Undecidable Problems - COPING WITH THE LIMITATIONS OF ALGORITHM POWERmuthukrishnavinayaga
This document discusses algorithms and their analysis. It begins by defining key properties of algorithms like their lower, upper, and tight bounds. It then discusses different techniques for determining algorithm lower bounds such as trivial, information theoretical, adversary, and reduction arguments. Decision trees are presented as a model for representing algorithms that use comparisons. Lower bounds proofs are given for sorting and searching algorithms. The document also covers polynomial time versus non-polynomial time problems, as well as NP-complete problems. Specific algorithms are analyzed like knapsack, traveling salesman, and approximation algorithms.
Undecidable Problems and Approximation AlgorithmsMuthu Vinayagam
The document discusses algorithm limitations and approximation algorithms. It begins by explaining that some problems have no algorithms or cannot be solved in polynomial time. It then discusses different algorithm bounds and how to derive lower bounds through techniques like decision trees. The document also covers NP-complete problems, approximation algorithms for problems like traveling salesman, and techniques like branch and bound. It provides examples of approximation algorithms that provide near-optimal solutions when an optimal solution is impossible or inefficient to find.
This document discusses lower bounds and limitations of algorithms. It begins by defining lower bounds and providing examples of problems where tight lower bounds have been established, such as sorting requiring Ω(nlogn) comparisons. It then discusses methods for establishing lower bounds, including trivial bounds, decision trees, adversary arguments, and problem reduction. The document covers several examples to illustrate these techniques. It also discusses the complexity classes P, NP, and NP-complete problems. Finally, it discusses approaches for tackling difficult combinatorial problems that are NP-hard, including exact and approximation algorithms.
This document discusses lower bounds and limitations of algorithms. It begins by defining lower bounds and providing examples of problems where tight lower bounds have been established, such as sorting requiring Ω(nlogn) comparisons. It then discusses methods for establishing lower bounds, including trivial bounds, decision trees, adversary arguments, and problem reduction. The document explores different classes of problems based on complexity, such as P, NP, and NP-complete problems. It concludes by examining approaches for tackling difficult combinatorial problems that are NP-hard, such as using exact algorithms, approximation algorithms, and local search heuristics.
The document discusses NP-complete problems and polynomial-time reductions between them. It analyzes permutation problems like Hamiltonian path/cycle and vertex coloring. It also covers subset problems like vertex cover, independent set, and satisfiability. The document proposes algorithms that use a decision box to solve these problems in polynomial time, even though directly finding optimal solutions is NP-complete. It shows how problems can be reduced to each other via the decision box approach.
The document discusses NP-complete problems and polynomial-time reductions between them. It summarizes several permutation and subset problems that are known to be NP-complete, including Hamiltonian path/cycle, vertex cover, and 3-SAT. It then describes polynomial-time algorithms for solving some of these problems exactly using a "decision box" that can determine in polynomial time whether an instance has a solution. For example, it presents an O(n) algorithm for finding a minimum vertex cover using a decision box to iteratively test subset sizes.
Lower bound theory Np hard & Np completenessyvtinsane
The document provides an overview of lower bound theory, NP-hardness, and NP-completeness in algorithm design, focusing on the characteristics of decision problems. It explains the distinctions between polynomial and non-polynomial time algorithms, deterministic and non-deterministic algorithms, and the concepts of tractable and intractable problems, including specific examples like sorting and satisfiability. Additionally, it highlights key decision problems that are NP-complete, such as hamiltonian cycles and the maximum clique problem.
Analysis and design of algorithms part 4Deepak John
The document discusses the analysis and design of algorithms, focusing on various problem types such as optimization and decision problems. It explains complexity classes like P and NP, and provides insight into NP-complete problems and approximation algorithms. Specific algorithms for bin packing, graph coloring, and the traveling salesman problem are detailed, highlighting their challenges and strategies.
NP-completeness means that many important computational problems cannot be solved quickly in polynomial time. NP-complete problems are a class of problems for which no efficient solution algorithms have been found. Many significant problems in computer science, such as the traveling salesman problem, satisfiability problems, and graph covering problems belong to this class. A problem is considered NP-complete if it can be solved by a non-deterministic algorithm in polynomial time and every problem in NP can be reduced to it in polynomial time. Showing that a problem is NP-complete involves proving that it is in NP and that a known NP-complete problem can be reduced to it in polynomial time. If any NP-complete problem could be solved in polynomial time
The document summarizes the concepts of P vs NP complexity classes. It states that P problems can be solved in polynomial time, like searching an array, while NP problems are solved in non-deterministic polynomial time, like the knapsack problem. It then defines different types of algorithms and complexity classes. The key classes discussed are P, NP, NP-Complete, and NP-Hard. It provides examples like sorting being in P, while the Hamiltonian problem is NP-Complete. A graphical representation is also included to illustrate the relationships between the complexity classes.
This document introduces Packed Computation, a new approach for solving NP-complete problems. It proposes reducing NP-complete problems to the Rules States Satisfiability (RSS) problem and using Packed Computation algorithms to solve RSS problems quickly. The document outlines the author's research from 2008-2014 in developing this approach. It defines RSS as an NP-complete problem and describes how Packed Computation algorithms work by sometimes selecting single states and sometimes selecting multiple "packed" states. While the algorithms show promise in testing, a mathematical proof of their polynomial runtime complexity is still outstanding. The document serves as an outline for the author's proposed new approach but does not yet prove its mathematical properties.
This document introduces algorithms that are polynomial time versus non-polynomial time and defines NP-complete problems. It discusses that NP-complete problems include Satisfiability (SAT), Traveling Salesman, Knapsack and Clique problems. These problems are difficult to solve in polynomial time and are mapped to each other through polynomial time reductions. While we can solve them in exponential time, finding a polynomial time algorithm would mean P=NP.
This document discusses NP-hard and NP-complete problems. It begins by defining the classes P, NP, NP-hard, and NP-complete. It then provides examples of NP-hard problems like the traveling salesperson problem, satisfiability problem, and chromatic number problem. It explains that to show a problem is NP-hard, one shows it is at least as hard as another known NP-hard problem. The document concludes by discussing how restricting NP-hard problems can result in problems that are solvable in polynomial time.
This document discusses branch and bound algorithms and NP-hard and NP-complete problems. It provides examples and proofs related to these topics.
1) Branch and bound is an algorithm that systematically enumerates candidate solutions by discarding subsets that are provably suboptimal. The knapsack problem is used as an example problem.
2) NP-hard and NP-complete problems are those whose best known algorithms run in non-polynomial time. If a problem can be solved in polynomial time, then all NP-complete problems can be. Proving problems NP-complete involves reducing other known NP-complete problems to the target problem.
3) Trees are connected graphs without cycles. They are used to represent hierarchies and
This document discusses NP-complete problems and nondeterministic algorithms. It begins by introducing NP-completeness as evidence that many important problems cannot be solved quickly. It then defines different complexity classes like P, NP, NP-hard, and NP-complete. It provides examples of NP-complete problems like maximum clique and 0/1 knapsack. It also discusses how to show that a problem is NP-complete using polynomial-time reductions from another known NP-complete problem. In particular, it shows that the clique decision problem is NP-complete by reducing SAT to it.
This document discusses the famous P vs NP problem in computer science. It begins by defining efficient ("P") vs inefficient ("NP") algorithms. P problems can be solved quickly in polynomial time, while NP problems are easy to check but hard to solve with no known fast algorithms. Examples are given like sorting vs integer factorization. The document goes on to explain NP-completeness, that if any NP problem like 3-SAT could be solved quickly, then all NP problems would be considered P. It concludes that proving whether P equals NP or not is extremely challenging and would have profound implications.
This document provides an introduction to NP-completeness, including: definitions of key concepts like decision problems, classes P and NP, and polynomial time reductions; examples of NP-complete problems like satisfiability and the traveling salesman problem; and approaches to dealing with NP-complete problems like heuristic algorithms, approximation algorithms, and potential help from quantum computing in the future. The document establishes NP-completeness as a central concept in computational complexity theory.
The document covers concepts in computational complexity theory, including classifications of problems such as P, NP, NP-complete, and NP-hard. It explains deterministic vs non-deterministic algorithms, the significance of polynomial-time problems, and reductions in algorithms. Various examples and definitions illustrate the relationship between these complexity classes and highlight the ongoing debate regarding whether P equals NP.
The document introduces the concepts of P, NP, and NP-completeness. It states that P problems can be solved in polynomial time, while NP problems can be verified in polynomial time using a non-deterministic algorithm. NP-complete problems are the hardest problems in NP and if any NP-complete problem could be solved in polynomial time, then all NP problems could be solved that way. As an example, it describes how the Traveling Salesman Problem is NP-complete as the number of possible routes grows exponentially with the number of cities.
Confidence interval two tail tests-lower bounds upperboundsssuser3c3f88
The document discusses lower bounds in computational complexity, emphasizing their significance in determining the minimum work required to solve problems such as sorting, searching, and matrix multiplication. It details methods for establishing these bounds, including decision trees and adversary arguments, and introduces complexity classes like P and NP, along with the concept of NP-completeness. The document raises the pivotal question of whether P equals NP, suggesting that if one NP-complete problem can be solved in polynomial time, all NP problems could similarly be solved in such time.
Brute Force and Divide & Conquer Techniquessusered62011
The document provides an overview of various problem-solving techniques such as brute force, exhaustive search, and divide-and-conquer methods, along with their applications and algorithms like Traveling Salesman Problem, Knapsack Problem, and sorting techniques like MergeSort and QuickSort. It discusses the advantages and disadvantages of these approaches, emphasizing their efficiency and complexity. Additionally, it includes specific algorithm procedures and examples to illustrate how these techniques are implemented.
This document discusses brute force and divide-and-conquer algorithms. It covers various brute force algorithms like computing an, string matching, closest pair problem, convex hull problems, and exhaustive search algorithms like the traveling salesman problem and knapsack problem. It then discusses divide-and-conquer techniques like merge sort, quicksort, and multiplication of large integers. Examples and analysis of algorithms like selection sort, bubble sort, and mergesort are provided.
The document discusses the theory of NP-completeness. It begins by classifying problems as solvable, unsolvable, tractable, or intractable. It then defines deterministic and nondeterministic algorithms, and how nondeterministic algorithms can be expressed. The document introduces the complexity classes P and NP. It discusses reducing one problem to another to prove NP-completeness via transitivity. Several classic NP-complete problems are proven to be NP-complete, such as 3SAT, 3-coloring, and subset sum. The document also discusses how to cope with NP-complete problems in practice by sacrificing optimality, generality, or efficiency.
This document provides an overview of brute force and divide-and-conquer algorithms. It discusses various brute force algorithms like computing an, string matching, closest pair problem, convex hull problems, and exhaustive search algorithms like the traveling salesman problem and knapsack problem. It also analyzes the time efficiency of these brute force algorithms. The document then discusses the divide-and-conquer approach and provides examples like merge sort, quicksort, and matrix multiplication. It provides pseudocode and analysis for mergesort. In summary, the document covers brute force and divide-and-conquer techniques for solving algorithmic problems.
The document discusses brute force algorithms and exhaustive search techniques. It provides examples of problems that can be solved using these approaches, such as computing powers and factorials, sorting, string matching, polynomial evaluation, the traveling salesman problem, knapsack problem, and the assignment problem. For each problem, it describes generating all possible solutions and evaluating them to find the best one. Most brute force algorithms have exponential time complexity, evaluating all possible combinations or permutations of the input.
1) The document discusses the complexity classes P, NP, NP-hard and NP-complete. P refers to problems that can be solved in polynomial time, while NP includes problems that can be verified in polynomial time.
2) NP-hard problems are at least as hard as the hardest problems in NP. NP-complete problems are the hardest problems in NP. If any NP-complete problem could be solved in polynomial time, then P would be equal to NP.
3) Common NP-complete problems discussed include the traveling salesman problem and integer knapsack problem. Reductions are used to show that one problem is at least as hard as another.
Deep Learning for Natural Language Processing_FDP on 16 June 2025 MITS.pptxresming1
This gives an introduction to how NLP has evolved from the time of World War II till this date through the advances in approaches, architectures and word representations. From rule based approaches, it advanced to statistical approaches. from traditional machine learning algorithms it advanced to deep neural network architectures. Deep neural architectures include recurrent neural networks, long short term memory, gated recurrent units, seq2seq models, encoder decoder models, transformer architecture, upto large language models and vision language models which are multimodal in nature.
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NP-completeness means that many important computational problems cannot be solved quickly in polynomial time. NP-complete problems are a class of problems for which no efficient solution algorithms have been found. Many significant problems in computer science, such as the traveling salesman problem, satisfiability problems, and graph covering problems belong to this class. A problem is considered NP-complete if it can be solved by a non-deterministic algorithm in polynomial time and every problem in NP can be reduced to it in polynomial time. Showing that a problem is NP-complete involves proving that it is in NP and that a known NP-complete problem can be reduced to it in polynomial time. If any NP-complete problem could be solved in polynomial time
The document summarizes the concepts of P vs NP complexity classes. It states that P problems can be solved in polynomial time, like searching an array, while NP problems are solved in non-deterministic polynomial time, like the knapsack problem. It then defines different types of algorithms and complexity classes. The key classes discussed are P, NP, NP-Complete, and NP-Hard. It provides examples like sorting being in P, while the Hamiltonian problem is NP-Complete. A graphical representation is also included to illustrate the relationships between the complexity classes.
This document introduces Packed Computation, a new approach for solving NP-complete problems. It proposes reducing NP-complete problems to the Rules States Satisfiability (RSS) problem and using Packed Computation algorithms to solve RSS problems quickly. The document outlines the author's research from 2008-2014 in developing this approach. It defines RSS as an NP-complete problem and describes how Packed Computation algorithms work by sometimes selecting single states and sometimes selecting multiple "packed" states. While the algorithms show promise in testing, a mathematical proof of their polynomial runtime complexity is still outstanding. The document serves as an outline for the author's proposed new approach but does not yet prove its mathematical properties.
This document introduces algorithms that are polynomial time versus non-polynomial time and defines NP-complete problems. It discusses that NP-complete problems include Satisfiability (SAT), Traveling Salesman, Knapsack and Clique problems. These problems are difficult to solve in polynomial time and are mapped to each other through polynomial time reductions. While we can solve them in exponential time, finding a polynomial time algorithm would mean P=NP.
This document discusses NP-hard and NP-complete problems. It begins by defining the classes P, NP, NP-hard, and NP-complete. It then provides examples of NP-hard problems like the traveling salesperson problem, satisfiability problem, and chromatic number problem. It explains that to show a problem is NP-hard, one shows it is at least as hard as another known NP-hard problem. The document concludes by discussing how restricting NP-hard problems can result in problems that are solvable in polynomial time.
This document discusses branch and bound algorithms and NP-hard and NP-complete problems. It provides examples and proofs related to these topics.
1) Branch and bound is an algorithm that systematically enumerates candidate solutions by discarding subsets that are provably suboptimal. The knapsack problem is used as an example problem.
2) NP-hard and NP-complete problems are those whose best known algorithms run in non-polynomial time. If a problem can be solved in polynomial time, then all NP-complete problems can be. Proving problems NP-complete involves reducing other known NP-complete problems to the target problem.
3) Trees are connected graphs without cycles. They are used to represent hierarchies and
This document discusses NP-complete problems and nondeterministic algorithms. It begins by introducing NP-completeness as evidence that many important problems cannot be solved quickly. It then defines different complexity classes like P, NP, NP-hard, and NP-complete. It provides examples of NP-complete problems like maximum clique and 0/1 knapsack. It also discusses how to show that a problem is NP-complete using polynomial-time reductions from another known NP-complete problem. In particular, it shows that the clique decision problem is NP-complete by reducing SAT to it.
This document discusses the famous P vs NP problem in computer science. It begins by defining efficient ("P") vs inefficient ("NP") algorithms. P problems can be solved quickly in polynomial time, while NP problems are easy to check but hard to solve with no known fast algorithms. Examples are given like sorting vs integer factorization. The document goes on to explain NP-completeness, that if any NP problem like 3-SAT could be solved quickly, then all NP problems would be considered P. It concludes that proving whether P equals NP or not is extremely challenging and would have profound implications.
This document provides an introduction to NP-completeness, including: definitions of key concepts like decision problems, classes P and NP, and polynomial time reductions; examples of NP-complete problems like satisfiability and the traveling salesman problem; and approaches to dealing with NP-complete problems like heuristic algorithms, approximation algorithms, and potential help from quantum computing in the future. The document establishes NP-completeness as a central concept in computational complexity theory.
The document covers concepts in computational complexity theory, including classifications of problems such as P, NP, NP-complete, and NP-hard. It explains deterministic vs non-deterministic algorithms, the significance of polynomial-time problems, and reductions in algorithms. Various examples and definitions illustrate the relationship between these complexity classes and highlight the ongoing debate regarding whether P equals NP.
The document introduces the concepts of P, NP, and NP-completeness. It states that P problems can be solved in polynomial time, while NP problems can be verified in polynomial time using a non-deterministic algorithm. NP-complete problems are the hardest problems in NP and if any NP-complete problem could be solved in polynomial time, then all NP problems could be solved that way. As an example, it describes how the Traveling Salesman Problem is NP-complete as the number of possible routes grows exponentially with the number of cities.
Confidence interval two tail tests-lower bounds upperboundsssuser3c3f88
The document discusses lower bounds in computational complexity, emphasizing their significance in determining the minimum work required to solve problems such as sorting, searching, and matrix multiplication. It details methods for establishing these bounds, including decision trees and adversary arguments, and introduces complexity classes like P and NP, along with the concept of NP-completeness. The document raises the pivotal question of whether P equals NP, suggesting that if one NP-complete problem can be solved in polynomial time, all NP problems could similarly be solved in such time.
Brute Force and Divide & Conquer Techniquessusered62011
The document provides an overview of various problem-solving techniques such as brute force, exhaustive search, and divide-and-conquer methods, along with their applications and algorithms like Traveling Salesman Problem, Knapsack Problem, and sorting techniques like MergeSort and QuickSort. It discusses the advantages and disadvantages of these approaches, emphasizing their efficiency and complexity. Additionally, it includes specific algorithm procedures and examples to illustrate how these techniques are implemented.
This document discusses brute force and divide-and-conquer algorithms. It covers various brute force algorithms like computing an, string matching, closest pair problem, convex hull problems, and exhaustive search algorithms like the traveling salesman problem and knapsack problem. It then discusses divide-and-conquer techniques like merge sort, quicksort, and multiplication of large integers. Examples and analysis of algorithms like selection sort, bubble sort, and mergesort are provided.
The document discusses the theory of NP-completeness. It begins by classifying problems as solvable, unsolvable, tractable, or intractable. It then defines deterministic and nondeterministic algorithms, and how nondeterministic algorithms can be expressed. The document introduces the complexity classes P and NP. It discusses reducing one problem to another to prove NP-completeness via transitivity. Several classic NP-complete problems are proven to be NP-complete, such as 3SAT, 3-coloring, and subset sum. The document also discusses how to cope with NP-complete problems in practice by sacrificing optimality, generality, or efficiency.
This document provides an overview of brute force and divide-and-conquer algorithms. It discusses various brute force algorithms like computing an, string matching, closest pair problem, convex hull problems, and exhaustive search algorithms like the traveling salesman problem and knapsack problem. It also analyzes the time efficiency of these brute force algorithms. The document then discusses the divide-and-conquer approach and provides examples like merge sort, quicksort, and matrix multiplication. It provides pseudocode and analysis for mergesort. In summary, the document covers brute force and divide-and-conquer techniques for solving algorithmic problems.
The document discusses brute force algorithms and exhaustive search techniques. It provides examples of problems that can be solved using these approaches, such as computing powers and factorials, sorting, string matching, polynomial evaluation, the traveling salesman problem, knapsack problem, and the assignment problem. For each problem, it describes generating all possible solutions and evaluating them to find the best one. Most brute force algorithms have exponential time complexity, evaluating all possible combinations or permutations of the input.
1) The document discusses the complexity classes P, NP, NP-hard and NP-complete. P refers to problems that can be solved in polynomial time, while NP includes problems that can be verified in polynomial time.
2) NP-hard problems are at least as hard as the hardest problems in NP. NP-complete problems are the hardest problems in NP. If any NP-complete problem could be solved in polynomial time, then P would be equal to NP.
3) Common NP-complete problems discussed include the traveling salesman problem and integer knapsack problem. Reductions are used to show that one problem is at least as hard as another.
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• Understand concepts of limit state and working stress method of design of structural steel members and various types of connections.
• Determine net area and effective sections in tension members, tension splices, lug angles and gussets.
• Execute design of compression members as per IS codal practice.
• Analyze concepts of design of flexural members.
• Design structural systems such as roof trusses, gantry girders as per provisions of IS 800 – 2007 of practice for limit state method.
OUTCOMES:
On successful completion of this course, the students will be able to,
• Analyze different types of bolted and welded connections.
• Develop skills to design tension members, splices, lug angles and gussets.
• Elaborate IS Code design practice of various compression members.
• Design laterally supported and unsupported beams, built-up beams, plate girders and stiffeners.
• Acquire knowledge about components of industrial structures, Gantry girders and roof trusses.
TEXT BOOKS:
1. Bhavikatti S S, “Design of Steel Structures”, By Limit State Method as per IS: 800 – 2007, IK International Publishing House Pvt. Ltd., 2019.
2. Subramanian N, “Design of Steel Structures”, Oxford University Press 2011.
REFERENCE BOOKS:
1. Duggal S K, “Limit State Design of Steel Structures”, Tata, McGraw Hill Education Pvt. Ltd., New Delhi, 2017.
2. Shiyekar M R, “Limit State Design in Structural Steel”, PHI Learning Private Limited, New Delhi, 2013.
3. IS: 800 – 2007, IS: 800 – 1984, General Construction in Steel – Code of Practice, BIS, New Delhi.
Structural steel types – Mechanical Properties of structural steel- Indian structural steel products- Steps involved in the Deign Process -Steel Structural systems and their Elements- -Type of Loads on Structures and Load combinations- Code of practices, Loading standards and Specifications - Concept of Allowable Stress Method, and Limit State Design Methods for Steel structures-Relative advantages and Limitations-Strengths and Serviceability Limit states.
Allowable stresses as per IS 800 section 11 -Concepts of Allowable stress design for bending and Shear –Check for Elastic deflection-Calculation of moment carrying capacity –Design of Laterally supported Solid Hot Rolled section beams-Allowable stress deign of Angle Tension and Compression Members and estimation of axial load carrying capacity.
Type of Fasteners- Bolts Pins and welds- Types of simple bolted and welded connections Relative advantages and Limitations-Modes of failure-the concept of Shear lag-efficiency of joints- Axially loaded bolted connections for Plates and Angle Members using bearing type bolts –Prying forces and Hanger connection– Design of Slip critical connections with High strength Friction Grip bolts.- Design of joints for combined shear and Tension- Eccentrically Loaded Bolted Bracket Connections- Welds-symbols and specifications- Effective area of welds-Fillet and but Welded connections-Axially Loaded connections for Plate and angle truss members and
Impurities of Water and their Significance.pptxdhanashree78
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7. • Polynomial-Time Algorithm: an algorithm whose order-of-magnitude time performance is bounded
from above by a polynomial function of n, where n is the size of its inputs.
• Exponential Algorithm: an algorithm whose order-of-magnitude time performance is not bounded
from above by a polynomial function of n.
• Tractable Problem: a problem that is solvable by a polynomial-time algorithm. The upper bound is
polynomial.
• • examples of tractable problems:
• – Searching an unordered list
• – Searching an ordered list
• – Sorting a list
• – Multiplication of integers
• -Finding a minimum spanning tree in a graph
• Intractable Problem: a problem that cannot be solved by a polynomial-time algorithm. The lower
bound is exponential.
8. • examples of intractable problems (ones that have been proven to have no
polynomial-time algorithm).
∗ Towers of Hanoi: we can prove that any algorithm that solves this problem
must have a worst-case running time that is at least 2n − 1.
∗ List all permutations (all possible orderings) of n numbers. – Others have
polynomial amounts of output, but still cannot be solved in polynomial time:
∗ For an n × n draughts board with an arrangement of pieces, determine
whether there is a winning strategy for White (i.e. a sequence of moves so
that, no matter what Black does, White is guaranteed to win). We can prove
that any algorithm that solves this problem must have a worst-case running
time that is at least 2n.
9. NP- algorithms- NP-hardness and NP-completeness
• NP Problem:
The NP problems set of problems whose solutions are hard to find but easy to verify and
are solved by Non-Deterministic Machine in polynomial time.
• NP-Hard Problem:
A Problem X is NP-Hard if there is an NP-Complete problem Y, such that Y is reducible to
X in polynomial time. NP-Hard problems are as hard as NP-Complete problems. NP-Hard
Problem need not be in NP class.
• A lot of times takes the particular problem solve and reducing different problems.
• example :
• Hamiltonian cycle .
• optimization problem .
• Shortest path
10. • NP-Complete Problem:
• A problem X is NP-Complete if there is an NP problem Y, such that Y is reducible to
X in polynomial time. NP-Complete problems are as hard as NP problems. A
problem is NP-Complete if it is a part of both NP and NP-Hard Problem. A non-
deterministic Turing machine can solve NP-Complete problem in polynomial time.
• A problem is np-complete when it is both np and np hard combines together.
• this means np complete problems can be verified in polynomial time.
• Example:
• Decision problems.
• Regular graphs
11. Difference between NP-Hard and NP-
Complete:
NP-hard NP-Complete
NP-Hard problems(say X) can be solved if and only if there is a NP-Complete
problem(say Y) that can be reducible into X in polynomial time.
NP-Complete problems can be solved by a non-deterministic Algorithm/Turing
Machine in polynomial time.
To solve this problem, it do not have to be in NP . To solve this problem, it must be both NP and NP-hard problems.
Time is unknown in NP-Hard. Time is known as it is fixed in NP-Hard.
NP-hard is not a decision problem. NP-Complete is exclusively a decision problem.
Not all NP-hard problems are NP-complete. All NP-complete problems are NP-hard
Do not have to be a Decision problem. It is exclusively a Decision problem.
It is optimization problem used. It is Decision problem used.
Example: Halting problem, Vertex cover problem, etc.
Example: Determine whether a graph has a Hamiltonian cycle, Determine
whether a Boolean formula is satisfiable or not, Circuit-satisfiability problem, etc.
12. • TSP is NP-Complete
• The traveling salesman problem consists of a salesman and a set of cities. The salesman has to
visit each one of the cities starting from a certain one and returning to the same city. The
challenge of the problem is that the traveling salesman wants to minimize the total length of
the trip
• Proof
• To prove TSP is NP-Complete, first we have to prove that TSP belongs to NP. In TSP, we find a
tour and check that the tour contains each vertex once. Then the total cost of the edges of the
tour is calculated. Finally, we check if the cost is minimum. This can be completed in polynomial
time. Thus TSP belongs to NP.
• Secondly, we have to prove that TSP is NP-hard. To prove this, one way is to show
that Hamiltonian cycle ≤p TSP (as we know that the Hamiltonian cycle problem is NPcomplete).
• Assume G = (V, E) to be an instance of Hamiltonian cycle.
13. • Hence, an instance of TSP is constructed. We create the complete
graph G'
= (V, E'
), where
14. • Now, suppose that a Hamiltonian cycle h exists in G. It is clear that the
cost of each edge in h is 0 in G'
as each edge belongs to E.
Therefore, h has a cost of 0 in G'
. Thus, if graph G has a Hamiltonian
cycle, then graph G'
has a tour of 0 cost.
• Conversely, we assume that G'
has a tour h'
of cost at most 0. The cost
of edges in E'
are 0 and 1 by definition. Hence, each edge must have a
cost of 0 as the cost of h'
is 0. We therefore conclude that h'
contains
only edges in E.
• We have thus proven that G has a Hamiltonian cycle, if and only
if G'
has a tour of cost at most 0. TSP is NP-complete.
15. APPROXIMATION ALGORITHM
TSP
• Nearest neighbor
• Step1:choose an arbitrary city as start
• Step2:Repeat the following operation until all the cities have been
visited ,go to the unvisited city nearest the one visited last
• Step3:return to the starting city
17. Multifragement
• Step1:Sort the edges in non decreasing order of weights .Intialize the
set of tour edges to be constructed to empty set
• Setp2:Add the nest edge on the sorted list to the tour ,skipping those
whose addition would have created a vertex of degree 3 or a cycle of
length leass than n .Repeat this step until a tour of length n obtained.
• a-b,c-d,b-c,a-c,b-d,a-d
• a-b,c-d b-c,d-a
19. Knapsack problem
• Step1:Order the items in decreasing order of relative values
• Step2:Select the items in this order skipping those that don’t fit into
the knapsack.
• Knapsack Capacity:16
• I1,I2,I4,I3
• I1,I2,I3=2+5+5=12
21. Randomized Algorithm
What is Randomized Algorithm?
An algorithm that uses random numbers to decide what to do next anywhere in its
logic is called Randomized Algorithm.
For example, in Randomized Quick Sort, we use a random number to pick the next
pivot (or we randomly shuffle the array).
Typically, this randomness is used to reduce time complexity or space complexity in
other standard algorithms.
22. Randomized Quick Sort Algorithm
• The algorithm exactly follows the standard algorithm except it
randomizes the pivot selection.
23. • Example
• Let us look at an example to understand how randomized quicksort
works in avoiding the worst case time complexity. Since, we are
designing randomized algorithms to decrease the occurence of worst
cases in time complexity lets take a sorted list as an input for this
example.
• The sorted input list is 3, 5, 7, 8, 12, 15. We need to apply the quick
sort algorithm to sort the list.
