7.8 Intractability. Overview. Properties of Algorithms. Exponential Growth. Q. What is an algorithm? A. Definition formalized using Turing machines.
|
|
- Samuel Bradley
- 5 years ago
- Views:
Transcription
1 Overview 7.8 Intractability Q. What is an algorithm? A. Definition formalized using Turing machines. Q. Which problems can be solved on a computer? A. Computability. Q. Which algorithms will be useful in practice? A. Analysis of algorithms. Q. Which problems can be solved in practice? A. Intractability. Introducti on to Computer Sci ence Robert Sedgew i ck and Kevi n Wayne Copyri ght w w.cs.pri nceton.edu/introcs 2 Properties of Algorithms Exponential Growth Q. Which algorithms are useful in practice? Exponential growth dwarfs technological change. Suppose each electron in the universe had power of today's supercomputers And each works for the life of the universe in an effort to solve one TSP problem via brute force. A working definition. [von Neumann 1953, Godel 1956, Cobham 1960, Edmonds 1962] Measure running time as a function of input size N. Efficient = polynomial time for all inputs. Inefficient = "exponential time" for some inputs. Quantity Ex. Dynamic programming algorithm for edit distance takes N2 steps. Ex. Brute force algorithm for TSP takes N steps Age of universe in seconds 1017 Electrons in universe Theory. Definition is broad and robust; huge gulf between polynomial and exponential algorithms. Number Supercomputer instructions per second 1079 Estimated Will not help solve 1,000 city TSP problem via brute force >> >> 1079 " 1017 " 1013 Practice. Exponents and constants of polynomials that arise are small scales to huge problems. 3 4
2 P Extended Church-Turing Thesis Def. P is the set of all yes-no problems solvable in poly-time on a deterministic Turing machine. Problem Description Algorithm Yes No Extended Church-Turing thesis. P = yes-no problems solvable in poly-time in this universe. If computable by a piece of hardware in time T(N) for input of size N, then computable by TM in time (T(N)) k for some constant k. RELPRIME COMPOSITE Are x and y relatively prime? Does x have a factor other than 1 and itself? Euclid (300 BCE) Agarwal-Kayal- Saxena (2002) 34, , Evidence supporting thesis. True for all physical computers. k = 2 for random access machines. [your laptop] EDIT- DISTANCE LSOLVE Is the edit distance between strings x and y less than 5? Is there a vector x that satisfies Ax = b? Dynamic Programming Gauss-Edmonds elimination niether neither # 0 1 1& % ( 2 4 "2 % ( $ % ' (, # 4& % ( % 2 ( $ % 36' ( acgggt ttttta " 1 0 0% $ ' $ ' # $ 0 1 1& ', " 1% $ ' $ 1 ' # $ 1& ' Implication. To make future computers more efficient, only need to focus on improving implementation of existing designs. Law of physics? A new constraint on what is possible? Remark. Algorithm typically also solves the related search problem. Possible counterexample? Quantum computers. like 2nd law of thermodynamics Satisfiability Intractability of Literal. A Boolean variable or its negation. Clause. A disjunction of 3 distinct literals. x i or x i C j = x 1 " x 2 " x 3 Q. How to solve an instance of. A1. Try all 2 n truth assignments. A2. Do something substantially more clever??? Conjunctive normal form. A propositional formula # that is the conjunction of clauses. " = C 1 # C 2 # C 3 # C 4 # C 5 Conjecture. No poly-time algorithm for. "intractable". Given a CNF formula # consisting of k clauses over n literals, does it have a satisfying truth assignment? ( x 1 " x 2 " x 3 ) # ( x 1 " x 2 " x 3 ) # ( x 1 " x 2 " x 3 ) # ( x 1 " x 2 " x 4 ) # ( x 2 " x 3 " x 4 ) x 1 = true, x 2 = true, x 3 = false, x 4 = true a satisfiable formula with 5 clauses and 4 variables 7 8
3 Reductions Planar 3-Color Q. Which problems won't we be able to solve in practice? A. No easy answers, but theory helps. Given a planar map, can it be colored using 3 colors so that no adjacent regions have the same color? Def. Problem X polynomial reduces to problem Y if arbitrary instances of problem X can be solved using: Polynomial number of standard computational steps, plus One call to subroutine for Y as last step. Consequence. If no poly-time algorithm for X, then no poly-time algorithm for Y. your research problem YES instance Planar 3-Color Given a planar map, can it be colored using 3 colors so that no adjacent regions have the same color?. Given a graph (need not be planar), is there a way to color the vertices red, green, and blue so that no adjacent vertices have the same color? yes instance NO instance. Applications. Register allocation, Potts model in physics, 11 12
4 . Given a graph, is there a way to color the vertices red, green, and blue so that no adjacent vertices have the same color? Claim. polynomial reduces to. Pf. Given instance #, we construct an instance of that is 3-colorable iff # is satisfiable. Construction. i. Create one vertex for each literal. ii. Create 3 new vertices T, F, and B; connect them in a triangle, and connect each literal to B. iii. Connect each literal to its negation. iv. For each clause, attach a gadget of 6 vertices and 13 edges. to be described next Claim. Graph is 3-colorable iff # is satisfiable. Claim. Graph is 3-colorable iff # is satisfiable. Pf. Suppose graph is 3-colorable. Consider assignment that sets all T-colored literals to true. (ii) ensures each literal is same color as T or F. (iii) ensures a literal and its negation are opposites. Pf. Suppose graph is 3-colorable. Consider assignment that sets all T-colored literals to true. (ii) ensures each literal is same color as T or F. (iii) ensures a literal and its negation are opposites. (iv) ensures at least one literal in each clause is same color as T. true T false F B B base x 1 x 2 x 3 C i = x 1 V x 2 V x 3 6-node gadget x 1 x x 1 2 x 2 x 3 x 3 x n x n true T F false 15 16
5 Claim. Graph is 3-colorable iff # is satisfiable. Claim. Graph is 3-colorable iff # is satisfiable. Pf. Suppose graph is 3-colorable. Consider assignment that sets all T-colored literals to true. (ii) ensures each literal is same color as T or F. (iii) ensures a literal and its negation are opposites. (iv) ensures at least one literal in each clause is same color as T. Pf. $ Suppose formula # is satisfiable. Color all true literals same color as T. Color vertex below green node F, and vertex below that B. Color remaining middle row vertices B. Color remaining bottom vertices T or F as forced. B not 3-colorable if all are red B a literal set to true in assignment x 1 x 2 x 3 C i = x 1 V x 2 V x 3 x 1 x 2 x 3 C i = x 1 V x 2 V x 3 6-node gadget 6-node gadget contradiction true T F false true T F false More Poly-Time Reductions Still More Hard Computational Problems reduces to Aerospace engineering. Optimal mesh partitioning for finite elements. Biology. Protein folding. Chemical engineering. Heat exchanger network synthesis. Chemistry. Chemical synthesizability. 3DM VERTEX COVER Dick Karp Turing award (1985) Civil engineering. Equilibrium of urban traffic flow. Economics. Computation of arbitrage in financial markets with friction. Electrical engineering. VLSI layout. Environmental engineering. Optimal placement of contaminant sensors. EXACT COVER PLANAR- CLIQUE HAM-CYCLE Financial engineering. Find minimum risk portfolio of given return. Game theory. Find Nash equilibrium that maximizes social welfare. Genomics. Phylogeny reconstruction. Mathematics. Given integer a 1,, a n, compute SUBSET-SUM INDEPENDENT SET TSP HAM-PATH Mechanical engineering. Structure of turbulence in sheared flows. Medicine. Reconstructing 3-D shape from biplane angiocardiogram. Operations research. Traveling salesperson problem. Physics. Partition function of 3-D Ising model in statistical mechanics. PARTITION INTEGER PROGRAMMING Politics. Shapley-Shubik voting power. Pop culture. Minesweeper consistency. Statistics. Optimal experimental design. KNAPSACK BIN-PACKING Conjecture: no poly-time algorithm for. (and hence none of these problems) 6,000+ scientific papers per year
6 Implications of Intractability Cook's Theorem Proving a problem intractable guide scientific inquiry. 1926: Ising introduces simple model for phase transitions. 1944: Onsager find closed form solution to 2D case in tour de force. 19xx: Feynman and other top minds seek 3D solution. a holy grail of statistical physics reduces to 3DM VERTEX COVER Stephen Cook Turing award (1982) 2000: Istrail reduces to 3D version of problem. EXACT COVER PLANAR- CLIQUE HAM-CYCLE search for closed formula appears doomed SUBSET-SUM INDEPENDENT SET TSP HAM-PATH PARTITION INTEGER PROGRAMMING KNAPSACK BIN-PACKING All of these problems (any many more) polynomial reduce to Cook + Karp Complexity Classes reduces to reduces to 3DM VERTEX COVER P. Set of yes-no problems solvable in poly-time on a deterministic TM. EXP. Same as P, but in exponential-time. NP. Same as P, but on non-deterministic Turing machine. Cook-Levin Theorem (1960s). ALL NP problems reduce to. EXACT COVER PLANAR- CLIQUE HAM-CYCLE if we can solve, we can solve any of them NP-complete. All NP problems to which reduces. SUBSET-SUM PARTITION INTEGER PROGRAMMING INDEPENDENT SET TSP HAM-PATH Implications. If efficient algorithm for, then P = NP. if we can solve any of them, we can solve If efficient algorithm for any NP-complete problem, then P = NP. If no efficient algorithm for some NP problem, then none for. KNAPSACK BIN-PACKING All of these problems are different manifestations of one "really hard" problem
7 Certificates Certificates Alternate (and Equivalent) Definition. NP is set of all yes-no problems for which you can check in poly-time that it is a yes instance (given a certificate).. Given a graph, is there a way to color the vertices red, green, and blue so that no adjacent vertices have the same color? Alt. Def. NP is set of all yes-no problems for which you can check in poly-time that it is a yes instance (given a certificate). FACTOR. Given two integers x and y, does x have a factor between 2 and y? x = 2773 y = 50 x = y = instance instance certificate certificate instance certificate The Main Question Implications of NP-Completeness Does P = NP? Is there a poly-time algorithm for SAT? Does nondeterminism help you solve problems faster? Clay $1 million prize. Classify problems according to their computational requirements. NP-complete: SAT, all Karp problems, thousands more. P: RELPRIME, COMPOSITE, LSOLVE. Unclassified: FACTOR is in NP, but unknown if NP-complete or in P. Jack Edmonds, 1962 Computational universality. EXP P NP NPcomplete EXP P = NP All known algorithms for NP-complete problems are exponential. If any NP-complete problem proved exponential, so are rest. If any NP-complete problem proved polynomial, so are rest. If P % NP If P = NP If yes: Efficient algorithms for, TSP, FACTOR,... If no: No efficient algorithms possible for, TSP,... Consensus opinion on P = NP? Probably no. would break modern cryptography and collapse economy Proving a problem is NP-complete can guide scientific inquiry. 1926: Ising introduces simple model for phase transitions. 1944: Onsager solves 2D case in tour de force. 19xx: Feynman and other top minds seek 3D solution. 2000: Istrail proves 3D problem NP-complete
8 Coping With Intractability Coping With Intractability Relax one of desired features. Solve the problem in poly-time. Solve the problem to optimality. Solve arbitrary instances of the problem. Relax one of desired features. Solve the problem in poly-time. Solve the problem to optimality. Solve arbitrary instances of the problem. Complexity theory deals with worst case behavior. Instance(s) you want to solve may be "easy." Concorde algorithm solved 13,509 US city TSP problem. Develop a heuristic, and hope it produces a good solution. No guarantees on quality of solution. Ex: TSP assignment heuristics. Ex: Metropolis algorithm, simulating annealing, genetic algorithms. Design an approximation algorithm. Guarantees to find a nearly-optimal solution. Ex: Euclidean TSP tour guaranteed to be within 1% of optimal. Active area of research, but not always possible Sanjeev Arora (1997) Coping With Intractability Summary Relax one of desired features. Solve the problem in poly-time. Solve the problem to optimality. Solve arbitrary instances of the problem. Exploit intractability. Cryptography. [stay tuned] Keep trying to prove P = NP. can do any 2 of 3 Many fundamental problems are intractable. TSP,,. 3D Ising model. Theory says: we probably won't be able to design efficient algorithms for them. You will encounter such problems in your scientific lives. If you know about intractability, you can identify them and avoid wasting time and energy
7.8: Intractability. Overview. Exponential Growth. Properties of Algorithms. What is an algorithm? Turing machine.
Overview 7.8: Intractability What is an algorithm? Turing machine. Which problems can be solved on a computer? Computability. Which ALGORITHMS will be useful in practice? Analysis of algorithms. Which
More information10.3: Intractability. Overview. Exponential Growth. Properties of Algorithms. What is an algorithm? Turing machine.
Overview 10.3: Intractability What is an algorithm? Turing machine. What problems can be solved on a computer? Computability. What ALGORITHMS will be useful in practice? Analysis of algorithms. Which PROBLEMS
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 31 P and NP Self-reducibility NP-completeness Adam Smith 12/1/2008 S. Raskhodnikova; based on slides by K. Wayne Central ideas we ll cover Poly-time as feasible most
More information4/22/12. NP and NP completeness. Efficient Certification. Decision Problems. Definition of P
Efficient Certification and completeness There is a big difference between FINDING a solution and CHECKING a solution Independent set problem: in graph G, is there an independent set S of size at least
More information4/19/11. NP and NP completeness. Decision Problems. Definition of P. Certifiers and Certificates: COMPOSITES
Decision Problems NP and NP completeness Identify a decision problem with a set of binary strings X Instance: string s. Algorithm A solves problem X: As) = yes iff s X. Polynomial time. Algorithm A runs
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURES 30-31 NP-completeness Definition NP-completeness proof for CIRCUIT-SAT Adam Smith 11/3/10 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova,
More informationA difficult problem. ! Given: A set of N cities and $M for gas. Problem: Does a traveling salesperson have enough $ for gas to visit all the cities?
Intractability A difficult problem Traveling salesperson problem (TSP) Given: A set of N cities and $M for gas. Problem: Does a traveling salesperson have enough $ for gas to visit all the cities? An algorithm
More informationIntractability. A difficult problem. Exponential Growth. A Reasonable Question about Algorithms !!!!!!!!!! Traveling salesperson problem (TSP)
A difficult problem Intractability A Reasonable Question about Algorithms Q. Which algorithms are useful in practice? A. [von Neumann 1953, Gödel 1956, Cobham 1964, Edmonds 1965, Rabin 1966] Model of computation
More information8.1 Polynomial-Time Reductions. Chapter 8. NP and Computational Intractability. Classify Problems
Chapter 8 8.1 Polynomial-Time Reductions NP and Computational Intractability Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 Classify Problems According to Computational
More informationNP and Computational Intractability
NP and Computational Intractability 1 Review Basic reduction strategies. Simple equivalence: INDEPENDENT-SET P VERTEX-COVER. Special case to general case: VERTEX-COVER P SET-COVER. Encoding with gadgets:
More informationCS 580: Algorithm Design and Analysis. Jeremiah Blocki Purdue University Spring 2018
CS 580: Algorithm Design and Analysis Jeremiah Blocki Purdue University Spring 2018 Recap Polynomial Time Reductions (X P Y ) View 1: A polynomial time algorithm for Y yields a polynomial time algorithm
More informationCS 580: Algorithm Design and Analysis. Jeremiah Blocki Purdue University Spring 2018
CS 580: Algorithm Design and Analysis Jeremiah Blocki Purdue University Spring 2018 Recap Polynomial Time Reductions (X P Y ) Key Problems Independent Set, Vertex Cover, Set Cover, 3-SAT etc Example Reductions
More information3/22/2018. CS 580: Algorithm Design and Analysis. 8.3 Definition of NP. Chapter 8. NP and Computational Intractability. Decision Problems.
CS 580: Algorithm Design and Analysis 8.3 Definition of NP Jeremiah Blocki Purdue University Spring 208 Recap Decision Problems Polynomial Time Reductions (X P Y ) Key Problems Independent Set, Vertex
More informationCS 580: Algorithm Design and Analysis
CS 580: Algorithm Design and Analysis Jeremiah Blocki Purdue University Spring 2018 Homework 5 due on March 29 th at 11:59 PM (on Blackboard) Recap Polynomial Time Reductions (X P Y ) P Decision problems
More information3/22/2018. CS 580: Algorithm Design and Analysis. Circuit Satisfiability. Recap. The "First" NP-Complete Problem. Example.
Circuit Satisfiability CS 580: Algorithm Design and Analysis CIRCUIT-SAT. Given a combinational circuit built out of AND, OR, and NOT gates, is there a way to set the circuit inputs so that the output
More informationCOP 4531 Complexity & Analysis of Data Structures & Algorithms
COP 4531 Complexity & Analysis of Data Structures & Algorithms Lecture 18 Reductions and NP-completeness Thanks to Kevin Wayne and the text authors who contributed to these slides Classify Problems According
More informationChapter 8. NP and Computational Intractability. CS 350 Winter 2018
Chapter 8 NP and Computational Intractability CS 350 Winter 2018 1 Algorithm Design Patterns and Anti-Patterns Algorithm design patterns. Greedy. Divide-and-conquer. Dynamic programming. Duality. Reductions.
More informationAnnouncements. Analysis of Algorithms
Announcements Analysis of Algorithms Piyush Kumar (Lecture 9: NP Completeness) Welcome to COP 4531 Based on Kevin Wayne s slides Programming Assignment due: April 25 th Submission: email your project.tar.gz
More informationPolynomial-Time Reductions
Reductions 1 Polynomial-Time Reductions Classify Problems According to Computational Requirements Q. Which problems will we be able to solve in practice? A working definition. [von Neumann 1953, Godel
More information8. INTRACTABILITY I. Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley. Last updated on 2/6/18 2:16 AM
8. INTRACTABILITY I poly-time reductions packing and covering problems constraint satisfaction problems sequencing problems partitioning problems graph coloring numerical problems Lecture slides by Kevin
More informationIntro to Theory of Computation
Intro to Theory of Computation LECTURE 25 Last time Class NP Today Polynomial-time reductions Adam Smith; Sofya Raskhodnikova 4/18/2016 L25.1 The classes P and NP P is the class of languages decidable
More informationAlgorithms Design & Analysis. Approximation Algorithm
Algorithms Design & Analysis Approximation Algorithm Recap External memory model Merge sort Distribution sort 2 Today s Topics Hard problem Approximation algorithms Metric traveling salesman problem A
More informationCOMPUTER SCIENCE. Computer Science. 16. Intractability. Computer Science. An Interdisciplinary Approach. Section 7.4.
COMPUTER SCIENCE S E D G E W I C K / W A Y N E PA R T I I : A L G O R I T H M S, M A C H I N E S, a n d T H E O R Y Computer Science Computer Science An Interdisciplinary Approach Section 7.4 ROBERT SEDGEWICK
More informationApproximation and Randomized Algorithms (ARA) Lecture 2, September 1, 2010
Approximation and Randomized Algorithms (ARA) Lecture 2, September 1, 2010 Last time Algorithm Revision Algorithms for the stable matching problem Five illustrative algorithm problems Computatibility Today
More informationNP and Computational Intractability
NP and Computational Intractability 1 Polynomial-Time Reduction Desiderata'. Suppose we could solve X in polynomial-time. What else could we solve in polynomial time? don't confuse with reduces from Reduction.
More informationComputer Science. 16. Intractability. 16. Intractability. Computer Science. Reasonable questions. P and NP Poly-time reductions NP-completeness
PA R T I I : A L G O R I T H M S, M A C H I N E S, a n d T H E O R Y PA R T I I : A L G O R I T H M S, M A C H I N E S, a n d T H E O R Y Computer Science 6. Intractability Computer Science Reasonable
More informationCS 583: Algorithms. NP Completeness Ch 34. Intractability
CS 583: Algorithms NP Completeness Ch 34 Intractability Some problems are intractable: as they grow large, we are unable to solve them in reasonable time What constitutes reasonable time? Standard working
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 26 Computational Intractability Polynomial Time Reductions Sofya Raskhodnikova S. Raskhodnikova; based on slides by A. Smith and K. Wayne L26.1 What algorithms are
More informationIntro to Theory of Computation
Intro to Theory of Computation LECTURE 24 Last time Relationship between models: deterministic/nondeterministic Class P Today Class NP Sofya Raskhodnikova Homework 9 due Homework 0 out 4/5/206 L24. I-clicker
More informationLecture 4: NP and computational intractability
Chapter 4 Lecture 4: NP and computational intractability Listen to: Find the longest path, Daniel Barret What do we do today: polynomial time reduction NP, co-np and NP complete problems some examples
More informationHarvard CS 121 and CSCI E-121 Lecture 22: The P vs. NP Question and NP-completeness
Harvard CS 121 and CSCI E-121 Lecture 22: The P vs. NP Question and NP-completeness Harry Lewis November 19, 2013 Reading: Sipser 7.4, 7.5. For culture : Computers and Intractability: A Guide to the Theory
More informationCS 301: Complexity of Algorithms (Term I 2008) Alex Tiskin Harald Räcke. Hamiltonian Cycle. 8.5 Sequencing Problems. Directed Hamiltonian Cycle
8.5 Sequencing Problems Basic genres. Packing problems: SET-PACKING, INDEPENDENT SET. Covering problems: SET-COVER, VERTEX-COVER. Constraint satisfaction problems: SAT, 3-SAT. Sequencing problems: HAMILTONIAN-CYCLE,
More informationEasy Problems vs. Hard Problems. CSE 421 Introduction to Algorithms Winter Is P a good definition of efficient? The class P
Easy Problems vs. Hard Problems CSE 421 Introduction to Algorithms Winter 2000 NP-Completeness (Chapter 11) Easy - problems whose worst case running time is bounded by some polynomial in the size of the
More information4/12/2011. Chapter 8. NP and Computational Intractability. Directed Hamiltonian Cycle. Traveling Salesman Problem. Directed Hamiltonian Cycle
Directed Hamiltonian Cycle Chapter 8 NP and Computational Intractability Claim. G has a Hamiltonian cycle iff G' does. Pf. Suppose G has a directed Hamiltonian cycle Γ. Then G' has an undirected Hamiltonian
More informationNP completeness and computational tractability Part II
Grand challenge: Classify Problems According to Computational Requirements NP completeness and computational tractability Part II Some Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All
More informationSAT, Coloring, Hamiltonian Cycle, TSP
1 SAT, Coloring, Hamiltonian Cycle, TSP Slides by Carl Kingsford Apr. 28, 2014 Sects. 8.2, 8.7, 8.5 2 Boolean Formulas Boolean Formulas: Variables: x 1, x 2, x 3 (can be either true or false) Terms: t
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 8.5 Sequencing Problems Basic genres.! Packing problems: SET-PACKING,
More information8.5 Sequencing Problems
8.5 Sequencing Problems Basic genres. Packing problems: SET-PACKING, INDEPENDENT SET. Covering problems: SET-COVER, VERTEX-COVER. Constraint satisfaction problems: SAT, 3-SAT. Sequencing problems: HAMILTONIAN-CYCLE,
More informationSAT, NP, NP-Completeness
CS 473: Algorithms, Spring 2018 SAT, NP, NP-Completeness Lecture 22 April 13, 2018 Most slides are courtesy Prof. Chekuri Ruta (UIUC) CS473 1 Spring 2018 1 / 57 Part I Reductions Continued Ruta (UIUC)
More informationNP Complete Problems. COMP 215 Lecture 20
NP Complete Problems COMP 215 Lecture 20 Complexity Theory Complexity theory is a research area unto itself. The central project is classifying problems as either tractable or intractable. Tractable Worst
More informationP and NP. Warm Up: Super Hard Problems. Overview. Problem Classification. Tools for classifying problems according to relative hardness.
Overview Problem classification Tractable Intractable P and NP Reductions Tools for classifying problems according to relative hardness Inge Li Gørtz Thank you to Kevin Wayne, Philip Bille and Paul Fischer
More information4/20/11. NP-complete problems. A variety of NP-complete problems. Hamiltonian Cycle. Hamiltonian Cycle. Directed Hamiltonian Cycle
A variety of NP-complete problems NP-complete problems asic genres. Packing problems: SE-PACKING, INDEPENDEN SE. Covering problems: SE-COVER, VEREX-COVER. Constraint satisfaction problems: SA, 3-SA. Sequencing
More informationChapter 8. NP and Computational Intractability
Chapter 8 NP and Computational Intractability Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. Acknowledgement: This lecture slide is revised and authorized from Prof.
More informationAlgorithms and Theory of Computation. Lecture 22: NP-Completeness (2)
Algorithms and Theory of Computation Lecture 22: NP-Completeness (2) Xiaohui Bei MAS 714 November 8, 2018 Nanyang Technological University MAS 714 November 8, 2018 1 / 20 Set Cover Set Cover Input: a set
More information8.5 Sequencing Problems. Chapter 8. NP and Computational Intractability. Hamiltonian Cycle. Hamiltonian Cycle
Chapter 8 NP and Computational Intractability 8.5 Sequencing Problems Basic genres. Packing problems: SET-PACKING, INDEPENDENT SET. Covering problems: SET-COVER, VERTEX-COVER. Constraint satisfaction problems:
More informationP and NP. Inge Li Gørtz. Thank you to Kevin Wayne, Philip Bille and Paul Fischer for inspiration to slides
P and NP Inge Li Gørtz Thank you to Kevin Wayne, Philip Bille and Paul Fischer for inspiration to slides 1 Overview Problem classification Tractable Intractable Reductions Tools for classifying problems
More informationReductions. Reduction. Linear Time Reduction: Examples. Linear Time Reductions
Reduction Reductions Problem X reduces to problem Y if given a subroutine for Y, can solve X. Cost of solving X = cost of solving Y + cost of reduction. May call subroutine for Y more than once. Ex: X
More informationTheory of Computation CS3102 Spring 2014 A tale of computers, math, problem solving, life, love and tragic death
Theory of Computation CS3102 Spring 2014 A tale of computers, math, problem solving, life, love and tragic death Nathan Brunelle Department of Computer Science University of Virginia www.cs.virginia.edu/~njb2b/theory
More informationChapter 2. Reductions and NP. 2.1 Reductions Continued The Satisfiability Problem (SAT) SAT 3SAT. CS 573: Algorithms, Fall 2013 August 29, 2013
Chapter 2 Reductions and NP CS 573: Algorithms, Fall 2013 August 29, 2013 2.1 Reductions Continued 2.1.1 The Satisfiability Problem SAT 2.1.1.1 Propositional Formulas Definition 2.1.1. Consider a set of
More informationAlgorithms 6.5 REDUCTIONS. designing algorithms establishing lower bounds classifying problems intractability
6.5 REDUCTIONS Algorithms F O U R T H E D I T I O N designing algorithms establishing lower bounds classifying problems intractability R O B E R T S E D G E W I C K K E V I N W A Y N E Algorithms, 4 th
More informationComputational Intractability 2010/4/15. Lecture 2
Computational Intractability 2010/4/15 Professor: David Avis Lecture 2 Scribe:Naoki Hatta 1 P and NP 1.1 Definition of P and NP Decision problem it requires yes/no answer. Example: X is a set of strings.
More information1. Introduction Recap
1. Introduction Recap 1. Tractable and intractable problems polynomial-boundness: O(n k ) 2. NP-complete problems informal definition 3. Examples of P vs. NP difference may appear only slightly 4. Optimization
More informationSome Algebra Problems (Algorithmic) CSE 417 Introduction to Algorithms Winter Some Problems. A Brief History of Ideas
Some Algebra Problems (Algorithmic) CSE 417 Introduction to Algorithms Winter 2006 NP-Completeness (Chapter 8) Given positive integers a, b, c Question 1: does there exist a positive integer x such that
More informationCOMP Analysis of Algorithms & Data Structures
COMP 3170 - Analysis of Algorithms & Data Structures Shahin Kamali Computational Complexity CLRS 34.1-34.4 University of Manitoba COMP 3170 - Analysis of Algorithms & Data Structures 1 / 50 Polynomial
More informationCSC 373: Algorithm Design and Analysis Lecture 15
CSC 373: Algorithm Design and Analysis Lecture 15 Allan Borodin February 13, 2013 Some materials are from Stephen Cook s IIT talk and Keven Wayne s slides. 1 / 21 Announcements and Outline Announcements
More informationCS/COE
CS/COE 1501 www.cs.pitt.edu/~nlf4/cs1501/ P vs NP But first, something completely different... Some computational problems are unsolvable No algorithm can be written that will always produce the correct
More informationCorrectness of Dijkstra s algorithm
Correctness of Dijkstra s algorithm Invariant: When vertex u is deleted from the priority queue, d[u] is the correct length of the shortest path from the source s to vertex u. Additionally, the value d[u]
More informationSummer School on Introduction to Algorithms and Optimization Techniques July 4-12, 2017 Organized by ACMU, ISI and IEEE CEDA.
Summer School on Introduction to Algorithms and Optimization Techniques July 4-12, 2017 Organized by ACMU, ISI and IEEE CEDA NP Completeness Susmita Sur-Kolay Advanced Computing and Microelectronics Unit
More informationNP-Completeness. NP-Completeness 1
NP-Completeness Reference: Computers and Intractability: A Guide to the Theory of NP-Completeness by Garey and Johnson, W.H. Freeman and Company, 1979. NP-Completeness 1 General Problems, Input Size and
More informationCS 580: Algorithm Design and Analysis
CS 580: Algorithm Design and Analysis Jeremiah Blocki Purdue University Spring 2018 Homework 5 due tonight at 11:59 PM (on Blackboard) Midterm 2 on April 4 th at 8PM (MATH 175) Practice Midterm Released
More informationNP-Complete Reductions 1
x x x 2 x 2 x 3 x 3 x 4 x 4 CS 4407 2 22 32 Algorithms 3 2 23 3 33 NP-Complete Reductions Prof. Gregory Provan Department of Computer Science University College Cork Lecture Outline x x x 2 x 2 x 3 x 3
More informationLimitations of Algorithm Power
Limitations of Algorithm Power Objectives We now move into the third and final major theme for this course. 1. Tools for analyzing algorithms. 2. Design strategies for designing algorithms. 3. Identifying
More informationNP Completeness and Approximation Algorithms
Winter School on Optimization Techniques December 15-20, 2016 Organized by ACMU, ISI and IEEE CEDA NP Completeness and Approximation Algorithms Susmita Sur-Kolay Advanced Computing and Microelectronic
More informationChapter 11. Approximation Algorithms. Slides by Kevin Wayne Pearson-Addison Wesley. All rights reserved.
Chapter 11 Approximation Algorithms Slides by Kevin Wayne. Copyright @ 2005 Pearson-Addison Wesley. All rights reserved. 1 P and NP P: The family of problems that can be solved quickly in polynomial time.
More information4/30/14. Chapter Sequencing Problems. NP and Computational Intractability. Hamiltonian Cycle
Chapter 8 NP and Computational Intractability Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 2 Hamiltonian Cycle 8.5 Sequencing Problems HAM-CYCLE: given an undirected
More informationComputational Complexity and Intractability: An Introduction to the Theory of NP. Chapter 9
1 Computational Complexity and Intractability: An Introduction to the Theory of NP Chapter 9 2 Objectives Classify problems as tractable or intractable Define decision problems Define the class P Define
More information6.080 / Great Ideas in Theoretical Computer Science Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationSome Algebra Problems (Algorithmic) CSE 417 Introduction to Algorithms Winter Some Problems. A Brief History of Ideas
CSE 417 Introduction to Algorithms Winter 2007 Some Algebra Problems (Algorithmic) Given positive integers a, b, c Question 1: does there exist a positive integer x such that ax = c? NP-Completeness (Chapter
More informationIntroduction to Complexity Theory
Introduction to Complexity Theory Read K & S Chapter 6. Most computational problems you will face your life are solvable (decidable). We have yet to address whether a problem is easy or hard. Complexity
More informationLecture 19: Finish NP-Completeness, conp and Friends
6.045 Lecture 19: Finish NP-Completeness, conp and Friends 1 Polynomial Time Reducibility f : Σ* Σ* is a polynomial time computable function if there is a poly-time Turing machine M that on every input
More informationNP-COMPLETE PROBLEMS. 1. Characterizing NP. Proof
T-79.5103 / Autumn 2006 NP-complete problems 1 NP-COMPLETE PROBLEMS Characterizing NP Variants of satisfiability Graph-theoretic problems Coloring problems Sets and numbers Pseudopolynomial algorithms
More informationCHAPTER 3 FUNDAMENTALS OF COMPUTATIONAL COMPLEXITY. E. Amaldi Foundations of Operations Research Politecnico di Milano 1
CHAPTER 3 FUNDAMENTALS OF COMPUTATIONAL COMPLEXITY E. Amaldi Foundations of Operations Research Politecnico di Milano 1 Goal: Evaluate the computational requirements (this course s focus: time) to solve
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION "Winter" 2018 http://cseweb.ucsd.edu/classes/wi18/cse105-ab/ Today's learning goals Sipser Ch 7 Define NP-completeness Give examples of NP-complete problems Use polynomial-time
More informationNP-Completeness. Andreas Klappenecker. [based on slides by Prof. Welch]
NP-Completeness Andreas Klappenecker [based on slides by Prof. Welch] 1 Prelude: Informal Discussion (Incidentally, we will never get very formal in this course) 2 Polynomial Time Algorithms Most of the
More informationEssential facts about NP-completeness:
CMPSCI611: NP Completeness Lecture 17 Essential facts about NP-completeness: Any NP-complete problem can be solved by a simple, but exponentially slow algorithm. We don t have polynomial-time solutions
More informationMore NP-Complete Problems
CS 473: Algorithms, Spring 2018 More NP-Complete Problems Lecture 23 April 17, 2018 Most slides are courtesy Prof. Chekuri Ruta (UIUC) CS473 1 Spring 2018 1 / 57 Recap NP: languages/problems that have
More informationApplied Computer Science II Chapter 7: Time Complexity. Prof. Dr. Luc De Raedt. Institut für Informatik Albert-Ludwigs Universität Freiburg Germany
Applied Computer Science II Chapter 7: Time Complexity Prof. Dr. Luc De Raedt Institut für Informati Albert-Ludwigs Universität Freiburg Germany Overview Measuring complexity The class P The class NP NP-completeness
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 informationComplexity, P and NP
Complexity, P and NP EECS 477 Lecture 21, 11/26/2002 Last week Lower bound arguments Information theoretic (12.2) Decision trees (sorting) Adversary arguments (12.3) Maximum of an array Graph connectivity
More informationNP-complete problems. CSE 101: Design and Analysis of Algorithms Lecture 20
NP-complete problems CSE 101: Design and Analysis of Algorithms Lecture 20 CSE 101: Design and analysis of algorithms NP-complete problems Reading: Chapter 8 Homework 7 is due today, 11:59 PM Tomorrow
More informationCSC373: Algorithm Design, Analysis and Complexity Fall 2017
CSC373: Algorithm Design, Analysis and Complexity Fall 2017 Allan Borodin October 25, 2017 1 / 36 Week 7 : Annoucements We have been grading the test and hopefully they will be available today. Term test
More informationDesign and Analysis of Algorithms
Design and Analysis of Algorithms CSE 5311 Lecture 25 NP Completeness Junzhou Huang, Ph.D. Department of Computer Science and Engineering CSE5311 Design and Analysis of Algorithms 1 NP-Completeness Some
More informationCS154, Lecture 13: P vs NP
CS154, Lecture 13: P vs NP The EXTENDED Church-Turing Thesis Everyone s Intuitive Notion of Efficient Algorithms Polynomial-Time Turing Machines More generally: TM can simulate every reasonable model of
More informationChapter 34: NP-Completeness
Graph Algorithms - Spring 2011 Set 17. Lecturer: Huilan Chang Reference: Cormen, Leiserson, Rivest, and Stein, Introduction to Algorithms, 2nd Edition, The MIT Press. Chapter 34: NP-Completeness 2. Polynomial-time
More informationNP-Complete Problems. More reductions
NP-Complete Problems More reductions Definitions P: problems that can be solved in polynomial time (typically in n, size of input) on a deterministic Turing machine Any normal computer simulates a DTM
More informationComputability and Complexity
Computability and Complexity Complexity and Turing Machines. P vs NP Problem CAS 705 Ryszard Janicki Department of Computing and Software McMaster University Hamilton, Ontario, Canada janicki@mcmaster.ca
More informationNP and NP Completeness
CS 374: Algorithms & Models of Computation, Spring 2017 NP and NP Completeness Lecture 23 April 20, 2017 Chandra Chekuri (UIUC) CS374 1 Spring 2017 1 / 44 Part I NP Chandra Chekuri (UIUC) CS374 2 Spring
More informationPolynomial-time Reductions
Polynomial-time Reductions Disclaimer: Many denitions in these slides should be taken as the intuitive meaning, as the precise meaning of some of the terms are hard to pin down without introducing the
More informationAnnouncements. Friday Four Square! Problem Set 8 due right now. Problem Set 9 out, due next Friday at 2:15PM. Did you lose a phone in my office?
N P NP Completeness Announcements Friday Four Square! Today at 4:15PM, outside Gates. Problem Set 8 due right now. Problem Set 9 out, due next Friday at 2:15PM. Explore P, NP, and their connection. Did
More informationAlgorithms. NP -Complete Problems. Dong Kyue Kim Hanyang University
Algorithms NP -Complete Problems Dong Kyue Kim Hanyang University dqkim@hanyang.ac.kr The Class P Definition 13.2 Polynomially bounded An algorithm is said to be polynomially bounded if its worst-case
More informationP P P NP-Hard: L is NP-hard if for all L NP, L L. Thus, if we could solve L in polynomial. Cook's Theorem and Reductions
Summary of the previous lecture Recall that we mentioned the following topics: P: is the set of decision problems (or languages) that are solvable in polynomial time. NP: is the set of decision problems
More informationUndecidable Problems. Z. Sawa (TU Ostrava) Introd. to Theoretical Computer Science May 12, / 65
Undecidable Problems Z. Sawa (TU Ostrava) Introd. to Theoretical Computer Science May 12, 2018 1/ 65 Algorithmically Solvable Problems Let us assume we have a problem P. If there is an algorithm solving
More informationNP Completeness. CS 374: Algorithms & Models of Computation, Spring Lecture 23. November 19, 2015
CS 374: Algorithms & Models of Computation, Spring 2015 NP Completeness Lecture 23 November 19, 2015 Chandra & Lenny (UIUC) CS374 1 Spring 2015 1 / 37 Part I NP-Completeness Chandra & Lenny (UIUC) CS374
More informationTheory of Computation Chapter 9
0-0 Theory of Computation Chapter 9 Guan-Shieng Huang May 12, 2003 NP-completeness Problems NP: the class of languages decided by nondeterministic Turing machine in polynomial time NP-completeness: Cook
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Spring 2016 http://cseweb.ucsd.edu/classes/sp16/cse105-ab/ Today's learning goals Sipser Ch 7.2, 7.3 Distinguish between polynomial and exponential DTIME Define nondeterministic
More informationCS 5114: Theory of Algorithms. Tractable Problems. Tractable Problems (cont) Decision Problems. Clifford A. Shaffer. Spring 2014
Department of Computer Science Virginia Tech Blacksburg, Virginia Copyright c 2014 by Clifford A. Shaffer : Theory of Algorithms Title page : Theory of Algorithms Clifford A. Shaffer Spring 2014 Clifford
More informationCSE 135: Introduction to Theory of Computation NP-completeness
CSE 135: Introduction to Theory of Computation NP-completeness Sungjin Im University of California, Merced 04-15-2014 Significance of the question if P? NP Perhaps you have heard of (some of) the following
More informationAlgorithms: COMP3121/3821/9101/9801
NEW SOUTH WALES Algorithms: COMP3121/3821/9101/9801 Aleks Ignjatović School of Computer Science and Engineering University of New South Wales LECTURE 9: INTRACTABILITY COMP3121/3821/9101/9801 1 / 29 Feasibility
More informationOn the Computational Hardness of Graph Coloring
On the Computational Hardness of Graph Coloring Steven Rutherford June 3, 2011 Contents 1 Introduction 2 2 Turing Machine 2 3 Complexity Classes 3 4 Polynomial Time (P) 4 4.1 COLORED-GRAPH...........................
More informationCSE 421 NP-Completeness
CSE 421 NP-Completeness Yin at Lee 1 Cook-Levin heorem heorem (Cook 71, Levin 73): 3-SA is NP-complete, i.e., for all problems A NP, A p 3-SA. (See CSE 431 for the proof) So, 3-SA is the hardest problem
More information