An IPS for TQBF Intro to Approximability
|
|
- Oswald Brooks
- 5 years ago
- Views:
Transcription
1 An IPS for TQBF Intro to Approximability
2 Outline Proof that TQBF (the language of true (valid) quantified Boolean formulas) is in the class IP Introduction to approximation algorithms for NP optimization problems
3 Recall from last time Simplification Lemma: Any TQBF instance ϕ can be converted in polynomial time into a simple TQBF instance ϕ such that ϕ is valid if and only if ϕ is Arithmetization Lemma: A simple TQBF instance ϕ is valid iff v(aϕ) > 0, where Aϕ is the arithmetization of ϕ
4 Recall from last time Simplification Lemma: Any TQBF instance ϕ can be converted in polynomial time into a simple TQBF instance ϕ such that ϕ is valid if and only if ϕ is Arithmetization Lemma: A simple TQBF instance ϕ is valid iff v(aϕ) > 0, where Aϕ is the arithmetization of ϕ
5 Recall from last time Simplification Lemma: Any TQBF instance ϕ can be converted in polynomial time into a simple TQBF instance ϕ such that ϕ is valid if and only if ϕ is Arithmetization Lemma: A simple TQBF instance ϕ is valid iff v(aϕ) > 0, where Aϕ is the arithmetization of ϕ Computing Mod a Prime Lemma: v(aϕ) > 0 iff there is a prime p in [2 A ϕ, 2 3 A ϕ ] such that v(aϕ) 0 mod p
6 An IPS for TQBF Input: a QBF ϕ; let ϕ have m quantifiers and without loss of generality assume that ϕ is simple Arithmetize ϕ to obtain Aϕ ; let A0 = Aϕ // ϕ is valid iff v(aϕ) > 0 Guess p in the range in [2 A ϕ, 2 3 A ϕ ]; reject if p is not prime Guess a 0 in the range [1,,p-1] // check that v(a0) = a 0 mod p
7 An IPS for TQBF, continued for i from 1 to m do // m is # quantifiers of ϕ 1. let Ai-1 = ci + ci (Ou Ai(u)), where Ou is the leftmost or symbol 2. guess a polynomial αi(u) of degree at most 2 Aϕ 3. check that ci + ci (Ou αi(u)) = ai-1 mod p; if not, reject 4. choose ri randomly and uniformly in the range [0... p-1] 5. let ai = αi(ri) mod p 6. let Ai be the expression Ai(ri) check that v(am) = am mod p; if not, reject and otherwise accept
8 Proof of correctness (outline) All steps of the algorithm run in polynomial time (easy)
9 Proof of correctness (outline) A strategy S(ϕ) is a choice of αi(u) at step 2 of each iteration i of the protocol Claim 1: If v(aϕ) = a 0 mod p then for some strategy S(ϕ), the IPS accepts with probability 1 Claim 2: If v(aϕ) a 0 mod p then for all strategies S(ϕ), the IPS accepts with probability at most 1- (1-2n/2 n ) n Correctness follows, since ϕ is valid iff v(aϕ) 0 mod p, for some prime p in the range specified in the IPS
10 Proof of correctness (outline) Claim 1: If v(aϕ) = a 0 mod p then for some strategy S(ϕ), the IPS accepts with probability 1
11 Proof of correctness (outline) Claim 1: If v(aϕ) = a 0 mod p then for some strategy S(ϕ), the IPS accepts with probability 1 Proof ideas: The strategy S(ϕ) simply returns the polynomial αi(u) that is equal to Ai(u) (mod p) Use the following (proved last time): Low degree lemma: Let ϕ be simple and let Aϕ = Qx Ai (x). Then the degree of polynomial Ai(x) is at most 2 α
12 Proof of correctness (outline) Claim 2: If v(aϕ) a 0 mod p then for all strategies S(ϕ), the IPS accepts with probability at most (1-2n/2 n ) n
13 Proof of correctness (outline) Claim 2: If v(aϕ) a 0 mod p then for all strategies S(ϕ), the IPS accepts with probability at most 1 - (1-2n/2 n ) n Proof ideas: Fix any strategy S = S(ϕ). For each i between 0 and m, let Ei(ϕ,S) be the event that v(ai) ai mod p, or that the protocol rejects before round i+1 is reached (or, if i=m, that the loop terminates) Show by induction that Prob[Ei(ϕ,S)] (1-2n/2 n ) i, where the probability is taken over the choice of ri
14 Summary We ve shown an interactive proof system that accepts TQBF Thus, IP = PSPACE: for any language L in PSPACE a prover can convince a coin-flipping verifier in polynomial time that a yes-instance x is indeed in L, and can fool the verifier with low probability when x is a no-instance of L
15 Summary The IP = PSPACE result raises other questions: If all of PSPACE can be proved (with low error probability) to a computationally limited coinflipping verifier, can we limit the verifier further when proving membership in an NP language with low error probability? We ll come back to this question after a detour to approximation algorithms for NP-hard problems
16 Approximation algorithms
17 Approximation algorithms Motivating example: Max SAT: Given a Boolean formula ϕ in conjunctive normal form, find the maximum number of clauses that can be simultaneously satisfied This is an optimization version of the classical SAT decision problem Exercise: suggest simple algorithms that aim to satisfy as many clauses as possible
18 Approximation algorithms for Max SAT Greedy algorithm: assign a truth value to the variables in turn, choosing a value for variable xi that satisfies at least half of the not-yet-satisfied clauses in which xi appears Even simpler: either the all-true or all-false assignment will satisfies at least half of the clauses (why?)
19 Approximation algorithms for Min Vertex Cover Given an undirected graph G = (V,E), find a minimum vertex cover for G. A vertex cover is a set of nodes that are incident on all edges of G Exercise: suggest simple algorithms that aim to find the smallest possible vertex cover
20 Approximation algorithms for Min Vertex Cover Greedy algorithm: Start with S = Repeat until the graph has no edges: Pick the vertex v that is incident on the most edges (breaking ties arbitrarily), add v to S and remove its incident edges from the graph
21 Approximation algorithms for Min Vertex Cover From The Nature of Computation by Chris Moore
22 Approximation algorithms for Min Vertex Cover Conservative algorithm: Start with S = Repeat until the graph has no edges: Pick any edge e1 of E, and add both its endpoints to S. Delete these two vertices from the graph as well as all edges adjacent to them This algorithm finds a vertex cover of size at most twice the minimum why?
23 Optimization problems An optimization problem has the following properties: Corresponding to an instance I of the problem is a set of solutions SI. Corresponding to each solution δ SI is a value, which is a positive rational number. is either a maximization problem, in which case we want to find the solution with maximum value, or a minimization problem. Let Opt(I) be the value of the optimal solution to I.
24 Optimization problems An algorithm A is an approximation algorithm for if given an instance I of, A computes a solution of I. Let A(I) denote the value of the solution computed by A on instance I. Let RA(I) = max{ A(I)/Opt(I), Opt(I)/A(I) }. Note that 1 RA(I) and the closer RA(I) is to 1, the better A performs on input I. Algorithm A has approximation ratio RA if RA RA(I) for all instances I of.
25 Next Class More on approximation algorithms How to show hardness of approximation Reading: Arora-Barak, 18.1, 18.2
W4231: Analysis of Algorithms
W4231: Analysis of Algorithms 11/23/99 NP-completeness of 3SAT, Minimum Vertex Cover, Maximum Independent Set, Boolean Formulae A Boolean formula is an expression that we can build starting from Boolean
More informationIntroduction to Algorithms / Algorithms I Lecturer: Michael Dinitz Topic: Approximation algorithms Date: 11/18/14
600.363 Introduction to Algorithms / 600.463 Algorithms I Lecturer: Michael Dinitz Topic: Approximation algorithms Date: 11/18/14 23.1 Introduction We spent last week proving that for certain problems,
More information/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Approximation algorithms Date: 11/27/18
601.433/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Approximation algorithms Date: 11/27/18 22.1 Introduction We spent the last two lectures proving that for certain problems, we can
More informationCS154, Lecture 18: PCPs, Hardness of Approximation, Approximation-Preserving Reductions, Interactive Proofs, Zero-Knowledge, Cold Fusion, Peace in
CS154, Lecture 18: PCPs, Hardness of Approximation, Approximation-Preserving Reductions, Interactive Proofs, Zero-Knowledge, Cold Fusion, Peace in the Middle East There are thousands of NP-complete problems
More information8 NP-complete problem Hard problems: demo
Ch8 NPC Millennium Prize Problems http://en.wikipedia.org/wiki/millennium_prize_problems 8 NP-complete problem Hard problems: demo NP-hard (Non-deterministic Polynomial-time hard), in computational complexity
More informationNP-complete Reductions
NP-complete Reductions 1. Prove that 3SAT P DOUBLE-SAT, i.e., show DOUBLE-SAT is NP-complete by reduction from 3SAT. The 3-SAT problem consists of a conjunction of clauses over n Boolean variables, where
More information1. Suppose you are given a magic black box that somehow answers the following decision problem in polynomial time:
1. Suppose you are given a magic black box that somehow answers the following decision problem in polynomial time: Input: A CNF formula ϕ with n variables x 1, x 2,..., x n. Output: True if there is an
More informationVertex Cover Approximations
CS124 Lecture 20 Heuristics can be useful in practice, but sometimes we would like to have guarantees. Approximation algorithms give guarantees. It is worth keeping in mind that sometimes approximation
More information9.1 Cook-Levin Theorem
CS787: Advanced Algorithms Scribe: Shijin Kong and David Malec Lecturer: Shuchi Chawla Topic: NP-Completeness, Approximation Algorithms Date: 10/1/2007 As we ve already seen in the preceding lecture, two
More informationWhere Can We Draw The Line?
Where Can We Draw The Line? On the Hardness of Satisfiability Problems Complexity 1 Introduction Objectives: To show variants of SAT and check if they are NP-hard Overview: Known results 2SAT Max2SAT Complexity
More informationProve, where is known to be NP-complete. The following problems are NP-Complete:
CMPSCI 601: Recall From Last Time Lecture 21 To prove is NP-complete: Prove NP. Prove, where is known to be NP-complete. The following problems are NP-Complete: SAT (Cook-Levin Theorem) 3-SAT 3-COLOR CLIQUE
More informationComputability Theory
CS:4330 Theory of Computation Spring 2018 Computability Theory Other NP-Complete Problems Haniel Barbosa Readings for this lecture Chapter 7 of [Sipser 1996], 3rd edition. Sections 7.4 and 7.5. The 3SAT
More informationval(y, I) α (9.0.2) α (9.0.3)
CS787: Advanced Algorithms Lecture 9: Approximation Algorithms In this lecture we will discuss some NP-complete optimization problems and give algorithms for solving them that produce a nearly optimal,
More informationExample of a Demonstration that a Problem is NP-Complete by reduction from CNF-SAT
20170926 CNF-SAT: CNF-SAT is a problem in NP, defined as follows: Let E be a Boolean expression with m clauses and n literals (literals = variables, possibly negated), in which - each clause contains only
More informationNP Completeness. Andreas Klappenecker [partially based on slides by Jennifer Welch]
NP Completeness Andreas Klappenecker [partially based on slides by Jennifer Welch] Overview We already know the following examples of NPC problems: SAT 3SAT We are going to show that the following are
More informationRandomness and Computation March 25, Lecture 5
0368.463 Randomness and Computation March 25, 2009 Lecturer: Ronitt Rubinfeld Lecture 5 Scribe: Inbal Marhaim, Naama Ben-Aroya Today Uniform generation of DNF satisfying assignments Uniform generation
More informationExercises Computational Complexity
Exercises Computational Complexity March 22, 2017 Exercises marked with a are more difficult. 1 Chapter 7, P and NP Exercise 1. Suppose some pancakes are stacked on a surface such that no two pancakes
More informationNP-Completeness. Algorithms
NP-Completeness Algorithms The NP-Completeness Theory Objective: Identify a class of problems that are hard to solve. Exponential time is hard. Polynomial time is easy. Why: Do not try to find efficient
More informationReductions and Satisfiability
Reductions and Satisfiability 1 Polynomial-Time Reductions reformulating problems reformulating a problem in polynomial time independent set and vertex cover reducing vertex cover to set cover 2 The Satisfiability
More informationPCP and Hardness of Approximation
PCP and Hardness of Approximation January 30, 2009 Our goal herein is to define and prove basic concepts regarding hardness of approximation. We will state but obviously not prove a PCP theorem as a starting
More informationCS 151 Complexity Theory Spring Final Solutions. L i NL i NC 2i P.
CS 151 Complexity Theory Spring 2017 Posted: June 9 Final Solutions Chris Umans 1. (a) The procedure that traverses a fan-in 2 depth O(log i n) circuit and outputs a formula runs in L i this can be done
More informationGraph Definitions. In a directed graph the edges have directions (ordered pairs). A weighted graph includes a weight function.
Graph Definitions Definition 1. (V,E) where An undirected graph G is a pair V is the set of vertices, E V 2 is the set of edges (unordered pairs) E = {(u, v) u, v V }. In a directed graph the edges have
More informationCopyright 2000, Kevin Wayne 1
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. SHORTEST-PATH: Given a graph G = (V, E), does there exists a simple
More informationLecture 1. 2 Motivation: Fast. Reliable. Cheap. Choose two.
Approximation Algorithms and Hardness of Approximation February 19, 2013 Lecture 1 Lecturer: Ola Svensson Scribes: Alantha Newman 1 Class Information 4 credits Lecturers: Ola Svensson (ola.svensson@epfl.ch)
More informationChapter 8. NP and Computational Intractability. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
Chapter 8 NP and Computational Intractability Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 Algorithm Design Patterns and Anti-Patterns Algorithm design patterns.
More informationMore on Polynomial Time and Space
CpSc 8390 Goddard Fall15 More on Polynomial Time and Space 20.1 The Original NP-Completeness Proof A configuration/snapshot of a machine is a representation of its current state (what info would be needed
More informationCMPSCI 311: Introduction to Algorithms Practice Final Exam
CMPSCI 311: Introduction to Algorithms Practice Final Exam Name: ID: Instructions: Answer the questions directly on the exam pages. Show all your work for each question. Providing more detail including
More informationNP-Complete Reductions 2
x 1 x 1 x 2 x 2 x 3 x 3 x 4 x 4 12 22 32 CS 447 11 13 21 23 31 33 Algorithms NP-Complete Reductions 2 Prof. Gregory Provan Department of Computer Science University College Cork 1 Lecture Outline NP-Complete
More informationNP and computational intractability. Kleinberg and Tardos, chapter 8
NP and computational intractability Kleinberg and Tardos, chapter 8 1 Major Transition So far we have studied certain algorithmic patterns Greedy, Divide and conquer, Dynamic programming to develop efficient
More information8.1 Polynomial-Time Reductions
8.1 Polynomial-Time Reductions Classify Problems According to Computational Requirements Q. Which problems will we be able to solve in practice? A working definition. Those with polynomial-time algorithms.
More information6.842 Randomness and Computation September 25-27, Lecture 6 & 7. Definition 1 Interactive Proof Systems (IPS) [Goldwasser, Micali, Rackoff]
6.84 Randomness and Computation September 5-7, 017 Lecture 6 & 7 Lecturer: Ronitt Rubinfeld Scribe: Leo de Castro & Kritkorn Karntikoon 1 Interactive Proof Systems An interactive proof system is a protocol
More informationTraveling Salesman Problem (TSP) Input: undirected graph G=(V,E), c: E R + Goal: find a tour (Hamiltonian cycle) of minimum cost
Traveling Salesman Problem (TSP) Input: undirected graph G=(V,E), c: E R + Goal: find a tour (Hamiltonian cycle) of minimum cost Traveling Salesman Problem (TSP) Input: undirected graph G=(V,E), c: E R
More informationLecture 10 October 7, 2014
6.890: Algorithmic Lower Bounds: Fun With Hardness Proofs Fall 2014 Lecture 10 October 7, 2014 Prof. Erik Demaine Scribes: Fermi Ma, Asa Oines, Mikhail Rudoy, Erik Waingarten Overview This lecture begins
More information31.6 Powers of an element
31.6 Powers of an element Just as we often consider the multiples of a given element, modulo, we consider the sequence of powers of, modulo, where :,,,,. modulo Indexing from 0, the 0th value in this sequence
More informationWhat Can We Do? CS125 Lecture 20 Fall 2014
CS125 Lecture 20 Fall 2014 We have defined the class of NP-complete problems, which have the property that if there is a polynomial time algorithm for any one of these problems, there is a polynomial time
More informationCSE200: Computability and complexity Interactive proofs
CSE200: Computability and complexity Interactive proofs Shachar Lovett January 29, 2018 1 What are interactive proofs Think of a prover trying to convince a verifer that a statement is correct. For example,
More information6 Randomized rounding of semidefinite programs
6 Randomized rounding of semidefinite programs We now turn to a new tool which gives substantially improved performance guarantees for some problems We now show how nonlinear programming relaxations can
More informationOutline. CS38 Introduction to Algorithms. Approximation Algorithms. Optimization Problems. Set Cover. Set cover 5/29/2014. coping with intractibility
Outline CS38 Introduction to Algorithms Lecture 18 May 29, 2014 coping with intractibility approximation algorithms set cover TSP center selection randomness in algorithms May 29, 2014 CS38 Lecture 18
More informationCPSC 536N: Randomized Algorithms Term 2. Lecture 10
CPSC 536N: Randomized Algorithms 011-1 Term Prof. Nick Harvey Lecture 10 University of British Columbia In the first lecture we discussed the Max Cut problem, which is NP-complete, and we presented a very
More informationApproximation Algorithms
15-251: Great Ideas in Theoretical Computer Science Spring 2019, Lecture 14 March 5, 2019 Approximation Algorithms 1 2 SAT 3SAT Clique Hamiltonian- Cycle given a Boolean formula F, is it satisfiable? same,
More information2SAT Andreas Klappenecker
2SAT Andreas Klappenecker The Problem Can we make the following boolean formula true? ( x y) ( y z) (z y)! Terminology A boolean variable is a variable that can be assigned the values true (T) or false
More informationNP Completeness. Andreas Klappenecker [partially based on slides by Jennifer Welch]
NP Completeness Andreas Klappenecker [partially based on slides by Jennifer Welch] Dealing with NP-Complete Problems Dealing with NP-Completeness Suppose the problem you need to solve is NP-complete. What
More informationA Mathematical Proof. Zero Knowledge Protocols. Interactive Proof System. Other Kinds of Proofs. When referring to a proof in logic we usually mean:
A Mathematical Proof When referring to a proof in logic we usually mean: 1. A sequence of statements. 2. Based on axioms. Zero Knowledge Protocols 3. Each statement is derived via the derivation rules.
More informationZero Knowledge Protocols. c Eli Biham - May 3, Zero Knowledge Protocols (16)
Zero Knowledge Protocols c Eli Biham - May 3, 2005 442 Zero Knowledge Protocols (16) A Mathematical Proof When referring to a proof in logic we usually mean: 1. A sequence of statements. 2. Based on axioms.
More informationPaths, Flowers and Vertex Cover
Paths, Flowers and Vertex Cover Venkatesh Raman, M.S. Ramanujan, and Saket Saurabh Presenting: Hen Sender 1 Introduction 2 Abstract. It is well known that in a bipartite (and more generally in a Konig)
More information8.1 Polynomial-Time Reductions
Algorithm Design Patterns and Anti-Patterns Analysis of Algorithms Algorithm design patterns. Ex. Greed. O(n 2 ) Dijkstra s SSSP (dense) Divide-and-conquer. O(n log n) FFT. Dynamic programming. O(n 2 )
More information11. APPROXIMATION ALGORITHMS
11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Lecture slides by Kevin Wayne Copyright 2005
More informationCMPSCI611: The SUBSET-SUM Problem Lecture 18
CMPSCI611: The SUBSET-SUM Problem Lecture 18 We begin today with the problem we didn t get to at the end of last lecture the SUBSET-SUM problem, which we also saw back in Lecture 8. The input to SUBSET-
More information1 Introduction. 1. Prove the problem lies in the class NP. 2. Find an NP-complete problem that reduces to it.
1 Introduction There are hundreds of NP-complete problems. For a recent selection see http://www. csc.liv.ac.uk/ ped/teachadmin/comp202/annotated_np.html Also, see the book M. R. Garey and D. S. Johnson.
More informationNP-Hardness. We start by defining types of problem, and then move on to defining the polynomial-time reductions.
CS 787: Advanced Algorithms NP-Hardness Instructor: Dieter van Melkebeek We review the concept of polynomial-time reductions, define various classes of problems including NP-complete, and show that 3-SAT
More informationChapter 10 Part 1: Reduction
//06 Polynomial-Time Reduction Suppose we could solve Y in polynomial-time. What else could we solve in polynomial time? don't confuse with reduces from Chapter 0 Part : Reduction Reduction. Problem X
More informationCh9: Exact Inference: Variable Elimination. Shimi Salant, Barak Sternberg
Ch9: Exact Inference: Variable Elimination Shimi Salant Barak Sternberg Part 1 Reminder introduction (1/3) We saw two ways to represent (finite discrete) distributions via graphical data structures: Bayesian
More informationDual-fitting analysis of Greedy for Set Cover
Dual-fitting analysis of Greedy for Set Cover We showed earlier that the greedy algorithm for set cover gives a H n approximation We will show that greedy produces a solution of cost at most H n OPT LP
More informationHomework 4 Solutions
CS3510 Design & Analysis of Algorithms Section A Homework 4 Solutions Uploaded 4:00pm on Dec 6, 2017 Due: Monday Dec 4, 2017 This homework has a total of 3 problems on 4 pages. Solutions should be submitted
More informationBest known solution time is Ω(V!) Check every permutation of vertices to see if there is a graph edge between adjacent vertices
Hard Problems Euler-Tour Problem Undirected graph G=(V,E) An Euler Tour is a path where every edge appears exactly once. The Euler-Tour Problem: does graph G have an Euler Path? Answerable in O(E) time.
More informationGraph Theory and Optimization Approximation Algorithms
Graph Theory and Optimization Approximation Algorithms Nicolas Nisse Université Côte d Azur, Inria, CNRS, I3S, France October 2018 Thank you to F. Giroire for some of the slides N. Nisse Graph Theory and
More informationPaths, Flowers and Vertex Cover
Paths, Flowers and Vertex Cover Venkatesh Raman M. S. Ramanujan Saket Saurabh Abstract It is well known that in a bipartite (and more generally in a König) graph, the size of the minimum vertex cover is
More informationGreedy algorithms Or Do the right thing
Greedy algorithms Or Do the right thing March 1, 2005 1 Greedy Algorithm Basic idea: When solving a problem do locally the right thing. Problem: Usually does not work. VertexCover (Optimization Version)
More informationSolving Linear Recurrence Relations (8.2)
EECS 203 Spring 2016 Lecture 18 Page 1 of 10 Review: Recurrence relations (Chapter 8) Last time we started in on recurrence relations. In computer science, one of the primary reasons we look at solving
More informationPolynomial-Time Approximation Algorithms
6.854 Advanced Algorithms Lecture 20: 10/27/2006 Lecturer: David Karger Scribes: Matt Doherty, John Nham, Sergiy Sidenko, David Schultz Polynomial-Time Approximation Algorithms NP-hard problems are a vast
More informationLecture Overview. 2 Shortest s t path. 2.1 The LP. 2.2 The Algorithm. COMPSCI 530: Design and Analysis of Algorithms 11/14/2013
COMPCI 530: Design and Analysis of Algorithms 11/14/2013 Lecturer: Debmalya Panigrahi Lecture 22 cribe: Abhinandan Nath 1 Overview In the last class, the primal-dual method was introduced through the metric
More informationApproximation Algorithms: The Primal-Dual Method. My T. Thai
Approximation Algorithms: The Primal-Dual Method My T. Thai 1 Overview of the Primal-Dual Method Consider the following primal program, called P: min st n c j x j j=1 n a ij x j b i j=1 x j 0 Then the
More informationIn this lecture we discuss the complexity of approximation problems, and show how to prove they are NP-hard.
In this lecture we discuss the complexity of approximation problems, and show how to prove they are NP-hard. 1 We will show how one can prove such results and then apply this technique to some approximation
More informationTowards the Proof of the PCP Theorem
CS640 Computational Complexity Towards the Proof of the PCP Theorem Instructor: Manindra Agrawal Scribe: Ramprasad Saptharishi Last class we saw completed our discussion on expander graphs. We shall now
More information! Greed. O(n log n) interval scheduling. ! Divide-and-conquer. O(n log n) FFT. ! Dynamic programming. O(n 2 ) edit distance.
Algorithm Design Patterns and Anti-Patterns Chapter 8 NP and Computational Intractability Algorithm design patterns. Ex.! Greed. O(n log n) interval scheduling.! Divide-and-conquer. O(n log n) FFT.! Dynamic
More information1 Undirected Vertex Geography UVG
Geography Start with a chip sitting on a vertex v of a graph or digraph G. A move consists of moving the chip to a neighbouring vertex. In edge geography, moving the chip from x to y deletes the edge (x,
More informationChapter Design Techniques for Approximation Algorithms
Chapter 2 Design Techniques for Approximation Algorithms I N THE preceding chapter we observed that many relevant optimization problems are NP-hard, and that it is unlikely that we will ever be able to
More information1 Matching in Non-Bipartite Graphs
CS 369P: Polyhedral techniques in combinatorial optimization Instructor: Jan Vondrák Lecture date: September 30, 2010 Scribe: David Tobin 1 Matching in Non-Bipartite Graphs There are several differences
More information1 Definition of Reduction
1 Definition of Reduction Problem A is reducible, or more technically Turing reducible, to problem B, denoted A B if there a main program M to solve problem A that lacks only a procedure to solve problem
More informationCS270 Combinatorial Algorithms & Data Structures Spring Lecture 19:
CS270 Combinatorial Algorithms & Data Structures Spring 2003 Lecture 19: 4.1.03 Lecturer: Satish Rao Scribes: Kevin Lacker and Bill Kramer Disclaimer: These notes have not been subjected to the usual scrutiny
More information11.1 Facility Location
CS787: Advanced Algorithms Scribe: Amanda Burton, Leah Kluegel Lecturer: Shuchi Chawla Topic: Facility Location ctd., Linear Programming Date: October 8, 2007 Today we conclude the discussion of local
More informationInapproximability of the Perimeter Defense Problem
Inapproximability of the Perimeter Defense Problem Evangelos Kranakis Danny Krizanc Lata Narayanan Kun Xu Abstract We model the problem of detecting intruders using a set of infrared beams by the perimeter
More informationChapter 5 Graph Algorithms Algorithm Theory WS 2012/13 Fabian Kuhn
Chapter 5 Graph Algorithms Algorithm Theory WS 2012/13 Fabian Kuhn Graphs Extremely important concept in computer science Graph, : node (or vertex) set : edge set Simple graph: no self loops, no multiple
More informationNotes for Lecture 24
U.C. Berkeley CS170: Intro to CS Theory Handout N24 Professor Luca Trevisan December 4, 2001 Notes for Lecture 24 1 Some NP-complete Numerical Problems 1.1 Subset Sum The Subset Sum problem is defined
More information15-451/651: Design & Analysis of Algorithms November 4, 2015 Lecture #18 last changed: November 22, 2015
15-451/651: Design & Analysis of Algorithms November 4, 2015 Lecture #18 last changed: November 22, 2015 While we have good algorithms for many optimization problems, the previous lecture showed that many
More informationarxiv: v2 [cs.cc] 29 Mar 2010
On a variant of Monotone NAE-3SAT and the Triangle-Free Cut problem. arxiv:1003.3704v2 [cs.cc] 29 Mar 2010 Peiyush Jain, Microsoft Corporation. June 28, 2018 Abstract In this paper we define a restricted
More information6.856 Randomized Algorithms
6.856 Randomized Algorithms David Karger Handout #4, September 21, 2002 Homework 1 Solutions Problem 1 MR 1.8. (a) The min-cut algorithm given in class works because at each step it is very unlikely (probability
More informationarxiv: v1 [cs.ds] 3 Mar 2015
Binary Search in Graphs Ehsan Emamjomeh-Zadeh David Kempe October 30, 2018 arxiv:1503.00805v1 [cs.ds] 3 Mar 2015 Abstract We study the following natural generalization of Binary Search to arbitrary connected
More informationLinear Programming in Small Dimensions
Linear Programming in Small Dimensions Lekcija 7 sergio.cabello@fmf.uni-lj.si FMF Univerza v Ljubljani Edited from slides by Antoine Vigneron Outline linear programming, motivation and definition one dimensional
More informationUSING QBF SOLVERS TO SOLVE GAMES AND PUZZLES. Zhihe Shen. Advisor: Howard Straubing
Boston College Computer Science Senior Thesis USING QBF SOLVERS TO SOLVE GAMES AND PUZZLES Zhihe Shen Advisor: Howard Straubing Abstract There are multiple types of games, such as board games and card
More informationA Simplied NP-complete MAXSAT Problem. Abstract. It is shown that the MAX2SAT problem is NP-complete even if every variable
A Simplied NP-complete MAXSAT Problem Venkatesh Raman 1, B. Ravikumar 2 and S. Srinivasa Rao 1 1 The Institute of Mathematical Sciences, C. I. T. Campus, Chennai 600 113. India 2 Department of Computer
More informationCS 125 Section #4 RAMs and TMs 9/27/16
CS 125 Section #4 RAMs and TMs 9/27/16 1 RAM A word-ram consists of: A fixed set of instructions P 1,..., P q. Allowed instructions are: Modular arithmetic and integer division on registers; the standard
More informationComplexity Classes and Polynomial-time Reductions
COMPSCI 330: Design and Analysis of Algorithms April 19, 2016 Complexity Classes and Polynomial-time Reductions Lecturer: Debmalya Panigrahi Scribe: Tianqi Song 1 Overview In this lecture, we introduce
More information! Greed. O(n log n) interval scheduling. ! Divide-and-conquer. O(n log n) FFT. ! Dynamic programming. O(n 2 ) edit distance.
Algorithm Design Patterns and Anti-Patterns 8. NP and Computational Intractability Algorithm design patterns. Ex.! Greed. O(n log n) interval scheduling.! Divide-and-conquer. O(n log n) FFT.! Dynamic programming.
More informationMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science Algorithms For Inference Fall 2014
Suggested Reading: Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.438 Algorithms For Inference Fall 2014 Probabilistic Modelling and Reasoning: The Junction
More informationDecision Problems. Observation: Many polynomial algorithms. Questions: Can we solve all problems in polynomial time? Answer: No, absolutely not.
Decision Problems Observation: Many polynomial algorithms. Questions: Can we solve all problems in polynomial time? Answer: No, absolutely not. Definition: The class of problems that can be solved by polynomial-time
More informationApproximation Algorithms
Approximation Algorithms Given an NP-hard problem, what should be done? Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one of three desired features. Solve problem to optimality.
More informationALGORITHMS EXAMINATION Department of Computer Science New York University December 17, 2007
ALGORITHMS EXAMINATION Department of Computer Science New York University December 17, 2007 This examination is a three hour exam. All questions carry the same weight. Answer all of the following six questions.
More informationIntroduction to Approximation Algorithms
Introduction to Approximation Algorithms Dr. Gautam K. Das Departmet of Mathematics Indian Institute of Technology Guwahati, India gkd@iitg.ernet.in February 19, 2016 Outline of the lecture Background
More informationCSC2420 Fall 2012: Algorithm Design, Analysis and Theory An introductory (i.e. foundational) level graduate course.
CSC2420 Fall 2012: Algorithm Design, Analysis and Theory An introductory (i.e. foundational) level graduate course. Allan Borodin November 8, 2012; Lecture 9 1 / 24 Brief Announcements 1 Game theory reading
More informationCS369G: Algorithmic Techniques for Big Data Spring
CS369G: Algorithmic Techniques for Big Data Spring 2015-2016 Lecture 11: l 0 -Sampling and Introduction to Graph Streaming Prof. Moses Charikar Scribe: Austin Benson 1 Overview We present and analyze the
More informationLecture 7. s.t. e = (u,v) E x u + x v 1 (2) v V x v 0 (3)
COMPSCI 632: Approximation Algorithms September 18, 2017 Lecturer: Debmalya Panigrahi Lecture 7 Scribe: Xiang Wang 1 Overview In this lecture, we will use Primal-Dual method to design approximation algorithms
More informationThe Resolution Algorithm
The Resolution Algorithm Introduction In this lecture we introduce the Resolution algorithm for solving instances of the NP-complete CNF- SAT decision problem. Although the algorithm does not run in polynomial
More informationApproximation Algorithms
Chapter 8 Approximation Algorithms Algorithm Theory WS 2016/17 Fabian Kuhn Approximation Algorithms Optimization appears everywhere in computer science We have seen many examples, e.g.: scheduling jobs
More information11. APPROXIMATION ALGORITHMS
Coping with NP-completeness 11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: weighted vertex cover LP rounding: weighted vertex cover generalized load balancing knapsack problem
More information1 Matchings in Graphs
Matchings in Graphs J J 2 J 3 J 4 J 5 J J J 6 8 7 C C 2 C 3 C 4 C 5 C C 7 C 8 6 J J 2 J 3 J 4 J 5 J J J 6 8 7 C C 2 C 3 C 4 C 5 C C 7 C 8 6 Definition Two edges are called independent if they are not adjacent
More informationApproximation Algorithms
Approximation Algorithms Frédéric Giroire FG Simplex 1/11 Motivation Goal: Find good solutions for difficult problems (NP-hard). Be able to quantify the goodness of the given solution. Presentation of
More informationSolution for Homework set 3
TTIC 300 and CMSC 37000 Algorithms Winter 07 Solution for Homework set 3 Question (0 points) We are given a directed graph G = (V, E), with two special vertices s and t, and non-negative integral capacities
More informationLecture 7: Counting classes
princeton university cos 522: computational complexity Lecture 7: Counting classes Lecturer: Sanjeev Arora Scribe:Manoj First we define a few interesting problems: Given a boolean function φ, #SAT is the
More informationLecture 19 Thursday, March 29. Examples of isomorphic, and non-isomorphic graphs will be given in class.
CIS 160 - Spring 2018 (instructor Val Tannen) Lecture 19 Thursday, March 29 GRAPH THEORY Graph isomorphism Definition 19.1 Two graphs G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) are isomorphic, write G 1 G
More information