# Quantum Monte Carlo and the negative sign problem

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Quantum Monte Carlo and the negative sign problem or how to earn one million dollar Matthias Troyer, ETH Zürich Uwe-Jens Wiese, Universität Bern

2 Complexity of many particle problems Classical 1 particle: 6-dimensional ODE 3 position and 3 velocity coordinates N particles: 6N-dimensional ODE m d 2 x dt 2 = F Quantum 1 particle: 3-dimensional PDE N particles: 3N dimensional PDE Quantum or classical lattice model 1 site: q states N sites: q N states i Ψ( x ) t = 1 2m ΔΨ( x ) + V ( x )Ψ( x ) Effort grows exponentially with N How can we solve this exponential problem?

3 The Metropolis Algorithm (1953)

4 Mapping quantum to classical systems Classical: Quantum: A = A c e E c /T e E c /T c c A = Tr Ae H /T Tre H /T Calculate exponential by integrating a diffusion equation dψ dτ = HΨ Ψ(1/T) = e (1/T )H Ψ(0) Map to world lines of the trajectories of the particles use Monte Carlo samples these world lines imaginary time 1/Τ space

5 The negative sign problem In mapping of quantum to classical system Z =Tre βh = i p i there is a sign problem if some of the pi < 0 Appears e.g. in simulation of electrons when two electrons exchange places (Pauli principle) i 1 > i 4 > i 3 > i 2 > i 1 >

6 The negative sign problem Sample with respect to absolute values of the weights A = i A i p i p i = i Exponentially growing cancellation in the sign sign = i A i sgn p i p i Exponential growth of errors i i p i sgn p i p i i i p i p i A sign p sign p i p i = Z/Z p = e βv (f f p ) sign sign = sign2 sign 2 M sign eβv (f f p ) M NP-hard problem (no general solution) [Troyer and Wiese, PRL 2005]

7 Is the sign problem exponentially hard? The sign problem is basis-dependent Diagonalize the Hamiltonian matrix \$ \$ exp(!"# i ) A = Tr[ Aexp(!"H) ] Tr[ exp(!"h) ] = i A i i exp(!"# i ) All weights are positive But this is an exponentially hard problem since dim(h)=2 N! Good news: the sign problem is basis-dependent! But: the sign problem is still not solved Despite decades of attempts Reminiscent of the NP-hard problems No proof that they are exponentially hard No polynomial solution either i i

8 Complexity of decision problems Partial hierarchy of decision problems Undecidable ( This sentence is false ) Partially decidable (halting problem of Turing machines) EXPSPACE Exponential space and time complexity: diagonalization of Hamiltonian PSPACE NP P Exponential time, polynomial space complexity: Monte Carlo Polynomial complexity on non-deterministic machine Traveling salesman problem 3D Ising spin glass Polynomial complexity on Turing machine

9 Complexity of decision problems Some problems are harder than others: Complexity class P Can be solved in polynomial time on a Turing machine Eulerian circuit problem Minimum spanning Tree (decision version) Detecting primality Complexity class NP Polynomial complexity using non-deterministic algorithms Hamiltonian cirlce problem Traveling salesman problem (decision version) Factorization of integers 3D spin glasses

10 The complexity class P The Eulerian circuit problem Seven bridges in Königsberg (now Kaliningrad) crossed the river Pregel Can we do a roundtrip by crossing each bridge exactly once? Is there a closed walk on the graph going through each edge exactly once? Looks like an expensive task by testing all possible paths. Euler: Desired path exits only if the coordination of each edge is even. This is of order O(N 2 ) Concering Königsberg: NO!

11 The complexity class NP The Hamiltonian cycle problem Sir Hamilton's Icosian game: Is there a closed walk on going through each vertex exactly once? Looks like an expensive task by testing all possible paths. No polynomial algorithm is known, nor a proof that it cannot be constructed

12 The complexity class NP Polynomial time complexity on a nondeterministic machine Can execute both branches of an if-statement, but branches cannot merge again Has exponential number of CPUs but no communication It can in polynomial time Test all possible paths on the graph to see whether there is a Hamiltonian cycle Test all possible configurations of a spin glass for a configuration smaller than a given energy c : E c < E It cannot Calculate a partition function since the sum over all states cannot be performed Z = exp( βε c ) c

13 NP-hardness and NP-completeness Polynomial reduction Two decision problems Q and P: Q P: there is an polynomial algorithm for Q, provided there is one for P Typical proof: Use the algorithm for P as a subroutine in an algorithm for P Many problems have been reduced to other problems NP-hardness A problem P is NP-hard if Q NP : Q P This means that solving it in polynomial time solves all problems in NP too NP-completeness A problem P is NP-complete, if P is NP-hard and Most Problems in NP were shown to be NP-complete P NP

14 The P versus NP problem Hundreds of important NP-complete problems in computer science Despite decades of research no polynomial time algorithm was found Exponential complexity has not been proven either The P versus NP problem Is P=NP or is P NP? One of the millenium challenges of the Clay Math Foundation 1 million US\$ for proving either P=NP or P NP? The situation is similar to the sign problem

15 The Ising spin glass: NP-complete 3D Ising spin glass H = J ij σ j σ j with J ij = 0,±1 The NP-complete question is: Is there a configuration with energy E 0? i, j Solution by Monte Carlo: Perform a Monte Carlo simulation at β = N ln2 + ln N + ln Measure the energy: E < E if there exists a state with energy E 0 E > E 0 +1 otherwise A Monte Carlo simulation can decide the question

16 The Ising spin glass: NP-complete 3D Ising spin glass is NP-complete H = J ij σ j σ j with J ij = 0,±1 i, j Frustration leads to NP-hardness of Monte Carlo? Exponentially long tunneling and autocorrelation times c 1! c 2!...! c i! c i +1!... ΔA = ( A A ) 2 = Var A M (1 + 2τ A )

17 Antiferronmagnetic couplings on a triangle:? Frustration Leads to frustration, cannot have each bond in lowest energy state With random couplings finding the ground state is NP-hard Quantum mechanical: negative probabilities for a world line configuration Due to exchange of fermions Negative weight (-J) 3 -J! -J! -J!

18 What is a solution of the sign problem? Consider a fermionic quantum system with a sign problem (some p i < 0 ) A = Tr[ Aexp( βh )] Tr [ exp( βh )] = A i p i i i p i Where the sampling of the bosonic system with respect to p i scales polynomially T ε 2 N n β m A solution of the sign problem is defined as an algorithm that can calculate the average with respect to p i also in polynomial time Note that changing basis to make all p i 0 might not be enough: the algorithm might still exhibit exponential scaling

19 Solving an NP-hard problem by QMC Take 3D Ising spin glass H = J ij σ j σ j with J ij = 0,±1 i, j View it as a quantum problem in basis where H it is not diagonal H (SG ) = J ij σ x jσ x j with J ij = 0,±1 i, j The randomness ends up in the sign of offdiagonal matrix elements Ignoring the sign gives the ferromagnet and loop algorithm is in P H (FM ) = σ x jσ x j The sign problem causes NP-hardness solving the sign problem solves all the NP-complete problems and prove NP=P i, j

20 Summary A solution to the sign problem solves all problems in NP Hence a general solution to the sign problem does not exist unless P=NP If you still find one and thus prove that NP=P you will get 1 million US \$! A Nobel prize? A Fields medal? What does this imply? A general method cannot exist Look for specific solutions to the sign problem or model-specific methods

21 The origin of the sign problem We sample with the wrong distribution by ignoring the sign! We simulate bosons and expect to learn about fermions? will only work in insulators and superfluids We simulate a ferromagnet and expect to learn something useful about a frustrated antiferromagnet? We simulate a ferromagnet and expect to learn something about a spin glass? This is the idea behind the proof of NP-hardness

22 Working around the sign problem 1. Simulate bosonic systems Bosonic atoms in optical lattices Helium-4 supersolids Nonfrustrated magnets 2. Simulate sign-problem free fermionic systems Attractive on-site interactions Half-filled Mott insulators 3. Restriction to quasi-1d systems Use the density matrix renormalization group method (DMRG) 4. Use approximate methods Dynamical mean field theory (DMFT)

### Computer Algorithms. NP-Complete Problems. CISC 4080 Yanjun Li

Computer Algorithms NP-Complete Problems NP-completeness The quest for efficient algorithms is about finding clever ways to bypass the process of exhaustive search, using clues from the input in order

### Monte Carlo simulations and error analysis. Matthias Troyer, ETH Zürich

Monte Carlo simulations and error analysis Matthias Troyer, ETH Zürich Outline of the lecture 1. Monte Carlo integration 2. Generating random numbers 3. The Metropolis algorithm 4. Monte Carlo error analysis

### 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

### NP-Completeness. CptS 223 Advanced Data Structures. Larry Holder School of Electrical Engineering and Computer Science Washington State University

NP-Completeness CptS 223 Advanced Data Structures Larry Holder School of Electrical Engineering and Computer Science Washington State University 1 Hard Graph Problems Hard means no known solutions with

### Introduction to Logic in Computer Science: Autumn 2006

Introduction to Logic in Computer Science: Autumn 2006 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today Now that we have a basic understanding

### Complexity Theory. IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar

Complexity Theory IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar Outline Goals Computation of Problems Concepts and Definitions Complexity Classes and Problems Polynomial Time Reductions Examples

### 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

### CAD Algorithms. P and NP

CAD Algorithms The Classes P and NP Mohammad Tehranipoor ECE Department 6 September 2010 1 P and NP P and NP are two families of problems. P is a class which contains all of the problems we solve using

### 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

### The origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson)

Chapter 11 Graph Theory The origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson) (N. Let us start with a formal definition of what is a graph. Definition 72. A graph

### 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

### Complexity Classes P and NP

Complexity Classes P and NP MATH 3220 Supplemental Presentation by John Aleshunas The cure for boredom is curiosity. There is no cure for curiosity Dorothy Parker Computational Complexity Theory In computer

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### Monte Carlo Simulation technique. S. B. Santra Department of Physics Indian Institute of Technology Guwahati

Monte Carlo Simulation technique S. B. Santra Department of Physics Indian Institute of Technology Guwahati What is Monte Carlo (MC)? Introduction Monte Carlo method is a common name for a wide variety

### Open Problems in Quantum Information Processing. John Watrous Department of Computer Science University of Calgary

Open Problems in Quantum Information Processing John Watrous Department of Computer Science University of Calgary #1 Open Problem Find new quantum algorithms. Existing algorithms: Shor s Algorithm (+ extensions)

### 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

### 2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8]

Code No: R05220502 Set No. 1 1. (a) Describe the performance analysis in detail. (b) Show that f 1 (n)+f 2 (n) = 0(max(g 1 (n), g 2 (n)) where f 1 (n) = 0(g 1 (n)) and f 2 (n) = 0(g 2 (n)). [8+8] 2. (a)

### SIMS 255 Foundations of Software Design. Complexity and NP-completeness

SIMS 255 Foundations of Software Design Complexity and NP-completeness Matt Welsh November 29, 2001 mdw@cs.berkeley.edu 1 Outline Complexity of algorithms Space and time complexity ``Big O'' notation Complexity

### Tetris is Hard: An Introduction to P vs NP

Tetris is Hard: An Introduction to P vs NP Based on Tetris is Hard, Even to Approximate in COCOON 2003 by Erik D. Demaine (MIT) Susan Hohenberger (JHU) David Liben-Nowell (Carleton) What s Your Problem?

### Quantum and Non-deterministic computers facing NP-completeness

Quantum and Non-deterministic computers facing NP-completeness Thibaut University of Vienna Dept. of Business Administration Austria Vienna January 29th, 2013 Some pictures come from Wikipedia Introduction

### 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:

### A Working Knowledge of Computational Complexity for an Optimizer

A Working Knowledge of Computational Complexity for an Optimizer ORF 363/COS 323 Instructor: Amir Ali Ahmadi TAs: Y. Chen, G. Hall, J. Ye Fall 2014 1 Why computational complexity? What is computational

### 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

V. Adamchik 1 Graph Theory Victor Adamchik Fall of 2005 Plan 1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs Introduction A.Aho and J.Ulman acknowledge that Fundamentally, computer

### 1 Basic Definitions and Concepts in Graph Theory

CME 305: Discrete Mathematics and Algorithms 1 Basic Definitions and Concepts in Graph Theory A graph G(V, E) is a set V of vertices and a set E of edges. In an undirected graph, an edge is an unordered

### 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

### 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

### 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

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

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

### Forest-Root Formulas in Statistical Physics. John Imbrie University of Virginia

Forest-Root Formulas in Statistical Physics John Imbrie University of Virginia 1 An interpolation formula We d better assume f goes to zero at infinity. Note the minus sign! 2 Generalization: Chevron interpolation

### Introduction to computer science

Introduction to computer science Michael A. Nielsen University of Queensland Goals: 1. Introduce the notion of the computational complexity of a problem, and define the major computational complexity classes.

### Sucesses and limitations of dynamical mean field theory. A.-M. Tremblay G. Sordi, D. Sénéchal, K. Haule, S. Okamoto, B. Kyung, M.

Sucesses and limitations of dynamical mean field theory A.-M. Tremblay G. Sordi, D. Sénéchal, K. Haule, S. Okamoto, B. Kyung, M. Civelli MIT, 20 October, 2011 How to make a metal Courtesy, S. Julian r

### Why? A central concept in Computer Science. Algorithms are ubiquitous.

Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online

### The Classes P and NP. mohamed@elwakil.net

Intractable Problems The Classes P and NP Mohamed M. El Wakil mohamed@elwakil.net 1 Agenda 1. What is a problem? 2. Decidable or not? 3. The P class 4. The NP Class 5. TheNP Complete class 2 What is a

### Introduction to Algorithms Review information for Prelim 1 CS 4820, Spring 2010 Distributed Wednesday, February 24

Introduction to Algorithms Review information for Prelim 1 CS 4820, Spring 2010 Distributed Wednesday, February 24 The final exam will cover seven topics. 1. greedy algorithms 2. divide-and-conquer algorithms

### Lecture 7: NP-Complete Problems

IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 7: NP-Complete Problems David Mix Barrington and Alexis Maciel July 25, 2000 1. Circuit

### Answers to some of the exercises.

Answers to some of the exercises. Chapter 2. Ex.2.1 (a) There are several ways to do this. Here is one possibility. The idea is to apply the k-center algorithm first to D and then for each center in D

### 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

### Universality in the theory of algorithms and computer science

Universality in the theory of algorithms and computer science Alexander Shen Computational models The notion of computable function was introduced in 1930ies. Simplifying (a rather interesting and puzzling)

### Informatique Fondamentale IMA S8

Informatique Fondamentale IMA S8 Cours 4 : graphs, problems and algorithms on graphs, (notions of) NP completeness Laure Gonnord http://laure.gonnord.org/pro/teaching/ Laure.Gonnord@polytech-lille.fr Université

### Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

### 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

### Measuring Rationality with the Minimum Cost of Revealed Preference Violations. Mark Dean and Daniel Martin. Online Appendices - Not for Publication

Measuring Rationality with the Minimum Cost of Revealed Preference Violations Mark Dean and Daniel Martin Online Appendices - Not for Publication 1 1 Algorithm for Solving the MASP In this online appendix

### One last point: we started off this book by introducing another famously hard search problem:

S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani 261 Factoring One last point: we started off this book by introducing another famously hard search problem: FACTORING, the task of finding all prime factors

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### ON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu

ON THE COMPLEXITY OF THE GAME OF SET KAMALIKA CHAUDHURI, BRIGHTEN GODFREY, DAVID RATAJCZAK, AND HOETECK WEE {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ABSTRACT. Set R is a card game played with a

### Single machine parallel batch scheduling with unbounded capacity

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

### 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

### Euler Paths and Euler Circuits

Euler Paths and Euler Circuits An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. An Euler path starts and

### 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

### P vs NP problem in the field anthropology

Research Article P vs NP problem in the field anthropology Michael.A. Popov, Oxford, UK Email Michael282.eps@gmail.com Keywords P =?NP - complexity anthropology - M -decision - quantum -like game - game-theoretical

### Fall 2015 Midterm 1 24/09/15 Time Limit: 80 Minutes

Math 340 Fall 2015 Midterm 1 24/09/15 Time Limit: 80 Minutes Name (Print): This exam contains 6 pages (including this cover page) and 5 problems. Enter all requested information on the top of this page,

### Ian Stewart on Minesweeper

Ian Stewart on Minesweeper It's not often you can win a million dollars by analysing a computer game, but by a curious conjunction of fate, there's a chance that you might. However, you'll only pick up

### A Fast Algorithm For Finding Hamilton Cycles

A Fast Algorithm For Finding Hamilton Cycles by Andrew Chalaturnyk A thesis presented to the University of Manitoba in partial fulfillment of the requirements for the degree of Masters of Science in Computer

### Quantum Computability and Complexity and the Limits of Quantum Computation

Quantum Computability and Complexity and the Limits of Quantum Computation Eric Benjamin, Kenny Huang, Amir Kamil, Jimmy Kittiyachavalit University of California, Berkeley December 7, 2003 This paper will

### MGF 1107 CH 15 LECTURE NOTES Denson. Section 15.1

1 Section 15.1 Consider the house plan below. This graph represents the house. Consider the mail route below. This graph represents the mail route. 2 Definitions 1. Graph a structure that describes relationships.

### 3. Eulerian and Hamiltonian Graphs

3. Eulerian and Hamiltonian Graphs There are many games and puzzles which can be analysed by graph theoretic concepts. In fact, the two early discoveries which led to the existence of graphs arose from

### What s next? Reductions other than kernelization

What s next? Reductions other than kernelization Dániel Marx Humboldt-Universität zu Berlin (with help from Fedor Fomin, Daniel Lokshtanov and Saket Saurabh) WorKer 2010: Workshop on Kernelization Nov

### Graph Theory. Euler tours and Chinese postmen. John Quinn. Week 5

Graph Theory Euler tours and Chinese postmen John Quinn Week 5 Recap: connectivity Connectivity and edge-connectivity of a graph Blocks Kruskal s algorithm Königsberg, Prussia The Seven Bridges of Königsberg

### Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

### Lecture 19: Introduction to NP-Completeness Steven Skiena. Department of Computer Science State University of New York Stony Brook, NY 11794 4400

Lecture 19: Introduction to NP-Completeness Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Reporting to the Boss Suppose

### CS5314 Randomized Algorithms. Lecture 16: Balls, Bins, Random Graphs (Random Graphs, Hamiltonian Cycles)

CS5314 Randomized Algorithms Lecture 16: Balls, Bins, Random Graphs (Random Graphs, Hamiltonian Cycles) 1 Objectives Introduce Random Graph Model used to define a probability space for all graphs with

### Graph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency.

Mária Markošová Graph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency. Isomorphism of graphs. Paths, cycles, trials.

### Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs

MCS-236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set

### Computational complexity theory

Computational complexity theory Goal: A general theory of the resources needed to solve computational problems What types of resources? Time What types of computational problems? decision problem Decision

### Reductions & NP-completeness as part of Foundations of Computer Science undergraduate course

Reductions & NP-completeness as part of Foundations of Computer Science undergraduate course Alex Angelopoulos, NTUA January 22, 2015 Outline Alex Angelopoulos (NTUA) FoCS: Reductions & NP-completeness-

### 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:

### Transportation Polytopes: a Twenty year Update

Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,

### 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

### Quantum magnetism with few cold atoms. Gerhard Zürn Graduiertenkolleg Hannover,

Quantum magnetism with few cold atoms Gerhard Zürn Graduiertenkolleg Hannover, 17.12.2015 States of matter http://en.wikipedia.org/wiki/high-temperature_superconductivity What are the key ingredients for

### Berkeley CS191x: Quantum Mechanics and Quantum Computation Optional Class Project

Berkeley CS191x: Quantum Mechanics and Quantum Computation Optional Class Project This document describes the optional class project for the Fall 2013 offering of CS191x. The project will not be graded.

### Chapter 7 Uncomputability

Chapter 7 Uncomputability 190 7.1 Introduction Undecidability of concrete problems. First undecidable problem obtained by diagonalisation. Other undecidable problems obtained by means of the reduction

### Million Dollar Mathematics!

Million Dollar Mathematics! Alissa S. Crans Loyola Marymount University Southern California Undergraduate Math Day University of California, San Diego April 30, 2011 This image is from the Wikipedia article

### CoNP and Function Problems

CoNP and Function Problems conp By definition, conp is the class of problems whose complement is in NP. NP is the class of problems that have succinct certificates. conp is therefore the class of problems

### On the Relationship between Classes P and NP

Journal of Computer Science 8 (7): 1036-1040, 2012 ISSN 1549-3636 2012 Science Publications On the Relationship between Classes P and NP Anatoly D. Plotnikov Department of Computer Systems and Networks,

### FORMAL LANGUAGES, AUTOMATA AND COMPUTATION

FORMAL LANGUAGES, AUTOMATA AND COMPUTATION REDUCIBILITY ( LECTURE 16) SLIDES FOR 15-453 SPRING 2011 1 / 20 THE LANDSCAPE OF THE CHOMSKY HIERARCHY ( LECTURE 16) SLIDES FOR 15-453 SPRING 2011 2 / 20 REDUCIBILITY

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

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

### IE 680 Special Topics in Production Systems: Networks, Routing and Logistics*

IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* Rakesh Nagi Department of Industrial Engineering University at Buffalo (SUNY) *Lecture notes from Network Flows by Ahuja, Magnanti

### Ultracold few-fermion systems with tunable interactions Selim Jochim Physikalisches Institut Universität Heidelberg

Ultracold few-fermion systems with tunable interactions Selim Jochim Physikalisches Institut Universität Heidelberg The matter we deal with T=40nK 1µK Density n=10 9 10 14 cm -3 Pressures as low as 10-17

### Theoretical Computer Science (Bridging Course) Complexity

Theoretical Computer Science (Bridging Course) Complexity Gian Diego Tipaldi A scenario You are a programmer working for a logistics company Your boss asks you to implement a program that optimizes the

### The Metropolis Algorithm

The Metropolis Algorithm Statistical Systems and Simulated Annealing Physics 170 When studying systems with a great many particles, it becomes clear that the number of possible configurations becomes exceedingly

### 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

### Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras

Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding

### Chapter 29: Atomic Structure. What will we learn in this chapter?

Chapter 29: Atomic Structure What will we learn in this chapter? Contents: Electrons in atoms Wave functions Electron spin Pauli exclusion principle Atomic structure Periodic table W. Pauli & N. Bohr Both

### Boulder Dash is NP hard

Boulder Dash is NP hard Marzio De Biasi marziodebiasi [at] gmail [dot] com December 2011 Version 0.01:... now the difficult part: is it NP? Abstract Boulder Dash is a videogame created by Peter Liepa and

### Heuristic Methods. Part #1. João Luiz Kohl Moreira. Observatório Nacional - MCT COAA. Observatório Nacional - MCT 1 / 14

Heuristic Methods Part #1 João Luiz Kohl Moreira COAA Observatório Nacional - MCT Observatório Nacional - MCT 1 / Outline 1 Introduction Aims Course's target Adviced Bibliography 2 Problem Introduction

### NP-completeness and the real world. NP completeness. NP-completeness and the real world (2) NP-completeness and the real world

-completeness and the real world completeness Course Discrete Biological Models (Modelli Biologici Discreti) Zsuzsanna Lipták Imagine you are working for a biotech company. One day your boss calls you

### Scheduling. Open shop, job shop, flow shop scheduling. Related problems. Open shop, job shop, flow shop scheduling

Scheduling Basic scheduling problems: open shop, job shop, flow job The disjunctive graph representation Algorithms for solving the job shop problem Computational complexity of the job shop problem Open

### A New Nature-inspired Algorithm for Load Balancing

A New Nature-inspired Algorithm for Load Balancing Xiang Feng East China University of Science and Technology Shanghai, China 200237 Email: xfeng{@ecusteducn, @cshkuhk} Francis CM Lau The University of

### Chapter 4: Trees. 2. Theorem: Let T be a graph with n vertices. Then the following statements are equivalent:

9 Properties of Trees. Definitions: Chapter 4: Trees forest - a graph that contains no cycles tree - a connected forest. Theorem: Let T be a graph with n vertices. Then the following statements are equivalent:

### Strong correlations in few-atom systems. Selim Jochim, Universität Heidelberg

Strong correlations in few-atom systems Selim Jochim, Universität Heidelberg Control over few atoms IBM, 1989: Control over individual atoms Our vision: Control also all correlations between atoms 1 2

### MCMC Using Hamiltonian Dynamics

5 MCMC Using Hamiltonian Dynamics Radford M. Neal 5.1 Introduction Markov chain Monte Carlo (MCMC) originated with the classic paper of Metropolis et al. (1953), where it was used to simulate the distribution

### The Classes P and NP

The Classes P and NP We now shift gears slightly and restrict our attention to the examination of two families of problems which are very important to computer scientists. These families constitute the

### Branislav K. Nikolić

Monte Carlo Simulations in Statistical Physics: Magnetic PhaseTransitions in the Ising Model Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware, U.S.A. PHYS 460/660: Computational

### Can linear programs solve NP-hard problems?

Can linear programs solve NP-hard problems? p. 1/9 Can linear programs solve NP-hard problems? Ronald de Wolf Linear programs Can linear programs solve NP-hard problems? p. 2/9 Can linear programs solve

### Consider a problem in which we are given a speed function

Fast marching methods for the continuous traveling salesman problem June Andrews and J. A. Sethian* Department of Mathematics, University of California, Berkeley, CA 94720 Communicated by Alexandre J.

### 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

### Mathematics for Algorithm and System Analysis

Mathematics for Algorithm and System Analysis for students of computer and computational science Edward A. Bender S. Gill Williamson c Edward A. Bender & S. Gill Williamson 2005. All rights reserved. Preface