 NP (complexity) Wikipedia In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete.In his 1972 paper, "Reducibility Among Combinatorial Problems", Richard Karp used Stephen Cook's 1971 theorem that the boolean satisfiability problem is NP-complete (also called the Cook-Levin theorem) to show that there is a polynomial time many-one reduction from …

## Karp's 21 NP-complete problems Wikipedia

Class NP-complete and NP-hard problems. Clearly, all problems in P are also in NP, so P NP. NP-Complete problems are in a formal sense the hardest problems in NP|if any one of them can be solved in poly time, then they can all be solved in poly time; this would result in the set equality P = NP. Similarly, if any one of the NP-Complete problem can be shown to require exponential, An important notion in this context is the set of NP-complete decision problems, which is a subset of NP and might be informally described as the "hardest" problems in NP. If there is a polynomial-time algorithm for even one of them, then there is a polynomial-time algorithm for all the problems in NP..

An Annotated List of Selected NP-complete Problems. Université de Liverpool, Département d'informatique, COMP202. Pierluigi Crescenzi, Viggo Kann, Magnús Halldórsson, Marek Karpinski, and Gerhard Woeginger. A compendium of NP optimization problems. KTH NADA. Stockholm. Voir aussi. 21 problèmes NP-complets de Karp NP-complete problems off all shapes and colors. •These are universal NP-problems...if you can solve them efficiently, you can solve ANY problem in NP efficiently. •L is NP-complete if: –L is in NP –ANY other problem in NP reduces to L. •If you come up with an efficient algorithm to 3-color a map, then P=NP.

In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete.In his 1972 paper, "Reducibility Among Combinatorial Problems", Richard Karp used Stephen Cook's 1971 theorem that the boolean satisfiability problem is NP-complete (also called the Cook-Levin theorem) to show that there is a polynomial time many-one reduction from … 1 NP-complete problems In the previous lecture notes, we deﬁned the notion of completeness for a com-plexity class. A priori, it is not clear that complete problems even exist for natural complexity classes such as NP or PSPACE. Fascinatingly, completeness turns out to be a pervasive phenomenon - most natural problems in NP are

• NP-complete problems are always yes/no questions. • In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a function, f(x), of the input, x. • Optimization problems, strictly speaking, can’t be NP-complete (only NP-hard). NP-Hard and NP-Complete Problems For many of the problems we know and study, the best algorithms for their solution have computing times can be clustered into two groups- 1. Solutions are bounded by the polynomial 2. Solutions are bounded by a nonpolynomial . No one has been able to device an algorithm which is bounded by the polynomial of small degree for the problems belonging to the second

Optimization Problems NP-complete problems are always yes/no questions. In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a … Although I think "List of NP-complete problems" is an excellent article (especially as a source of new articles to write), I'm concerned that we may also be vulnerable to claims of copyright violation because we essentially steal the NP Guide's presentation.

Some First NP-complete problem We need to nd some rst NP-complete problem. Finding the rst NP-complete problem was the result of the Cook-Levin theorem. We’ll deal with this later. For now, trust me that: Independent Set is a packing problem and is NP-complete. Vertex Cover is a covering problem and is NP-complete. NP-Completeness How would you define NP-Complete? They are the “hardest” problems in NP Definition of NP-Complete Q is an NP-Complete problem if: 1) Q is in NP 2) every other NP problem polynomial time reducible to Q Getting Started How do you show that EVERY NP problem reduces to Q? One way would be to already have an NP-Complete problem

Optimization Problems NP-complete problems are always yes/no questions. In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a … A Some NP-Complete Problems To ask the hard question is simple. But what does it mean? What are we going to do? W. H. Auden In this appendix we present a brief list of NP-complete problems; we restrict

NP-complete problems are the hardest in NP: if any NP-complete problem is p-time solvable, then all problems in NP are p-time solvable How to formally compare easiness/hardness of problems? Reductions Reduce language L 1 to L 2 via function f: 1. Convert input x of L 1 to instance f(x) of L 2 2. Apply decision algorithm for L 2 to f(x) Running time = time to compute f + time to apply decision The Partition-Knapsack Problem This problem is what we originally referred to as “knapsack.” Given a list of integers L, can we partition it into two disjoint sets whose sums are equal? Example: L={3,4,5,6,14,18}, Solution: 3+4+18=5+6+16 Partition-Knapsack is NP-complete; reduction from Knapsack.

Since the original results, thousands of other problems have been shown to be NP-complete by reductions from other problems previously shown to be NP-complete; many of these problems are collected in Garey and Johnson's 1979 book Computers and Intractability: A Guide to the Theory of NP-Completeness. NP-Completeness How would you define NP-Complete? They are the “hardest” problems in NP Definition of NP-Complete Q is an NP-Complete problem if: 1) Q is in NP 2) every other NP problem polynomial time reducible to Q Getting Started How do you show that EVERY NP problem reduces to Q? One way would be to already have an NP-Complete problem

Computer Algorithms Design and Analysis Known NP-Complete Problem Garey & Johnson: Computer and Intractability: A Guide to the Theory of NP-Completeness, Freeman, 1979 About 300 problems i.e. SAT, Clique, Hamiltonian, Partition, Knapsack … Note: 0-1 knapsack problem is NPC problem, but it can be solved by using dynamic programming in polynomial time, think about why and Although I think "List of NP-complete problems" is an excellent article (especially as a source of new articles to write), I'm concerned that we may also be vulnerable to claims of copyright violation because we essentially steal the NP Guide's presentation.

Most Tensor Problems Are NP-Hard CHRISTOPHER J. HILLAR, Mathematical Sciences Research Institute LEK-HENG LIM, University of Chicago We prove that multilinear (tensor) analogues of many efﬁciently computable problems in numerical linear algebra are NP-hard. Our list includes: determining the feasibility of a system of bilinear equations, de- NP Complete (abbreviated as NPC) problems, standing at the crux of deciding whether P=NP, are among hardest problems in computer science and other related areas. Through decades, NPC problems are treated as one class. Observing that NPC problems have different natures, it is unlikely that they will have the same complexity. Our intensive study shows that NPC problems are not all equivalent in

Tutorial 8 NP-Complete Problems Nanjing University. • NP-complete problems are always yes/no questions. • In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a function, f(x), of the input, x. • Optimization problems, strictly speaking, can’t be NP-complete (only NP-hard)., NP-complete problem, any of a class of computational problems for which no efficient solution algorithm has been found. Many significant computer-science problems belong to this class—e.g., the traveling salesman problem, satisfiability problems, and graph-covering problems. So-called easy, or.

### NP-Completeness Annotated List of Selected NP-complete Problems. On dit qu’un problème de décision est « NP-complet » lorsque le langage correspondant est NP-complet. C’est une notion informelle car il existe plusieurs moyens de coder les instances d’un problème, mais cela ne pose pas de difficultés dans la mesure où on emploie un codage raisonnable du problème vers le langage considéré (voir la section NP-complétude faible)., 1 NP-complete problems In the previous lecture notes, we deﬁned the notion of completeness for a com-plexity class. A priori, it is not clear that complete problems even exist for natural complexity classes such as NP or PSPACE. Fascinatingly, completeness turns out to be a pervasive phenomenon - most natural problems in NP are.

1 NP-complete problems The University of Edinburgh. Connecting problems together. NP-Completeness What are the hardest problems in NP? The Cook-Levin Theorem A concrete NP-complete problem. Recap from Last Time. The Limits of Computability RE A HALT TM L D co-RE R ADD 0*1* A HALT TM L D EQ TM EQ TM. The Limits of Efficient Computation P NP R. P and NP Refresher The class P consists of all problems solvable in deterministic polynomial time. …, NP-complete problem, any of a class of computational problems for which no efficient solution algorithm has been found. Many significant computer-science problems belong to this class—e.g., the traveling salesman problem, satisfiability problems, and graph-covering problems. So-called easy, or.

### Tutorial 8 NP-Complete Problems Nanjing University Annotated List of Selected NP-complete Problems. NP Complete (abbreviated as NPC) problems, standing at the crux of deciding whether P=NP, are among hardest problems in computer science and other related areas. Through decades, NPC problems are treated as one class. Observing that NPC problems have different natures, it is unlikely that they will have the same complexity. Our intensive study shows that NPC problems are not all equivalent in A Useful List of NP-Complete Problems Graphs. Vertex Cover Decision Problem(VC): Given a graph G=(V,E) and a positive integer k, is there a subset V' of V of vertices which form a Vertex Cover for G with the size of V' no more than k. .(A Vertex Cover for G is a set of vertices with the property that every edge has at least one vertex from that set as an endpoint.). • NP-complete problems
• NP-complete problems
• More NP-Complete and NP-hard Problems

• LECTURE NOTES: NP-COMPLETE PROBLEMS 3 checked in polynomial time. W is known as the ”witness” or ”certiﬁcate.” At worst, all solutions w must be checked, giving exponential running time. Or trying giving `NP-complete' or `NP and complete' as a search term to http://liinwww.ira.uka.de/searchbib/index> (This is basically a bibliography database, but, you can click on the `on-line papers' button to list electronically readable full texts).

NP problems have their own significance in programming, but the discussion becomes quite hot when we deal with differences between NP, P , NP-Complete and NP-hard. P and NP- Many of us know the difference between them. P- Polynomial time solving. Problems … THE P VERSUS NP PROBLEM STEPHEN COOK 1. Statement of the Problem The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To deﬁne the problem precisely it is necessary to give a formal model of a

Most Tensor Problems Are NP-Hard CHRISTOPHER J. HILLAR, Mathematical Sciences Research Institute LEK-HENG LIM, University of Chicago We prove that multilinear (tensor) analogues of many efﬁciently computable problems in numerical linear algebra are NP-hard. Our list includes: determining the feasibility of a system of bilinear equations, de- 4 NP –HARD AND NP –COMPLETE PROBLEMS BASIC CONCEPTS •The computing times of algorithms fall into two groups. •Group1–consists of problems whose solutions are bounded by …

An Annotated List of Selected NP-complete Problems. Université de Liverpool, Département d'informatique, COMP202. Pierluigi Crescenzi, Viggo Kann, Magnús Halldórsson, Marek Karpinski, and Gerhard Woeginger. A compendium of NP optimization problems. KTH NADA. Stockholm. Voir aussi. 21 problèmes NP-complets de Karp • NP-complete problems are always yes/no questions. • In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a function, f(x), of the input, x. • Optimization problems, strictly speaking, can’t be NP-complete (only NP-hard).

NP-complete problem, any of a class of computational problems for which no efficient solution algorithm has been found. Many significant computer-science problems belong to this class—e.g., the traveling salesman problem, satisfiability problems, and graph-covering problems. So-called easy, or Optimization Problems NP-complete problems are always yes/no questions. In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a …

Most Tensor Problems Are NP-Hard CHRISTOPHER J. HILLAR, Mathematical Sciences Research Institute LEK-HENG LIM, University of Chicago We prove that multilinear (tensor) analogues of many efﬁciently computable problems in numerical linear algebra are NP-hard. Our list includes: determining the feasibility of a system of bilinear equations, de- NP Complete (abbreviated as NPC) problems, standing at the crux of deciding whether P=NP, are among hardest problems in computer science and other related areas. Through decades, NPC problems are treated as one class. Observing that NPC problems have different natures, it is unlikely that they will have the same complexity. Our intensive study shows that NPC problems are not all equivalent in

Most Tensor Problems Are NP-Hard CHRISTOPHER J. HILLAR, Mathematical Sciences Research Institute LEK-HENG LIM, University of Chicago We prove that multilinear (tensor) analogues of many efﬁciently computable problems in numerical linear algebra are NP-hard. Our list includes: determining the feasibility of a system of bilinear equations, de- Some First NP-complete problem We need to nd some rst NP-complete problem. Finding the rst NP-complete problem was the result of the Cook-Levin theorem. We’ll deal with this later. For now, trust me that: Independent Set is a packing problem and is NP-complete. Vertex Cover is a covering problem and is NP-complete.

NP problems have their own significance in programming, but the discussion becomes quite hot when we deal with differences between NP, P , NP-Complete and NP-hard. P and NP- Many of us know the difference between them. P- Polynomial time solving. Problems … A Useful List of NP-Complete Problems Graphs. Vertex Cover Decision Problem(VC): Given a graph G=(V,E) and a positive integer k, is there a subset V' of V of vertices which form a Vertex Cover for G with the size of V' no more than k. .(A Vertex Cover for G is a set of vertices with the property that every edge has at least one vertex from that set as an endpoint.)

P, NP, and NP-Completeness The Basics of Computational Complexity The focus of this book is the P versus NP Question and the theory of NP-completeness. It also provides adequate preliminaries regarding computational problems and compu-tational models. The P versus NP Question asks whether ﬁnding solutions is harder than checking NP-complete Problems and Physical Reality Scott Aaronson∗ Abstract Can NP-complete problems be solved eﬃciently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adia-batic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation,

NP-complete Problems and Physical Reality Scott Aaronson∗ Abstract Can NP-complete problems be solved eﬃciently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adia-batic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, NP-complete problems off all shapes and colors. •These are universal NP-problems...if you can solve them efficiently, you can solve ANY problem in NP efficiently. •L is NP-complete if: –L is in NP –ANY other problem in NP reduces to L. •If you come up with an efficient algorithm to 3-color a map, then P=NP. • NP-complete problems are always yes/no questions. • In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a function, f(x), of the input, x. • Optimization problems, strictly speaking, can’t be NP-complete (only NP-hard). P, NP, and NP-Completeness The Basics of Computational Complexity The focus of this book is the P versus NP Question and the theory of NP-completeness. It also provides adequate preliminaries regarding computational problems and compu-tational models. The P versus NP Question asks whether ﬁnding solutions is harder than checking

## NP-complete problem mathematics Britannica Combinatorial Optimization with Graph Convolutional. NP-complete Problems and Physical Reality Scott Aaronson∗ Abstract Can NP-complete problems be solved eﬃciently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adia-batic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation,, 16 NP-Hard Problems (December 3 and 5) 16.1 ‘E cient’ Problems A long time ago1, theoretical computer scientists like Steve Cook and Dick Karp decided that a minimum requirement of any e cient algorithm is that it runs in polynomial time: O(nc) for some constant c. People recognized early on that not all problems can be solved this quickly.

### NP-Complete Problems Virginia Tech

Karp's 21 NP-complete problems Wikipedia. LECTURE NOTES: NP-COMPLETE PROBLEMS 3 checked in polynomial time. W is known as the ”witness” or ”certiﬁcate.” At worst, all solutions w must be checked, giving exponential running time., Clearly, all problems in P are also in NP, so P NP. NP-Complete problems are in a formal sense the hardest problems in NP|if any one of them can be solved in poly time, then they can all be solved in poly time; this would result in the set equality P = NP. Similarly, if any one of the NP-Complete problem can be shown to require exponential.

Optimization Problems NP-complete problems are always yes/no questions. In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a … NP-Hard and NP-Complete Problems For many of the problems we know and study, the best algorithms for their solution have computing times can be clustered into two groups- 1. Solutions are bounded by the polynomial 2. Solutions are bounded by a nonpolynomial . No one has been able to device an algorithm which is bounded by the polynomial of small degree for the problems belonging to the second

Since the original results, thousands of other problems have been shown to be NP-complete by reductions from other problems previously shown to be NP-complete; many of these problems are collected in Garey and Johnson's 1979 book Computers and Intractability: A Guide to the Theory of NP-Completeness. NP-complete problems are closely related to each other and all can be reduced to each other in polynomial time. (Of course, not all such reductions are efﬁcient.) In this work we focus on four canonical NP-hard problems . Maximal Independent Set (MIS). Given an undirected graph, ﬁnd the largest subset of vertices in

Some First NP-complete problem We need to nd some rst NP-complete problem. Finding the rst NP-complete problem was the result of the Cook-Levin theorem. We’ll deal with this later. For now, trust me that: Independent Set is a packing problem and is NP-complete. Vertex Cover is a covering problem and is NP-complete. THE P VERSUS NP PROBLEM STEPHEN COOK 1. Statement of the Problem The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To deﬁne the problem precisely it is necessary to give a formal model of a

algorithm, then P = NP. If any problem in NP cannot be solved by a polynomial-time deterministic algorithm, then NP-complete problems are not in P. • This theorem makes NP-complete problems the focus of the P=NP question. • Most theoretical computer scientists believe that P ≠ NP. But no one has proved this yet. NP P NP P NP-complete NP NP-complete Problems and Physical Reality Scott Aaronson∗ Abstract Can NP-complete problems be solved eﬃciently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adia-batic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation,

NP-hard problem. Deciding if a graph have a MIS of size k, deciding if a subset is a MIS, are NP-complete problems (this is typically proven via a reduction to SAT). This problem was one of Richard Karp’s original 21 problems shown NP-complete in his 1972 seminal article . NP-Hard and NP-Complete Problems For many of the problems we know and study, the best algorithms for their solution have computing times can be clustered into two groups- 1. Solutions are bounded by the polynomial 2. Solutions are bounded by a nonpolynomial . No one has been able to device an algorithm which is bounded by the polynomial of small degree for the problems belonging to the second

2.Select a problem Z known to be NP-Complete. 3.Consider an arbitrary instance s Z of problem Z. Show how to construct, in polynomial time, an instance s X of problem X such that (a)If s Z 2 Z, then s X 2 X and (b)If s X 2 X, then sz 2 z. T. M. Murali December 2, 2009 CS 4104: NP-complete problems 1 NP-complete problems In the previous lecture notes, we deﬁned the notion of completeness for a com-plexity class. A priori, it is not clear that complete problems even exist for natural complexity classes such as NP or PSPACE. Fascinatingly, completeness turns out to be a pervasive phenomenon - most natural problems in NP are

16 NP-Hard Problems (December 3 and 5) 16.1 ‘E cient’ Problems A long time ago1, theoretical computer scientists like Steve Cook and Dick Karp decided that a minimum requirement of any e cient algorithm is that it runs in polynomial time: O(nc) for some constant c. People recognized early on that not all problems can be solved this quickly NP-complete Problems and Physical Reality Scott Aaronson∗ Abstract Can NP-complete problems be solved eﬃciently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adia-batic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation,

Clearly, all problems in P are also in NP, so P NP. NP-Complete problems are in a formal sense the hardest problems in NP|if any one of them can be solved in poly time, then they can all be solved in poly time; this would result in the set equality P = NP. Similarly, if any one of the NP-Complete problem can be shown to require exponential 1 NP-complete problems In the previous lecture notes, we deﬁned the notion of completeness for a com-plexity class. A priori, it is not clear that complete problems even exist for natural complexity classes such as NP or PSPACE. Fascinatingly, completeness turns out to be a pervasive phenomenon - most natural problems in NP are

In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete.In his 1972 paper, "Reducibility Among Combinatorial Problems", Richard Karp used Stephen Cook's 1971 theorem that the boolean satisfiability problem is NP-complete (also called the Cook-Levin theorem) to show that there is a polynomial time many-one reduction from … NP-complete problem, any of a class of computational problems for which no efficient solution algorithm has been found. Many significant computer-science problems belong to this class—e.g., the traveling salesman problem, satisfiability problems, and graph-covering problems. So-called easy, or

• NP-complete problems are always yes/no questions. • In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a function, f(x), of the input, x. • Optimization problems, strictly speaking, can’t be NP-complete (only NP-hard). LECTURE NOTES: NP-COMPLETE PROBLEMS 3 checked in polynomial time. W is known as the ”witness” or ”certiﬁcate.” At worst, all solutions w must be checked, giving exponential running time.

More NP-Complete Problems Stanford University. 1 NP-complete problems In the previous lecture notes, we deﬁned the notion of completeness for a com-plexity class. A priori, it is not clear that complete problems even exist for natural complexity classes such as NP or PSPACE. Fascinatingly, completeness turns out to be a pervasive phenomenon - most natural problems in NP are, Adiabatic quantum optimization fails for random instances of NP-complete problems Boris Altshuler,1,2, Hari Krovi, 2,yand Jeremie Roland z 1Columbia University 2NEC Laboratories America Inc.

### ProblГЁme NP-complet вЂ” WikipГ©dia NP-complete problems web.cse.msstate.edu. NP-Hard and NP-Complete Problems For many of the problems we know and study, the best algorithms for their solution have computing times can be clustered into two groups- 1. Solutions are bounded by the polynomial 2. Solutions are bounded by a nonpolynomial . No one has been able to device an algorithm which is bounded by the polynomial of small degree for the problems belonging to the second, NP-hard problem. Deciding if a graph have a MIS of size k, deciding if a subset is a MIS, are NP-complete problems (this is typically proven via a reduction to SAT). This problem was one of Richard Karp’s original 21 problems shown NP-complete in his 1972 seminal article ..

CMSC 451 Reductions & NP-completeness. NP-complete problems are the hardest in NP: if any NP-complete problem is p-time solvable, then all problems in NP are p-time solvable How to formally compare easiness/hardness of problems? Reductions Reduce language L 1 to L 2 via function f: 1. Convert input x of L 1 to instance f(x) of L 2 2. Apply decision algorithm for L 2 to f(x) Running time = time to compute f + time to apply decision, NP-Hard and NP-Complete Problems 2 – The problems in class NPcan be veriﬁed in polynomial time If we are given a certiﬁcate of a solution, we can verify that ….

### NP-Completeness Stanford University More NP-Complete Problems. Computer Algorithms Design and Analysis Known NP-Complete Problem Garey & Johnson: Computer and Intractability: A Guide to the Theory of NP-Completeness, Freeman, 1979 About 300 problems i.e. SAT, Clique, Hamiltonian, Partition, Knapsack … Note: 0-1 knapsack problem is NPC problem, but it can be solved by using dynamic programming in polynomial time, think about why and NP-complete problems are the hardest in NP: if any NP-complete problem is p-time solvable, then all problems in NP are p-time solvable How to formally compare easiness/hardness of problems? Reductions Reduce language L 1 to L 2 via function f: 1. Convert input x of L 1 to instance f(x) of L 2 2. Apply decision algorithm for L 2 to f(x) Running time = time to compute f + time to apply decision. • Liste de problГЁmes NP-complets вЂ” WikipГ©dia
• NP hard and NP Complete problems Basic Concepts
• NP hard and NP Complete problems Basic Concepts
• NP-Hard and NP-Complete Problems

• NP-Complete: can be solved in Polynomial time only using a Non-deterministic method. NP-Complete may not last. Oh, one more thing, it is believed that if anyone could *ever* solve an "NP-Complete" problem in "P" time, then *all* "NP-complete" problems could also be solved that way by using the same method, and the whole class of "NP-Complete NP-complete problems are closely related to each other and all can be reduced to each other in polynomial time. (Of course, not all such reductions are efﬁcient.) In this work we focus on four canonical NP-hard problems . Maximal Independent Set (MIS). Given an undirected graph, ﬁnd the largest subset of vertices in

NP Complete (abbreviated as NPC) problems, standing at the crux of deciding whether P=NP, are among hardest problems in computer science and other related areas. Through decades, NPC problems are treated as one class. Observing that NPC problems have different natures, it is unlikely that they will have the same complexity. Our intensive study shows that NPC problems are not all equivalent in algorithm, then P = NP. If any problem in NP cannot be solved by a polynomial-time deterministic algorithm, then NP-complete problems are not in P. • This theorem makes NP-complete problems the focus of the P=NP question. • Most theoretical computer scientists believe that P ≠ NP. But no one has proved this yet. NP P NP P NP-complete NP

NP Certification algorithm intuition. ・Certifier views things from "managerial" viewpoint. ・Certifier doesn't determine whether s ∈ X on its own; rather, it checks a proposed proof t that s ∈ X. Def. Algorithm C(s, t) is a certifier for problem X if for every string s, s ∈ X iff there exists a string t such that C(s, t) = yes. Def. NP is the set of problems for which there exists a An important notion in this context is the set of NP-complete decision problems, which is a subset of NP and might be informally described as the "hardest" problems in NP. If there is a polynomial-time algorithm for even one of them, then there is a polynomial-time algorithm for all the problems in NP.

NP-complete special cases include the edge dominating set problem, i.e., the dominating set problem in line graphs. NP-complete variants include the connected dominating set problem and the maximum leaf spanning tree problem. Clearly, all problems in P are also in NP, so P NP. NP-Complete problems are in a formal sense the hardest problems in NP|if any one of them can be solved in poly time, then they can all be solved in poly time; this would result in the set equality P = NP. Similarly, if any one of the NP-Complete problem can be shown to require exponential

Class NP, NP-complete, and NP-hard problems W. H¨am¨al¨ainen November 6, 2006 1 Class NP Class NP contains all computational problems such that the corre- sponding decision problem can be solved in a polynomial time by a nondeterministic Turing machine. algorithm, then P = NP. If any problem in NP cannot be solved by a polynomial-time deterministic algorithm, then NP-complete problems are not in P. • This theorem makes NP-complete problems the focus of the P=NP question. • Most theoretical computer scientists believe that P ≠ NP. But no one has proved this yet. NP P NP P NP-complete NP

Some First NP-complete problem We need to nd some rst NP-complete problem. Finding the rst NP-complete problem was the result of the Cook-Levin theorem. We’ll deal with this later. For now, trust me that: Independent Set is a packing problem and is NP-complete. Vertex Cover is a covering problem and is NP-complete. An Annotated List of Selected NP-complete Problems. Université de Liverpool, Département d'informatique, COMP202. Pierluigi Crescenzi, Viggo Kann, Magnús Halldórsson, Marek Karpinski, and Gerhard Woeginger. A compendium of NP optimization problems. KTH NADA. Stockholm. Voir aussi. 21 problèmes NP-complets de Karp

On dit qu’un problème de décision est « NP-complet » lorsque le langage correspondant est NP-complet. C’est une notion informelle car il existe plusieurs moyens de coder les instances d’un problème, mais cela ne pose pas de difficultés dans la mesure où on emploie un codage raisonnable du problème vers le langage considéré (voir la section NP-complétude faible). NP-Hard and NP-Complete Problems For many of the problems we know and study, the best algorithms for their solution have computing times can be clustered into two groups- 1. Solutions are bounded by the polynomial 2. Solutions are bounded by a nonpolynomial . No one has been able to device an algorithm which is bounded by the polynomial of small degree for the problems belonging to the second

NP Certification algorithm intuition. ・Certifier views things from "managerial" viewpoint. ・Certifier doesn't determine whether s ∈ X on its own; rather, it checks a proposed proof t that s ∈ X. Def. Algorithm C(s, t) is a certifier for problem X if for every string s, s ∈ X iff there exists a string t such that C(s, t) = yes. Def. NP is the set of problems for which there exists a NP-Completeness How would you define NP-Complete? They are the “hardest” problems in NP Definition of NP-Complete Q is an NP-Complete problem if: 1) Q is in NP 2) every other NP problem polynomial time reducible to Q Getting Started How do you show that EVERY NP problem reduces to Q? One way would be to already have an NP-Complete problem

2.Select a problem Z known to be NP-Complete. 3.Consider an arbitrary instance s Z of problem Z. Show how to construct, in polynomial time, an instance s X of problem X such that (a)If s Z 2 Z, then s X 2 X and (b)If s X 2 X, then sz 2 z. T. M. Murali December 2, 2009 CS 4104: NP-complete problems NP problems have their own significance in programming, but the discussion becomes quite hot when we deal with differences between NP, P , NP-Complete and NP-hard. P and NP- Many of us know the difference between them. P- Polynomial time solving. Problems …

algorithm, then P = NP. If any problem in NP cannot be solved by a polynomial-time deterministic algorithm, then NP-complete problems are not in P. • This theorem makes NP-complete problems the focus of the P=NP question. • Most theoretical computer scientists believe that P ≠ NP. But no one has proved this yet. NP P NP P NP-complete NP NP-complete problems are the hardest in NP: if any NP-complete problem is p-time solvable, then all problems in NP are p-time solvable How to formally compare easiness/hardness of problems? Reductions Reduce language L 1 to L 2 via function f: 1. Convert input x of L 1 to instance f(x) of L 2 2. Apply decision algorithm for L 2 to f(x) Running time = time to compute f + time to apply decision However, David Zuckerman showed in 1996 that every one of these 21 problems has a constrained optimization version that is impossible to approximate within any constant factor unless P = NP, by showing that Karp's approach to reduction generalizes to a specific type of approximability reduction. THE P VERSUS NP PROBLEM STEPHEN COOK 1. Statement of the Problem The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To deﬁne the problem precisely it is necessary to give a formal model of a