4/20/11. NP-complete problems. A variety of NP-complete problems. Hamiltonian Cycle. Hamiltonian Cycle. Directed Hamiltonian Cycle
|
|
- Ethan Lloyd
- 5 years ago
- Views:
Transcription
1 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 problems: HAMILONIAN-CYCLE, SP. Partitioning problems: 3D-MACHING, 3-COLOR. Numerical problems: SUSE-SUM, KNAPSACK. Randall Munro Hamiltonian Cycle HAM-CYCLE: given an undirected graph G = (V, E), does there exist a simple cycle Γ that contains every node in V. Hamiltonian Cycle HAM-CYCLE: given an undirected graph G = (V, E), does there exist a simple cycle Γ that contains every node in V. 1 1' 2 2' 3 3' 4 4' 5 YES: vertices and faces of a dodecahedron. NO: bipartite graph with odd number of nodes. 3 4 Directed Hamiltonian Cycle DIR-HAM-CYCLE: given a directed graph G = (V, E), does there exists a simple directed cycle Γ that contains every node in V? Claim. DIR-HAM-CYCLE P HAM-CYCLE. Proof. Given a directed graph G = (V, E), construct an undirected graph G' with 3n nodes. a b c G v d e a out b out c out v in G' v v out d in e in Directed Hamiltonian Cycle Claim. G has a Hamiltonian cycle iff G' does. Proof. Suppose G has a directed Hamiltonian cycle Γ. hen G' has an undirected Hamiltonian cycle (same order). Suppose G' has an undirected Hamiltonian cycle Γ'. Γ' must visit nodes in G' using one of following two orders:,, G, R,, G, R,, G, R,,,, R, G,, R, G,, R, G,, lue nodes in Γ' make up directed Hamiltonian cycle Γ in G, or reverse of one
2 3-SA Reduces to Directed Hamiltonian Cycle Claim. 3-SA P DIR-HAM-CYCLE. Proof. Given an instance Φ of 3-SA, we construct an instance of DIR-HAM-CYCLE that has a Hamiltonian cycle iff Φ is satisfiable. Construction. irst, create graph that has 2 n Hamiltonian cycles which correspond in a natural way to 2 n possible truth assignments. 3-SA Reduces to Directed Hamiltonian Cycle Construction. Given 3-SA instance Φ with n variables x i and k clauses. Construct G to have 2 n Hamiltonian cycles. Intuition: traverse path i from left to right set variable x i = 1. s x 2 x 3 t 7 3k SA Reduces to Directed Hamiltonian Cycle 3-SA Reduces to Directed Hamiltonian Cycle or each clause: add a node and 6 edges. Claim. Φ is satisfiable iff G has a Hamiltonian cycle. C clause node 1 = x1 V x2 V x3 clause node C 2 = x1 V x2 V x3 s Proof. Suppose 3-SA instance has satisfying assignment x. hen, define Hamiltonian cycle in G as follows: if x i = 1, traverse row i from left to right if x i = 0, traverse row i from right to left for each clause C j, there will be at least one row in which we are going in the "correct" direction to include node C j x 2 x 3 t SA Reduces to Directed Hamiltonian Cycle Claim. Φ is satisfiable iff G has a Hamiltonian cycle. Proof. Suppose G has a Hamiltonian cycle Γ. If Γ enters clause node C j, it must depart on mate edge. Set x i = 1 iff Γ traverses row i left to right. Since Γ visits each clause node C j, at least one of the paths is traversed in "correct" direction, and each clause is satisfied. raveling Salesperson Problem SP. Given a set of n cities and a distance function d(u, v), is there a tour of length D? All 13,509 cities in US with a population of at least 500 Reference:
3 raveling Salesperson Problem SP. Given a set of n cities and a distance function d(u, v), is there a tour of length D? raveling Salesperson Problem SP. Given a set of n cities and a pairwise distance function d(u, v), is there a tour of length D? HAM-CYCLE: given a graph G = (V, E), does there exists a simple cycle that contains every node in V? Claim. HAM-CYCLE P SP. Proof. Given instance G = (V, E) of HAM-CYCLE, create n cities with distance function $ 1 if (u, v) " E d(u, v) = % & 2 if (u, v) # E SP instance has tour of length n iff G is Hamiltonian. Optimal SP tour Reference: Graph Coloring 3-COLOR: Given an undirected graph G does there exists a way to color the nodes red, green, and blue so that no adjacent nodes have the same color? asic genres. Packing problems: SE-PACKING, INDEPENDEN SE. Covering problems: SE-COVER, VEREX-COVER. Constraint satisfaction problems: SA, 3-SA. Sequencing problems: HAMILONIAN-CYCLE, SP. Partitioning problems: 3D-MACHING, 3-COLOR. Numerical problems: SUSE-SUM, KNAPSACK. yes instance 16 K-Coloring and Register Allocation Generalization: k-coloring Arises in when trying to allocate resources in the presence of constraints K-Coloring Value of k affects the difficulty of the problem: A graph is 2-colorable iff it is bipartite Register allocation. Assign program variables to machine register so that no more than k registers are used and no two program variables that are needed at the same time are assigned to the same register. is NP-complete. Interference graph. Nodes are program variables names, edge between u and v if there exists an operation where both u and v are "live" at the same time. Observation. [Chaitin 1982] Can solve register allocation problem iff interference graph is k-colorable
4 Claim. 3-SA P 3-COLOR. Proof. Given 3-SA instance Φ, we construct an instance of 3-COLOR that is 3-colorable iff Φ is satisfiable. Initial construct: Claim. Graph is 3-colorable iff Φ is satisfiable. Proof. Suppose graph is 3-colorable. Consider assignment that sets all literals to. (ii) ensures each literal is or. (iii) ensures a literal and its negation are opposites. base x 2 x 2 x 3 x 3 x n x n Properties:,, each receive a different color, and literals receives the colors, he nodes for x i and x i each receive a different color (, or ) base x 2 x 2 x 3 x 3 x n x n Gadget that represents a clause: If the clause is not satisfied the gadget is not 3-colorable x 2 x 3 C i = V x 2 V x 3 6-node gadget x 2 x 3 C i = V x 2 V x 3 his node can t be colored! If the clause is satisfied the gadget is 3-colorable If the clause is satisfied the gadget is 3-colorable x 2 x 3 C i = V x 2 V x 3 x 2 x 3 C i = V x 2 V x
5 Claim. Graph is 3-colorable iff Φ is satisfiable. Proof. Suppose graph is 3-colorable. Consider assignment that sets all literals to. (ii) ensures each literal is or. (iii) ensures a literal and its negation are opposites. (iv) ensures at least one literal in each clause is. Claim. Graph is 3-colorable iff Φ is satisfiable. Proof. Suppose 3-SA formula Φ is satisfiable. Color all literals. Color node below green node, and node below that. Color remaining middle row nodes. Color remaining bottom nodes or as forced. a literal set to in 3-SA assignment x 2 x 3 C i = V x 2 V x 3 x 2 x 3 C i = V x 2 V x 3 6-node gadget Planar 3-Colorability PLANAR-3-COLOR. Given a planar map, can it be colored using 3 colors so that no adjacent regions have the same color? Planar 3-Colorability PLANAR-3-COLOR. Given a planar map, can it be colored using 3 colors so that no adjacent regions have the same color? YES instance. NO instance Planarity Def. A graph is planar if it can be embedded in the plane in such a way that no two edges cross. Applications: VLSI circuit design, computer graphics. Planarity esting Planarity testing. [Hopcroft-arjan 1974] O(n). Remark. Many intractable graph problems can be solved in polytime if the graph is planar; many tractable graph problems can be solved faster if the graph is planar. planar K 5 : non-planar K 3,3 : non-planar Kuratowski's heorem. An undirected graph G is non-planar iff it contains a subgraph homeomorphic to K 5 or K 3,3. homeomorphic to K 3,
6 Planar 3-Colorability and Graph 3-Colorability Claim. PLANAR-3-COLOR P PLANAR-GRAPH-3-COLOR. Proof sketch. Create a vertex for each region, and an edge between regions that share a nontrivial border. Planar k-colorability PLANAR-2-COLOR. Solvable in polynomial time. PLANAR-3-COLOR. NP-complete. PLANAR-4-COLOR. Solvable in O(1) time. heorem. [Appel-Haken, 1976] Every planar map is 4-colorable. Resolved century-old open problem. Used 50 days of computer time to deal with many special cases. irst major theorem to be proved using computer. alse intuition. If PLANAR-3-COLOR is hard, then so is PLANAR-4- COLOR Polynomial-ime Reductions constraint satisfaction 3-SA 3-SA reduces to INDEPENDEN SE Dick Karp (1972) 1985 uring Award INDEPENDEN SE DIR-HAM-CYCLE GRAPH 3-COLOR SUSE-SUM VEREX COVER HAM-CYCLE PLANAR 3-COLOR SCHEDULING SE COVER SP packing and covering sequencing partitioning numerical 33 6
CS 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationNP and NP-Completeness
0/2/206 Algorithms NP-Completeness 7- Algorithms NP-Completeness 7-2 Efficient Certification NP and NP-Completeness By a solution of a decision problem X we understand a certificate witnessing that an
More information7.8 Intractability. Overview. Properties of Algorithms. Exponential Growth. Q. What is an algorithm? A. Definition formalized using Turing machines.
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
More informationAlgorithms, Lecture 3 on NP : Nondeterminis7c Polynomial Time
Algorithms, Lecture 3 on NP : Nondeterminis7c Polynomial Time Last week: Defined Polynomial Time Reduc7ons: Problem X is poly 7me reducible to Y X P Y if can solve X using poly computa7on and a poly number
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 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 informationNP-complete Problems
NP-complete Problems HP, TSP, 3COL, 0/1IP Dimitris Diamantis µπλ November 6, 2014 Dimitris Diamantis (µπλ ) NP-complete Problems November 6, 2014 1 / 34 HAMILTON PATH is NP-Complete Definition Given an
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 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 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 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 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. 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 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 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 informationLecture 18: More NP-Complete Problems
6.045 Lecture 18: More NP-Complete Problems 1 The Clique Problem a d f c b e g Given a graph G and positive k, does G contain a complete subgraph on k nodes? CLIQUE = { (G,k) G is an undirected graph with
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 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 informationNP-Complete Problems
NP-Complete Problems Max Bisection Is NP-Complete max cut becomes max bisection if we require that S = V S. We shall reduce the more general max cut to max bisection. Add V isolated nodes to G to yield
More informationTheory of Computer Science. Theory of Computer Science. E5.1 Routing Problems. E5.2 Packing Problems. E5.3 Conclusion.
Theory of Computer Science May 31, 2017 E5. Some NP-Complete Problems, Part II Theory of Computer Science E5. Some NP-Complete Problems, Part II E5.1 Routing Problems Malte Helmert University of Basel
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 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 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 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 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 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 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 informationUC Berkeley CS 170: Efficient Algorithms and Intractable Problems Handout 22 Lecturer: David Wagner April 24, Notes 22 for CS 170
UC Berkeley CS 170: Efficient Algorithms and Intractable Problems Handout 22 Lecturer: David Wagner April 24, 2003 Notes 22 for CS 170 1 NP-completeness of Circuit-SAT We will prove that the circuit satisfiability
More informationNP-Completeness. Subhash Suri. May 15, 2018
NP-Completeness Subhash Suri May 15, 2018 1 Computational Intractability The classical reference for this topic is the book Computers and Intractability: A guide to the theory of NP-Completeness by Michael
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 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 informationVIII. NP-completeness
VIII. NP-completeness 1 / 15 NP-Completeness Overview 1. Introduction 2. P and NP 3. NP-complete (NPC): formal definition 4. How to prove a problem is NPC 5. How to solve a NPC problem: approximate algorithms
More informationData Structures in Java
Data Structures in Java Lecture 21: Introduction to NP-Completeness 12/9/2015 Daniel Bauer Algorithms and Problem Solving Purpose of algorithms: find solutions to problems. Data Structures provide ways
More informationAutomata Theory CS Complexity Theory I: Polynomial Time
Automata Theory CS411-2015-17 Complexity Theory I: Polynomial Time David Galles Department of Computer Science University of San Francisco 17-0: Tractable vs. Intractable If a problem is recursive, then
More informationNP-Completeness. ch34 Hewett. Problem. Tractable Intractable Non-computable computationally infeasible super poly-time alg. sol. E.g.
NP-Completeness ch34 Hewett Problem Tractable Intractable Non-computable computationally infeasible super poly-time alg. sol. E.g., O(2 n ) computationally feasible poly-time alg. sol. E.g., O(n k ) No
More informationmax bisection max cut becomes max bisection if we require that It has many applications, especially in VLSI layout.
max bisection max cut becomes max bisection if we require that S = V S. It has many applications, especially in VLSI layout. c 2011 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 330 max bisection
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 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 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 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 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 informationLecture 24 : Even more reductions
COMPSCI 330: Design and Analysis of Algorithms December 5, 2017 Lecture 24 : Even more reductions Lecturer: Yu Cheng Scribe: Will Wang 1 Overview Last two lectures, we showed the technique of reduction
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 informationCOMP 382. Unit 10: NP-Completeness
COMP 382 Unit 10: NP-Completeness Time complexity 1 log n n n 2 n 3 2 n n n Space complexity 1 log n n n 2 n 3 2 n n n Complexity theory Focus on decidability (yes/no) problems What is P? Deterministic,
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 informationToday: NP-Completeness (con t.)
Today: NP-Completeness (con t.) COSC 581, Algorithms April 22, 2014 Many of these slides are adapted from several online sources Reading Assignments Today s class: Chapter 34.5 (con t.) Recall: Proving
More informationCMSC 441: Algorithms. NP Completeness
CMSC 441: Algorithms NP Completeness Intractable & Tractable Problems Intractable problems: As they grow large, we are unable to solve them in reasonable time What constitutes reasonable time? Standard
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 informationCSCI3390-Lecture 17: A sampler of NP-complete problems
CSCI3390-Lecture 17: A sampler of NP-complete problems 1 List of Problems We now know that if L is any problem in NP, that L P SAT, and thus SAT is NP-hard. Since SAT is also in NP we find that SAT is
More informationCS154, Lecture 15: Cook-Levin Theorem SAT, 3SAT
CS154, Lecture 15: Cook-Levin Theorem SAT, 3SAT Definition: A language B is NP-complete if: 1. B NP 2. Every A in NP is poly-time reducible to B That is, A P B When this is true, we say B is NP-hard On
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 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 informationECS122A Handout on NP-Completeness March 12, 2018
ECS122A Handout on NP-Completeness March 12, 2018 Contents: I. Introduction II. P and NP III. NP-complete IV. How to prove a problem is NP-complete V. How to solve a NP-complete problem: approximate algorithms
More informationAdmin NP-COMPLETE PROBLEMS. Run-time analysis. Tractable vs. intractable problems 5/2/13. What is a tractable problem?
Admin Two more assignments No office hours on tomorrow NP-COMPLETE PROBLEMS Run-time analysis Tractable vs. intractable problems We ve spent a lot of time in this class putting algorithms into specific
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 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 information1.1 P, NP, and NP-complete
CSC5160: Combinatorial Optimization and Approximation Algorithms Topic: Introduction to NP-complete Problems Date: 11/01/2008 Lecturer: Lap Chi Lau Scribe: Jerry Jilin Le This lecture gives a general introduction
More informationCS21 Decidability and Tractability
CS21 Decidability and Tractability Lecture 20 February 23, 2018 February 23, 2018 CS21 Lecture 20 1 Outline the complexity class NP NP-complete probelems: Subset Sum NP-complete problems: NAE-3-SAT, max
More informationChapter 3: Proving NP-completeness Results
Chapter 3: Proving NP-completeness Results Six Basic NP-Complete Problems Some Techniques for Proving NP-Completeness Some Suggested Exercises 1.1 Six Basic NP-Complete Problems 3-SATISFIABILITY (3SAT)
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 informationCS311 Computational Structures. NP-completeness. Lecture 18. Andrew P. Black Andrew Tolmach. Thursday, 2 December 2010
CS311 Computational Structures NP-completeness Lecture 18 Andrew P. Black Andrew Tolmach 1 Some complexity classes P = Decidable in polynomial time on deterministic TM ( tractable ) NP = Decidable in polynomial
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 informationMore on NP and Reductions
Indian Institute of Information Technology Design and Manufacturing, Kancheepuram Chennai 600 127, India An Autonomous Institute under MHRD, Govt of India http://www.iiitdm.ac.in COM 501 Advanced Data
More 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 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 informationNP-completeness. Chapter 34. Sergey Bereg
NP-completeness Chapter 34 Sergey Bereg Oct 2017 Examples Some problems admit polynomial time algorithms, i.e. O(n k ) running time where n is the input size. We will study a class of NP-complete problems
More informationAlgorithms and Complexity Theory. Chapter 8: Introduction to Complexity. Computer Science - Durban - September 2005
Algorithms and Complexity Theory Chapter 8: Introduction to Complexity Jules-R Tapamo Computer Science - Durban - September 2005 Contents 1 Introduction 2 1.1 Dynamic programming...................................
More informationPlanar Graphs (1) Planar Graphs (3) Planar Graphs (2) Planar Graphs (4)
S-72.2420/T-79.5203 Planarity; Edges and Cycles 1 Planar Graphs (1) Topological graph theory, broadly conceived, is the study of graph layouts. Contemporary applications include circuit layouts on silicon
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 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 informationCS 320, Fall Dr. Geri Georg, Instructor 320 NP 1
NP CS 320, Fall 2017 Dr. Geri Georg, Instructor georg@colostate.edu 320 NP 1 NP Complete A class of problems where: No polynomial time algorithm has been discovered No proof that one doesn t exist 320
More informationClasses of Problems. CS 461, Lecture 23. NP-Hard. Today s Outline. We can characterize many problems into three classes:
Classes of Problems We can characterize many problems into three classes: CS 461, Lecture 23 Jared Saia University of New Mexico P is the set of yes/no problems that can be solved in polynomial time. Intuitively
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 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 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 information