n where (z.r) = z i r i Since r {0, 1} n, each r denotes a subset S [n] s.t. the i th element in [n] is selected in S if the bit r i = 1. Thus (z.r) = z i, S [n] defined by r. i S Proof construction: Now we explain the construction of the PCP proof system for QUADEQ. Let us consider an instance of QUADEQ as a model example: u 1 u 2 + u 3 u 5 + u 4 = 1 u 1 + u 3 u 4 = 0 u 2 + u 5 u 4 = 1 (27.1) The proof of satisfiablity of the above QUADEQ instance will consist of a vector u {0, 1} 5 denoting a satisfying assignment of the variables u 1, u 2,..., u 5. A polytime DTM can simply check whether each of the clauses is satisfied by putting the appropriate values to the variables. However this will require the verifier to read the entire certificate. But we want to read only a constant number of locations of the proof and verify the proof with high probability. So in our case the prover has to supply the verifier a special encoded version of vector u such that the verifier with p(n) random coins can read only a constant number of locations of the encoded proof and can decide with high probability if the proof is correct or not as required by the PCP theorem. So we will show that such an encoded proof does exist and demonstrate how the verifier will check the proof.

3 Lecture 27: Nov 17, PCP certificate description: The certificate π will consist of two parts g 1 and g 2, where g 1 = f u {0, 1} 2n g 2 = f u u {0, 1} 2n2. and f u f u u g 1 g 2 That is, the proof essentially consists of Walsh-Hadamard encoding of the string u and the string u u (note that the tensor product can be interpreted as just a vector instead of a matrix). Thus the total length of the proof will be 2 n + 2 n2. The main challenge in proving this weaker version of PCP theorem is to verify the proof using constant number of queries. There are two parts in verifying the given proof. First, the verifier needs to verify that the first part of given proof is indeed a Walsh-Hadamard encoding of some string u and second part of the proof is an encoding of u u (of course, with constant number of queries to the proof). After this, the verifier needs to verify that u is a satisfying assignment of the given QUADEQ instance. First, we describe how we can do the second part. That is, we assume that the given proof is valid (i.e., we assume that the first part of the given proof is a Walsh-Hadamard encoding of some string u and the second part is an encoding of u u), and we describe a PCP verifier. Observe that the Walsh-Hadamard encoding of u u essentially consists of sums of the form S u iu j where the set S is a subset of [n] [n]. Also, note that the LHS of the equations in the QUADEQ instance are of the same form. In other words, the LHS of equations are present at various locations of Walsh-Hadamard encoding of u u. However, we cannot read each one of those locations in the given proof and verify with the RHS of equations, since we are allowed to read constantly many locations in the proof. More interestingly, observe that any linear combination (over F 2 ) of LHS of the given set of equations is again in the form of LHS of a quadratic equation, and hence will be present at some location in the Walsh-Hadamard encoding of u u. This gives an idea as to how to read the proof using constantly many queries. For the sake of explanation, let us consider our QUADEQ instance Let w = u 1u 2 + u 3 u 5 + u 4 u 1 + u 3 u 4 (27.2) u 2 + u 5 u 4 and b = (27.3) Then the verifier need to check whether w = b by making constant number of queries to the proof. Let m denote the number of equations in the QUADEQ instance. First, we show the following lemma: Lemma (Random Subsum Lemma) If w b then P r R {0,1} m(w.r = b.r) = 1 2 (27.4) Proof. Observe that P r R {0,1} m(w.r = b.r) = P r R {0,1} m((w + b).r = 0)

4 Lecture 27: Nov 17, Now, consider w + b. If w b, then w + b 0. Let v = w + b. Then (w + b).r = m v i r i In the above expression, the RHS will be a sum (over F 2 ) of r i s depending on the non-zero positions of the string w + b. Let the above sum be r i1 + + r ik (where k is the number of terms in the sum). Now, the total number of m-length strings is 2 m. We need to find the fraction of strings (i.e., the probability) that makes the above expression 0. Note that, the degrees of freedom in choosing the r i s that make the above sum zero is (m 1), i.e., if we fix (k 1) number of terms in the sum, then the k th term automatically gets fixed to make the sum 0. Therefore, P((w + b).r = 0) = 2m 1 2 m = 1 2 Now, to check whether w = b, the verifier picks a random string r {0, 1} m, and computes w.r and b.r. Since w.r is again in the form of LHS of a quadratic equation, it queries the proof corresponding to the location w.r (note that w.r will be present in some location in the Walsh-Hadamard encoding of u u). If the given system of equations is satisfied by u, then clearly w.r = b.r and the verifier accepts. If not, then we have w b and by the above lemma, the verifier rejects with probability 1/2. Remark: We can reduce the probability of making an error by increasing the number of queries (but constantly many queries) to the proof by using independently picked random strings r. Thus, assuming that the given proof is indeed valid, we have a PCP verifier for QUADEQ that uses polynomially many random bits and constant number of queries to the proof. The remaining task is to verify that the proof is indeed valid. That is, we need to do the following: 1. First, the verifier needs to ensure that g 1 = f u for some u {0, 1} n and g 2 = f w for some w {0, 1} n2. 2. Second, the verifier needs to ensure that w = u u. Towards this task, we make the following definitions: Definition (Linear functions on q-length strings) A function h : {0, 1} q {0, 1} is a linear function if for every x, y F q 2, h(x + y) = h(x) + h(y), where addition (+) is over F 2. Definition (Equivalent definition of a linear function on q-length strings) A function h : {0, 1} q {0, 1} is a linear function if S [q] such that for every x = (x 1,..., x n ) F q 2, h(x) = i S x i (27.5) First, we show the following claim: Claim The above definitions of linear functions are equivalent

5 Lecture 27: Nov 17, Proof. It is straightforward to show that Definition 27.7 Definition By Definition 27.7, there exists S [q] such that h(x) = i S x i (27.6) for all x F q 2. Therefore, for all x, y Fq 2, we have h(x + y) = i S(x i + y i ) = i S x i + i S y i = h(x) + h(y) (27.7) where the last equality makes use of Definition Note that this is precisely Definition To show the converse, we represent a vector in terms of standard basis vectors, i.e., we write x as x = x 1 (1, 0,..., 0) T + x 2 (0, 1, 0,..., 0) T + + x q (0, 0,..., 0, 1) T (27.8) Then, by using Definition 27.6, for any x F q 2, we have h(x) = q x i h(e i ) (27.9) where e i is the standard basis vector with a 1 at i th position. Note that the above is true since, if x i = 0 then h(x i e i ) = 0, and if x i = 1 then h(x i e i ) = h(e i ). Observe that the above expression can be written as a sum of a subset of x i s depending on the value of h( ) for the basis vector e i s. That is, there exists a set S [q] such h(x) = i S x i (27.10) which is precisely Definition Having defined linear functions, we show the following: Claim Walsh-Hadamard codewords of length 2 q are precisely linear functions on q-length strings Remark: We say that a 2 q length string is a Walsh-Hadamard codeword if it is a Walsh-Hadamard encoding of some q-length string. Proof. For a fixed q-length string (say v), the Walsh-Hadamard codeword is obtained by taking the dot product of v with all q-length strings. That is, the codeword is given by ( q ((v.r)) r {0,1} q = = ( i S v r i v i r i ) ) r {0,1} q r {0,1} q where S v [q] depending on v. But, using Definition 27.7 of linear functions, we see that the sum in the above expression is precisely a linear function on q-length string r that is defined by S v. Thus, we conclude that Walsh- Hadamard codewrords of length 2 q are precisely linear functions on q-length strings.

6 Lecture 27: Nov 17, Thus, we have a nice characterization of Walsh-Hadamard codewords of length 2 q in terms of linear functions on q-length strings. Now, let us recall our PCP verifier s objective. The first step is to check whether the first part of the proof is a valid Walsh-Hadamard codeword, that is, to check if g 1 = f u for some u {0, 1} q. But, by Claim 27.9, it suffices to check whether the given 2 q -length string is a truth table for a linear function on q-length strings. But, it is not immediately clear how to do this by using constant number of queries to the proof. Let us first try a naive strategy. Suppose we pick two strings x, y {0, 1} q randomly and query the three locations in the proof corresponding to h(x), h(y) and h(x + y). We accept the proof if we find that h(x) + h(y) = h(x + y) and reject if not. Now, if h is a linear function, we accept with probability 1. However, if h is not linear, we need not reject with high probability. To see this, suppose that h is indeed a linear function and we define another function h by just flipping a single bit of h. Then, it is clear that h is no longer a linear function. But, h satisfies the condition in Definition 27.6 for every pair of x and y if none of x and y correspond to the bit that is flipped. Therefore, for randomly picked strings x and y, the probability that h (x) + h (y) h (x + y) is not high. Hence, this strategy does not server our purpose. However, it turns out that we need not check whether the given 2 q length string is exactly the truth table of a linear function, but it suffices to check whether the given string is close to a linear function. In this case, we can pretend the given proof as a slightly corrupted version of a Walsh-Hadamard codeword, and we can do local decoding of this codeword to verify whether u is a satisfying assignment for the QUADEQ instance. Note that even if the given proof is close to a linear function, we can make use of our random subsum lemma. Also, it turns out that one can test whether the given proof is close to a linear function by using constant number of queries to the proof. We will study the details in the next lecture. References [SKhot08] [ALM+] [AB09] KHOT, SUBHASH and ODED REGEV Vertex cover might be hard to approximate to within 2 - ɛ. Journal of Computer and System Sciences 74.3 (2008): ARORA, SANJEEV, CARSTEN LUND, RAJEEV MOTWANI, MADHU SUDAN and MARIO SZEGEDY Proof verification and the hardness of approximation problems. Journal of the ACM (JACM) 45, no. 3 (1998): ARORA, SANJEEV, and BARAK, BOAZ. Computational Complexity: A Modern Approach, Cambridge University Press, 2009.

### 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

### On the Unique Games Conjecture

On the Unique Games Conjecture Antonios Angelakis National Technical University of Athens June 16, 2015 Antonios Angelakis (NTUA) Theory of Computation June 16, 2015 1 / 20 Overview 1 Introduction 2 Preliminary

### (67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7

(67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition

### Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

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

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

### Lecture 3: Linear Programming Relaxations and Rounding

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

### Lecture 3: Finding integer solutions to systems of linear equations

Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture

### 1 The Line vs Point Test

6.875 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 5: Low Degree Testing Lecturer: Dana Moshkovitz Scribe: Gregory Minton and Dana Moshkovitz Having seen a probabilistic verifier for linearity

### Linearity and Group Homomorphism Testing / Testing Hadamard Codes

Title: Linearity and Group Homomorphism Testing / Testing Hadamard Codes Name: Sofya Raskhodnikova 1, Ronitt Rubinfeld 2 Affil./Addr. 1: Pennsylvania State University Affil./Addr. 2: Keywords: SumOriWork:

### 1 Formulating The Low Degree Testing Problem

6.895 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 5: Linearity Testing Lecturer: Dana Moshkovitz Scribe: Gregory Minton and Dana Moshkovitz In the last lecture, we proved a weak PCP Theorem,

### 1. Nondeterministically guess a solution (called a certificate) 2. Check whether the solution solves the problem (called verification)

Some N P problems Computer scientists have studied many N P problems, that is, problems that can be solved nondeterministically in polynomial time. Traditionally complexity question are studied as languages:

### Lecture 17 : Equivalence and Order Relations DRAFT

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

### Testing Low-Degree Polynomials over

Testing Low-Degree Polynomials over Noga Alon, Tali Kaufman, Michael Krivelevich, Simon Litsyn, and Dana Ron Department of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel and Institute for Advanced

### 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

### 2.1 Complexity Classes

15-859(M): Randomized Algorithms Lecturer: Shuchi Chawla Topic: Complexity classes, Identity checking Date: September 15, 2004 Scribe: Andrew Gilpin 2.1 Complexity Classes In this lecture we will look

### MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

### 3. Equivalence Relations. Discussion

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

### 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

### Definition 11.1. Given a graph G on n vertices, we define the following quantities:

Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define

### (x + a) n = x n + a Z n [x]. Proof. If n is prime then the map

22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find

### Solution to Homework 2

Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### Introduction to Algorithms. Part 3: P, NP Hard Problems

Introduction to Algorithms Part 3: P, NP Hard Problems 1) Polynomial Time: P and NP 2) NP-Completeness 3) Dealing with Hard Problems 4) Lower Bounds 5) Books c Wayne Goddard, Clemson University, 2004 Chapter

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### Lecture 16 : Relations and Functions DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence

### Change of Continuous Random Variable

Change of Continuous Random Variable All you are responsible for from this lecture is how to implement the Engineer s Way (see page 4) to compute how the probability density function changes when we make

### Discuss the size of the instance for the minimum spanning tree problem.

3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can

### Notes on NP Completeness

Notes on NP Completeness Rich Schwartz November 10, 2013 1 Overview Here are some notes which I wrote to try to understand what NP completeness means. Most of these notes are taken from Appendix B in Douglas

### 4 Domain Relational Calculus

4 Domain Relational Calculus We now present two relational calculi that we will compare to RA. First, what is the difference between an algebra and a calculus? The usual story is that the algebra RA is

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

### CSC 373: Algorithm Design and Analysis Lecture 16

CSC 373: Algorithm Design and Analysis Lecture 16 Allan Borodin February 25, 2013 Some materials are from Stephen Cook s IIT talk and Keven Wayne s slides. 1 / 17 Announcements and Outline Announcements

### CMPSCI611: Approximating MAX-CUT Lecture 20

CMPSCI611: Approximating MAX-CUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NP-hard problems. Today we consider MAX-CUT, which we proved to

### INTRODUCTION TO CODING THEORY: BASIC CODES AND SHANNON S THEOREM

INTRODUCTION TO CODING THEORY: BASIC CODES AND SHANNON S THEOREM SIDDHARTHA BISWAS Abstract. Coding theory originated in the late 1940 s and took its roots in engineering. However, it has developed and

### Lecture 5 - CPA security, Pseudorandom functions

Lecture 5 - CPA security, Pseudorandom functions Boaz Barak October 2, 2007 Reading Pages 82 93 and 221 225 of KL (sections 3.5, 3.6.1, 3.6.2 and 6.5). See also Goldreich (Vol I) for proof of PRF construction.

### NP-Completeness and Cook s Theorem

NP-Completeness and Cook s Theorem Lecture notes for COM3412 Logic and Computation 15th January 2002 1 NP decision problems The decision problem D L for a formal language L Σ is the computational task:

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### COMS4236: Introduction to Computational Complexity. Summer 2014

COMS4236: Introduction to Computational Complexity Summer 2014 Mihalis Yannakakis Lecture 17 Outline conp NP conp Factoring Total NP Search Problems Class conp Definition of NP is nonsymmetric with respect

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

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

### Lecture 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

### 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,

### 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

### Permutation Betting Markets: Singleton Betting with Extra Information

Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu

### P versus NP, and More

1 P versus NP, and More Great Ideas in Theoretical Computer Science Saarland University, Summer 2014 If you have tried to solve a crossword puzzle, you know that it is much harder to solve it than to verify

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### RANDOMIZATION IN APPROXIMATION AND ONLINE ALGORITHMS. Manos Thanos Randomized Algorithms NTUA

RANDOMIZATION IN APPROXIMATION AND ONLINE ALGORITHMS 1 Manos Thanos Randomized Algorithms NTUA RANDOMIZED ROUNDING Linear Programming problems are problems for the optimization of a linear objective function,

### 160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

### Linear Codes. Chapter 3. 3.1 Basics

Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length

### 2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system

1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables

### 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.,

### 5.7 Literal Equations

5.7 Literal Equations Now that we have learned to solve a variety of different equations (linear equations in chapter 2, polynomial equations in chapter 4, and rational equations in the last section) we

### Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

### ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS

ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS M. AXTELL, J. COYKENDALL, AND J. STICKLES Abstract. We recall several results of zero divisor graphs of commutative rings. We

### Combinatorial PCPs with ecient veriers

Combinatorial PCPs with ecient veriers Or Meir Abstract The PCP theorem asserts the existence of proofs that can be veried by a verier that reads only a very small part of the proof. The theorem was originally

### Diagonalization. Ahto Buldas. Lecture 3 of Complexity Theory October 8, 2009. Slides based on S.Aurora, B.Barak. Complexity Theory: A Modern Approach.

Diagonalization Slides based on S.Aurora, B.Barak. Complexity Theory: A Modern Approach. Ahto Buldas Ahto.Buldas@ut.ee Background One basic goal in complexity theory is to separate interesting complexity

### THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan

### Quadratic forms Cochran s theorem, degrees of freedom, and all that

Quadratic forms Cochran s theorem, degrees of freedom, and all that Dr. Frank Wood Frank Wood, fwood@stat.columbia.edu Linear Regression Models Lecture 1, Slide 1 Why We Care Cochran s theorem tells us

### We now explore a third method of proof: proof by contradiction.

CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement

### v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.

3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with

### 1 Scalars, Vectors and Tensors

DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH350 Classical Physics Handout 1 8.8.2009 1 Scalars, Vectors and Tensors In physics, we are interested in obtaining laws (in the form of mathematical

### Notes on Complexity Theory Last updated: August, 2011. Lecture 1

Notes on Complexity Theory Last updated: August, 2011 Jonathan Katz Lecture 1 1 Turing Machines I assume that most students have encountered Turing machines before. (Students who have not may want to look

### Victor Shoup Avi Rubin. fshoup,rubing@bellcore.com. Abstract

Session Key Distribution Using Smart Cards Victor Shoup Avi Rubin Bellcore, 445 South St., Morristown, NJ 07960 fshoup,rubing@bellcore.com Abstract In this paper, we investigate a method by which smart

### MATH 131 SOLUTION SET, WEEK 12

MATH 131 SOLUTION SET, WEEK 12 ARPON RAKSIT AND ALEKSANDAR MAKELOV 1. Normalisers We first claim H N G (H). Let h H. Since H is a subgroup, for all k H we have hkh 1 H and h 1 kh H. Since h(h 1 kh)h 1

### 88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small

### 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

### 3.4 Complex Zeros and the Fundamental Theorem of Algebra

86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and

### Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

### Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:

Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start

### OHJ-2306 Introduction to Theoretical Computer Science, Fall 2012 8.11.2012

276 The P vs. NP problem is a major unsolved problem in computer science It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a \$ 1,000,000 prize for the

### 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

### Page 1. CSCE 310J Data Structures & Algorithms. CSCE 310J Data Structures & Algorithms. P, NP, and NP-Complete. Polynomial-Time Algorithms

CSCE 310J Data Structures & Algorithms P, NP, and NP-Complete Dr. Steve Goddard goddard@cse.unl.edu CSCE 310J Data Structures & Algorithms Giving credit where credit is due:» Most of the lecture notes

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

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

### Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2

Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the

### 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

### . 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9

Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a

### NP-complete? NP-hard? Some Foundations of Complexity. Prof. Sven Hartmann Clausthal University of Technology Department of Informatics

NP-complete? NP-hard? Some Foundations of Complexity Prof. Sven Hartmann Clausthal University of Technology Department of Informatics Tractability of Problems Some problems are undecidable: no computer

### Reading 13 : Finite State Automata and Regular Expressions

CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

### Polarization codes and the rate of polarization

Polarization codes and the rate of polarization Erdal Arıkan, Emre Telatar Bilkent U., EPFL Sept 10, 2008 Channel Polarization Given a binary input DMC W, i.i.d. uniformly distributed inputs (X 1,...,

### Notes 11: List Decoding Folded Reed-Solomon Codes

Introduction to Coding Theory CMU: Spring 2010 Notes 11: List Decoding Folded Reed-Solomon Codes April 2010 Lecturer: Venkatesan Guruswami Scribe: Venkatesan Guruswami At the end of the previous notes,

### Lecture 2: Complexity Theory Review and Interactive Proofs

600.641 Special Topics in Theoretical Cryptography January 23, 2007 Lecture 2: Complexity Theory Review and Interactive Proofs Instructor: Susan Hohenberger Scribe: Karyn Benson 1 Introduction to Cryptography

### Near Optimal Solutions

Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.

### NP-Completeness I. Lecture 19. 19.1 Overview. 19.2 Introduction: Reduction and Expressiveness

Lecture 19 NP-Completeness I 19.1 Overview In the past few lectures we have looked at increasingly more expressive problems that we were able to solve using efficient algorithms. In this lecture we introduce

### Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### Tiers, Preference Similarity, and the Limits on Stable Partners

Tiers, Preference Similarity, and the Limits on Stable Partners KANDORI, Michihiro, KOJIMA, Fuhito, and YASUDA, Yosuke February 7, 2010 Preliminary and incomplete. Do not circulate. Abstract We consider

### Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits

Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique

### CSE 135: Introduction to Theory of Computation Decidability and Recognizability

CSE 135: Introduction to Theory of Computation Decidability and Recognizability Sungjin Im University of California, Merced 04-28, 30-2014 High-Level Descriptions of Computation Instead of giving a Turing

### The last three chapters introduced three major proof techniques: direct,

CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements

### DIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents

DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the Cauchy-Riemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition

### Lecture 4: AC 0 lower bounds and pseudorandomness

Lecture 4: AC 0 lower bounds and pseudorandomness Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: Jason Perry and Brian Garnett In this lecture,

The Hadamard Product Elizabeth Million April 12, 2007 1 Introduction and Basic Results As inexperienced mathematicians we may have once thought that the natural definition for matrix multiplication would

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product

Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.

### R u t c o r Research R e p o r t. Boolean functions with a simple certificate for CNF complexity. Ondřej Čepeka Petr Kučera b Petr Savický c

R u t c o r Research R e p o r t Boolean functions with a simple certificate for CNF complexity Ondřej Čepeka Petr Kučera b Petr Savický c RRR 2-2010, January 24, 2010 RUTCOR Rutgers Center for Operations

### Vector Spaces II: Finite Dimensional Linear Algebra 1

John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

### Tutorial 8. NP-Complete Problems

Tutorial 8 NP-Complete Problems Decision Problem Statement of a decision problem Part 1: instance description defining the input Part 2: question stating the actual yesor-no question A decision problem

### SCORE SETS IN ORIENTED GRAPHS

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