I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantummechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, malamenthogarth spacetimes, quantum gravity, closed timelike curves, and anthropic computing. In his 1972 paper, reducibility among combinatorial problems, richard karp used stephen cooks 1971 theorem that the boolean satisfiability problem is np complete also called the cooklevin theorem to show that there is a polynomial time manyone reduction from the. More npcomplete problems np hard problems tautology problem node cover knapsack. You can help by expanding it with reliably sourced entries. The focus of this book is the pversusnp question and the theory of npcompleteness. Average case complete problems siam journal on computing. The complexity class p is fully contained in the class np since it. Many interesting combinatorial problems were found to be npcomplete. This set of notes on number theory was originally written in 1995 for students at the imo level. This is a dynamic list and may never be able to satisfy particular standards for completeness. Usually we focus on length of the output from the transducer, because.
The \hardest problems in np are the np complete problems. The problem is known to be np hard with the nondiscretized euclidean metric. It is also the home of one of the most fundamental open problems in mathematics, namely the famous np versus p problem. This is the book to have if you are looking at p v np problems, and np completeness. Many significant computerscience problems belong to this classe. In complexity theory, the abbreviation np stands for. It is not intended to be an exact definition, but should help you to understand the concept. So when we prove this, we prove that there is basically. Nphard problems like npcomplete problems, but not necessarily in np.
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. 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. In his 1972 paper, reducibility among combinatorial problems, richard karp used stephen cooks 1971 theorem that the boolean satisfiability problem is npcomplete also called the cooklevin theorem to show that there is a polynomial time manyone reduction from. As there are hundreds of such problems known, this list is in no way. A guide to the theory of np completeness michael r. If an nphard problem belongs to set np, then it is npcomplete. The np complete problems represent the hardest problems in np. Conpcomplete problems are defined in the usual way.
The \hardest problems in np are the npcomplete problems. In computational complexity theory, karps 21 np complete problems are a set of computational problems which are np complete. Because if we prove npcompletenessi mean, really we care about nphardness, but we might as well do npcompleteness. It is not known whether or not conp and np are the same classes of problems. Np complete problems are in essence the hardest of the np problems. Have you ever heard a software engineer refer to a problem as npcomplete. A strong argument that you cannot solve the optimization version of an npcomplete problem in polytime. Analytic number theory lecture notes based on davenports book andreas str ombergsson these lecture notes follow to a large extent davenports book 15, b ut with things reordered and often expanded. There are many problems already known to be np complete, and listed in the bible of the subject. The point of these notes is not in t he rst place to serve as. Today, pure and applied number theory is an exciting mix of simultaneously broad and deep theory, which is constantly informed and motivated. Conversely, if you show that one particular problem in np is intractable, then all npcomplete problems would. The waiters problem is npcomplete, since a given orders price can be found and checked quickly, but finding an order to match a price is much harder. This is a rough guide to the meaning of np complete.
If an np hard problem belongs to set np, then it is np complete. Other resources, such as memory, parallelism and more are studied in computational complexity, but we will not treat them here. For example, since sat is in np, its complement x x is a formula that is not satisfiable is in conp. Show that, if np and conp are different, then p and np are also different. David johnson also runs a column in the journal journal of algorithms in the hcl. For example, here are some problems in number theory that remain unsolved. Since there is little hope to solve them fast in the worst case, researchers look for algorithms which are fast just on average. Computers and intractability a guide to the theory of npcompleteness. Most of the problems that well see that are nphard are also npcomplete. 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. The main source of this knowledge was the theory of computation community, which has been my academic and social home throughout this period. Sat boolean satisfiability problem is the first npcomplete problem proved by cook see clrs book for proof. A problem h is nphard if and only if there is an npcomplete problem l that is polynomial time turingreducible to h i. The problem for points on the plane is np complete with the discretized euclidean metric and rectilinear metric.
Example binary search olog n, sorting on log n, matrix multiplication 0n 2. In short, particular guesses in npcomplete problems can be checked easily, but systematically finding solutions is far more difficult. This is a list of some of the more commonly known problems that are np complete when expressed as decision problems. 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. Classes p and np are two frequently studied classes of problems in computer science. This list is in no way comprehensive there are more than 3000 known npcomplete problems.
Algebraic numbers, finite continued fractions, infinite continued fractions, periodic continued fractions, lagrange and pell, eulers totient function, quadratic residues and nonresidues, sums of squares and quadratic forms. There must be some first npcomplete problem proved by definition of npcomplete problems. The set of np complete problems is often denoted by np c or npc. This matter is sensitive to the choice of a particular npcomplete problem and a. This text is meant to be a reference, and not a replacement but rather a supplement to a number theory textbook.
Mathematics and computation a theory revolutionizing technology and science avi wigderson princeton university press princeton and oxford. To answer the rest of question, you first need to understand which np hard problems are also np complete. Note that these problems are simple to state just because a topic is accessibile does not mean that it is easy. An annotated list of selected np complete problems. A problem is npcomplete if it is both nphard and in np. We prove that p is not np in this theory if and only if p is not np in the bss theory over the reals. Conversely, if you show that one particular problem in np is intractable, then all np complete problems would. This text is meant to be a reference, and not a replacement but rather a supplement to. The complexity class p is fully contained in the class np since it takes polynomial time to solve the problem, it also. Complexity theory is the appropriate setting for the study of such problems. If any np complete problem has a polynomial time algorithm, all problems in np do.
Olympiad number theory through challenging problems. The problem for graphs is np complete if the edge lengths are assumed integers. This book aims to describe such recent achievements of complexity theory in the context of the classical results. To belong to set np, a problem needs to be i a decision problem, ii the number of solutions to the problem should be finite and each solution should be of polynomial. The p versus np problem clay mathematics institute. What are the current approaches for solving np complete. It also provides adequate preliminaries regarding computational problems and computational models. A course on number theory pdf 9p this note explains the following topics. Can npcomplete problems be solved efficiently in the physical universe. If you can solve a problem quickly you can also verify the solution quickly. It is always useful to know about npcompleteness even for engineers.
Posted in algebra and number theory tagged 3sat, an10, difficulty 8, nondivisibility of a product polynomial, root of modulus 1 protected. The book 15 provides an excellent reference to the subject, with 300 npcomplete problems listed in the appendix. The input size depends on a condition number, which is not assumed known by the machine. Algebra and number theory discussions of npcomplete. The pversusnp question asks whether or not finding solutions is harder than checking the correctness of solutions. Np complete problems problem a is npcomplete ifa is in np polytime to verify proposed solution any problem in np reduces to a second condition says. If a problem is np and all other np problems are polynomialtime reducible to it, the problem is npcomplete. A cut is a set of edges whose removal leaves a graph disconnected. In computational complexity theory, karps 21 npcomplete problems are a set of computational problems which are npcomplete. 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. Nov 15, 2008 a blog by jeff atwood on programming and human factors.
An example is checking whether universal product codes upc or international standard book number isbn codes are legitimate. P, np and mathematics a computational complexity perspective. P versus np a gift to mathematics from computer science. Have you ever heard a software engineer refer to a problem as np complete.
Group1consists of problems whose solutions are bounded by the polynomial of small degree. Computers and intractability a guide to the theory of np completeness. Although a solution to an np complete problem can be verified quickly, there is no known way to find a solution quickly. Recall that a prime number is an integer greater than 1 whose only positive factors are 1 and the number itself. It covers the theory, and then provides a list of a problems that are npcomplete, broken down into sections, with details of how the problem has been proved to be npcomplete, i. It covers the basic background material that an imo student should be familiar with.
To answer the rest of question, you first need to understand which nphard problems are also npcomplete. Divisibility is an extremely fundamental concept in number theory, and has applications including puzzles, encrypting messages, computer security, and many algorithms. The systematic study of number theory was initiated around 300b. It grew out of undergraduate courses that the author taught at harvard, uc san diego, and the university of washington. Solutions to practice questions for csci 6420 exam 2. Nathanson, graduate texts in mathematics 165, springer 1996 arithmetic, geometry and coding theory, proceedings of a conference at cirm luminy, june 28july 2, 1993. Inverse problems and the geometry of sumsets, melvyn b. The theory admits deterministic and nondeterministic polynomial time recognizable problems. This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. If you suspect a problem youre looking at is np complete, the first step is to look for it in garey and johnson. Np set of decision problems for which there exists a polytime certifier.
What we care about, from an algorithmic perspective, is proving the problems are npcomplete. Most of the problems in this list are taken from garey and johnsons seminal book. These are just my personal ideas and are not meant to be rigorous. Np hard and np complete problems basic concepts the computing times of algorithms fall into two groups.
We shall focus on time number of elementary operations3 performed as the primary resource of algorithms, when studying their e. Can np complete problems be solved efficiently in the physical universe. Note that all the p problems are also np problems, or to put it another way, the class p is contained within the class np. What are some of the np hardnp complete problems in algebra. List of npcomplete problems from wikipedia, the free encyclopedia here are some of the more commonly known problems that are np complete when expressed as decision problems. Npcomplete problem, any of a class of computational problems for which no efficient solution algorithm has been found.
More precisely, each input to the problem should be associated with a set of. This is a rough guide to the meaning of npcomplete. And in real life, npcomplete problems are fairly common, especially in large scheduling tasks. Then we develop a theory with weak and strong approximate compu. What are the differences between np, npcomplete and nphard. Npcomplete problems are in essence the hardest of the np problems. At worst, all solutions w must be checked, giving exponential running time.