Most of the problems that well see that are np hard are also np complete. Recall that a prime number is an integer greater than 1 whose only positive factors are 1 and the number itself. Strategy 3sat sequencing problemspartitioning problemsother problems proving other problems npcomplete i claim. An np complete problem is one that can be converted into any other np problem in a reasonable, i. It is not intended to be an exact definition, but should help you to understand the concept. Sat boolean satisfiability problem is the first npcomplete problem proved by cook see clrs book for proof. P versus np a gift to mathematics from computer science. Computers and intractability a guide to the theory of npcompleteness. What we care about, from an algorithmic perspective, is proving the problems are np complete. P, np and mathematics a computational complexity perspective. Theory of npcompleteness show that many of the problems with no polynomial time algorithms are computationally related the group of problems is further subdivided into two classes npcomplete.
Nov 15, 2008 a blog by jeff atwood on programming and human factors. Amazingly enough, npcomplete problems exist, and furthermore hundreds of natural computational problems arising in many different areas of mathematics. In order to get a problem which is np hard but not np complete, it suffices to find a computational class which a has complete problems, b provably contains np, and c is provably different from np. These are just my personal ideas and are not meant to be rigorous. The main source of this knowledge was the theory of computation community, which has been my academic and social home throughout this period. Traveling salesman problem given n cities 1, 2, 3, and integer distances. What npcomplete problems are most similar to integer factoring. The np complete problems represent the hardest problems in np. Many of the problems for which there is no polynomial time algorithm available are. Note that these problems are simple to state just because a topic is accessibile does not mean that it is easy. Problem description algorithm yes no multiple is x a multiple of y. The focus of this book is the pversusnp question and the theory of npcompleteness.
The problems are culled from various sources, many from actual contests. There must be some first npcomplete problem proved by definition of npcomplete problems. So if you have a p algorithm for any npcomplete problem, then you have a p algorithm for all np problems and npp. If a problem is proved to be npc, there is no need to waste time on trying to find an efficient algorithm for it. The consideration of np complete problems typically involves enumerative problems in graph theory, one example being the existance of a hamiltonian cycle in a graph. The center for discrete mathematics and theoretical computer science. Freeman, 1979 david johnson also runs a column in the journal journal of algorithms in the hcl. Have you ever heard a software engineer refer to a problem as np complete.
This is a list of some of the more commonly known problems that are np complete when expressed as decision problems. The problem for graphs is npcomplete if the edge lengths are assumed integers. It is always useful to know about npcompleteness even for engineers. In computational complexity theory, a problem is npcomplete when it can be solved by a restricted class of brute force search algorithms and it can be used to simulate any other problem with a similar algorithm. While pspace contains np, and has complete problems, the containment is. It is widely believed that showing a problem to be np complete is tantamount to proving its computational intractability. The perhaps first really good book that illustrates these conversions is by garey and.
New nphard and npcomplete polynomial and integer divisibility. If a problem is np and all other np problems are polynomialtime reducible to it, the problem is npcomplete. So, my question is, how were the first npcomplete problems proven to be npcomplete. It is always useful to know about np completeness even for engineers. Show that a problem already known to be npcomplete can be reduced to the new problem. Nphard nondeterministic polynomialtime hard, in computational complexity theory, is a class of problems that are, informally, at least as hard as the hardest problems in np. A list of books and papers ill reference as i discuss the problems. If any np complete problem has a polynomial time algorithm, all problems in np do. Satisfiability the theory of np completeness started from satisfiability, which was proved to be np complete. The focus of this book is to teach the reader how to identify, deal with, and understand the essence of npcomplete problems. So when we prove this, we prove that there is basically. Np or p np nphardproblems are at least as hard as an npcomplete problem, but npcomplete technically refers only to decision problems,whereas.
It was the first book exclusively on the theory of npcompleteness and. Show that a problem already known to be np complete can be reduced to the new problem. One example of the problems they show to be npcomplete is the following decision problem. The set of np complete problems is often denoted by np c or npc. Two npcomplete problems in coding theory with an application in code based cryptography abstract. Npcomplete problem, any of a class of computational problems for which no efficient solution algorithm has been found. Conversely, if you show that one particular problem in np is intractable, then all npcomplete problems would. Proofs are given when appropriate, or when they illustrate some insight or important idea.
An annotated list of selected np complete problems. The title text refers to the fact that npcomplete problems have no known polynomial time general solutions, and it is unknown if such a solution can ever be found. Satisfiability the theory of npcompleteness started from satisfiability, which was proved to be npcomplete. Many significant computerscience problems belong to this classe.
The authors very intuitive account of the most fundamental issues in complexity theory makes the book suitable for a firstyear computational complexity class. What are some of the most notable npcomplete problems. Foundations of np completeness the complexity of theorem proving procedures, stephen cook, 1971. In this paper it is shown that the reconstruction of a punctured code from a given code is an np complete problem. The problem is known to be nphard with the nondiscretized euclidean metric. Schaums outline of theory and problems of combinatorics. Proving np completeness by reduction to prove a problem is np complete, use the earlier observation. Computers and intractability a guide to the theory of np completeness. Modern approach, by arora and barak gives such example. The book includes three additional undercurrents, in the form of three series of separate. And in real life, npcomplete problems are fairly common, especially in large scheduling tasks. Important things of this paper cook suggested other problems in \ np \ might be as hard as the satisfiability problem and be members of the \ np \ class i. I regret that, because of both space and cognitive limitations, i was unable to discuss every paper related to the solvability of np complete problems in the physical world.
Although a solution to an np complete problem can be verified quickly, there is no known way to find a solution quickly. Important things of this paper cook suggested other problems in \np\ might be as hard as the satisfiability problem and be members of the \np\ class i. The theory of the npcompleteness does not provide any method of obtaining polynomial time algorithms for the problems of the second group. This book is a classic, developing the theory, then cataloguing many npcomplete problems.
Two np complete problems in coding theory with an application in code based cryptography abstract. For math students that have never learned number theory and want to learn it quickly and actively, this is still your best choice. Some simplified npcomplete problems proceedings of the. In this paper it is shown that the reconstruction of a punctured code from a. Number theory for beginners by andre weil is the slickest,most concise yet best written introduction to number theory ive ever seenits withstood the test of time very well. Np complete problems are defined in a precise sense as the hardest problems in p. As it happens, we end the story exactly where we started it, with shors quantum algorithm for factoring. David johnson also runs a column in the journal journal of algorithms in the hcl.
The primary topics in this part of the specialization are. If sis np complete, t2np and s p t, then tis np complete. It can be used as an introduction as either a textbook or for selfstudy, or to experts, since it provides expositions of the various subareas of complexity theory. Most of the problems in this list are taken from garey and johnsons seminal book.
Instead, we can focus on design approximation algorithm. The \hardest problems in np are the npcomplete problems. Npcompleteness, various heuristics, as well as quantum algorithms, perhaps the most advanced and modern topic. This is a list of some of the more commonly known problems that are npcomplete when. The problem for points on the plane is npcomplete with the discretized euclidean metric and rectilinear metric.
That is the np in nphard does not mean nondeterministic polynomial time. P, np, and npcompleteness guide books acm digital library. Two npcomplete problems in coding theory with an application. The complexity class p is fully contained in the class np since it. If sis npcomplete, t2np and s p t, then tis npcomplete. So, my question is, how were the first np complete problems proven to be np complete. The pversusnp question asks whether or not finding solutions is harder than checking the correctness of solutions. Optimization problems 3 that is enough to show that if the optimization version of an npcomplete problem can be solved in polytime, then p np. Most of the problems that well see that are nphard are also npcomplete. If y is npcomplete and x 2npsuch that y p x, then x is npcomplete. A guide to the theory of npcompleteness is an influential textbook by michael garey and david s. A guide to the theory of npcompleteness series of books in the mathematical sciences at.
March 27, 2018 acknowledgments in this book i tried to present some of the knowledge and understanding i acquired in my four decades in the eld. This is a list of some of the more commonly known problems that are np complete when. Valenza, graduate text 186, springer 1999 fermats last theorem for amateurs, p. A guide to the theory of npcompleteness, and are here presented in the same order and organization. What are some of the np hardnp complete problems in algebra. This is a rough guide to the meaning of npcomplete. Decision problems for which there is a polytime algorithm. This problem is known to be in np, but it is unknown if it is npcomplete. The pversusnp question asks whether or not finding solutions. For example, here are some problems in number theory that remain unsolved. At one time, the set of known np complete problems must have been zero, and this would have made it impossible to resort to step 2 in the above process. There must be some first np complete problem proved by definition of np complete problems.
Proving npcompleteness by reduction to prove a problem is npcomplete, use the earlier observation. Do you stand by the statement testing whether a solved n. Example of a problem that is nphard but not npcomplete. The title text refers to the fact that np complete problems have no known polynomial time general solutions, and it is unknown if such a solution can ever be found. In order to get a problem which is nphard but not npcomplete, it suffices to find a computational class which a has complete problems, b provably contains np, and c is provably different from np. A related problem is to find a partition that is optimal terms of the number of edges between parts. Schrijver, wiley 1999 fourier analysis on number fields, d.
It is widely believed that showing a problem to be npcomplete is tantamount to proving its computational intractability. At one time, the set of known npcomplete problems must have been zero, and this would have made it impossible to resort to step 2 in the above process. If the waiter can find an efficient general solution to this, he will have solved one of the most famous problems in mathematics. I regret that, because of both space and cognitive limitations, i was unable to discuss every paper related to the solvability of npcomplete problems in the physical world. Strategy 3sat sequencing problemspartitioning problemsother problems proving other problems np complete i claim. Sat boolean satisfiability problem is the first np complete problem proved by cook see clrs book for proof. Oct 29, 2009 part of the questions allure is that the vast majority of np problems whose solutions seem to require exponential time are whats called npcomplete, meaning that a polynomialtime solution to one can be adapted to solve all the others. Np completeness the theory of np completeness is a solution to the practical problem of applying complexity theory to individual problems. Because if we prove npcompletenessi mean, really we care about nphardness, but we might as well do npcompleteness. In this paper we show that a number of npcomplete problems remain npcomplete even when their domains are substantially restricted. Have you ever heard a software engineer refer to a problem as npcomplete. It is the the advent of the computer though that has had the greatest influence on combinatorics, and vice versa. More precisely, each input to the problem should be associated with a set of solutions of polynomial length, whose validity can be tested quickly in polynomial time, such that the output for any input is yes if the solution set is nonempty and no if it is empty.
Jan 25, 2018 np hard and np complete problems watch more videos at. If y is np complete and x 2npsuch that y p x, then x is np complete. See pages 249251 of garey and johnson, computers and intractability, for a dozen np complete problems in number theory. They define the decision problem integer factoring on input of three. A strong argument that you cannot solve the optimization version of an npcomplete problem in polytime.
An npcomplete problem is one that can be converted into any other np problem in a reasonable, i. This book offers a comprehensive perspective to modern topics in complexity theory. So if you have a p algorithm for any np complete problem, then you have a p algorithm for all np problems and npp. Foundations of npcompleteness the complexity of theorem proving procedures, stephen cook, 1971. Np hard and np complete problems watch more videos at. Many papers just say that the optimization problem they have is np hard and. In computer science, more specifically computational complexity theory, computers and intractability. Because if we prove np completenessi mean, really we care about np hardness, but we might as well do np completeness. In this paper we show that a number of np complete problems remain np complete even when their domains are substantially restricted. This list is in no way comprehensive there are more than 3000 known npcomplete problems.
The problems are culled from various sources, many from actual contests and olympiads, and in general are very di. Following are some np complete problems, for which no polynomial time algorithm. As there are hundreds of such problems known, this list is in no way comprehensive. A problem that is npcomplete can be solved in polynomial time iff all other npcomplete problems can also be solved in polynomial time nphard. It also provides adequate preliminaries regarding computational problems and computational models. Interactions of computational complexity theory and mathematics. The pversus np question asks whether or not finding solutions is harder than checking the correctness of solutions. At worst, all solutions w must be checked, giving exponential running time. The focus of this book is the pversus np question and the theory of np completeness. In the list of npcomplete problems below, the form of a typical entry is as follows. The complexity class p is fully contained in the class np since it takes polynomial time to solve the problem, it also.