Completeness problem solving in automata, languages. Winter2005 young cs 599 npcompleteness so, if an npcomplete problem is in p then pnp if p. Introduction to theory of computation p, np, and np completeness sungjin im university of california, merced 04232015. P b cs3343analysisofalgorithms npcompleteness 4 problem reduction sequence cs3343analysisofalgorithms npcompleteness 5 3 3sat cs3343analysisofalgorithms npcompleteness 6 clique. Introduction to theory of computation p, np, and npcompleteness sungjin im university of california, merced. In the 1970s, theoretical computer scientists showed hun. P, np, and npcompleteness siddhartha sen questions. Tractability polynomial time ptime onk, where n is the input size and k is a constant.
Suppose we know sat is npcomplete somehow, which gives us for every problem a in np, we have a sat. Solving npcomplete problemssmall vertex coverstreesload balancingset cover t. Np complete the group of problems which are both in np and np hard are known as np complete problem. Proving a problem in np to be np complete tells us that it is as hard to solve as any other np problem. Np then all npcomplete problems are in npp question. Pdf npcomplete problems and proof methodology researchgate.
Completeness problem solving in automata, languages, and. Eppsteins longer, but very good introduction to np completeness, with sections like why should we care. Np and npcompleteness np np is a class of languages that contains all of p, but which most people think also contains many languages that arent in p. Informally, a language lis in np if there is a \guessandcheck algorithm for l. A problem p in np is npcomplete if every other problem in np can be transformed or reduced into p in polynomial time. Murali april 28, may 3, 2016 coping with npcompleteness. It is especially useful to understand np completeness.
It is clear that any np complete problem can be reduced to this one. Instead, we can focus on design approximation algorithm. With the verifier definition of np, one way of phrasing this question is if a solution to a problem can be checked efficiently, can that problem be solved efficiently. The p versus np problem is a major unsolved problem in computer science. Winter2005 young cs 599 npcompleteness 10 npcomplete problems. A problem that is npcomplete can be solved in polynomial time iff all other np. A problem p in np is np complete if every other problem in np can be transformed or.
A problem is np hard if all problems in np are polynomial time reducible to it, even though it may not be in np itself. Furthermore np is not a subset of nphard, since not every problem in np is hard. Np may be equivalently defined as the set of decision problems that can be solved in polynomial time on a nondeterministic turing machine. Many algorithms in security depends on the fact that no known solutions exist for np hard problems. Now suppose we have a np complete problem r and it is reducible to q then q is at least as hard as r and since r is an np hard problem.
The proof above of npcompleteness for bounded halting is great for the theory of npcompleteness, but doesnt help us understand other more abstract problems such as the hamiltonian cycle problem. If x is np complete and a deterministic, polynomialtime algorithm exists that can solve all instances of. Notes on npcompleteness this is a summary of what we did in class on the topic of npcompleteness. The proof above of np completeness for bounded halting is great for the theory of np completeness, but doesnt help us understand other more abstract problems such as the hamiltonian cycle problem. A compendium of np optimization problems edited by pierluigi crescenzi and viggo kann. My favorite np complete problem is the minesweeper problem. What is the definition of p, np, npcomplete and nphard. Most proofs of np completeness dont look like the one above.
The satisfiability problem sat study of boolean functions generally is concerned with the set of truth assignments assignments of 0 or 1 to each of the variables that make the function true. Np question is the most important question in theoretical computer science. Npcomplete the group of problems which are both in np and nphard are known as npcomplete problem. Since our choice of l was arbitrary, any language l. We will show that if there is such an approximation algorithm, then we could solve a known npcomplete problem hamiltonian cycle in polynomial time, so p would equal np. Ofn in np on the order of at most fn at most as hard as an npcomplete problem.
For many interesting problems we cannot nd a polynomial time algorithm we cannot prove that no polynomial time algorithm exists the best we can do is formalize a class of npcomplete problems that. It was set up this way because its easier to compare the difficulty of decision problems than that of optimization problems. Solving np complete problemssmall vertex coverstreestreewidthapproximation algorithms coping with np completeness t. P set of decision problems for which there exists a polytime algorithm.
What are the differences between np, npcomplete and nphard. Npcomplete problem, any of a class of computational problems for which no efficient solution algorithm has been found. Np complete problems are in np, the set of all decision problems whose solutions can be verified in polynomial time. Currently, there are many known problems in np, and there is no solution to show anyone of them in p.
Over the past seven chapters we have developed algorithms for finding shortest paths and minimum spanning. If an npcomplete problem can be solved in polynomial time, then all problems in np can be, and thus p np. A problem a is npcomplete if a 2np and every problem x 2np is reducible to a. For any problem y in np, there is a reduction from y to x. To prove that it is npcomplete, we will reduce the. A problem language l is nphard if every problem in np can be reduced to l in polynomial time. Decision problems for which there is a polytime certifier. Npcompleteness applies to the realm of decision problems. If we check npcompleteness by reducing from a known npcomplete problem, then how do we start. Now suppose we have a npcomplete problem r and it is reducible to q then q is at least as hard as r and since r is an nphard problem. Np completeness andreas klappenecker partially based on slides by jennifer welch dealing with npcomplete problems.
That means that a solution to any one npcomplete problem is a solution to all np problems. Winter2005 young cs 599 npcompleteness 9 one of the most important open problem in theoretical compute science. The other part is to show that another problem p, which is known to. It asks whether every problem whose solution can be quickly verified can also be solved quickly.
Every abstract problem has to be represented somehow for the computer to work on it ultimately a binary representation consider the problem. Another essential part of an npcompleteness proof is showing the problem is in np. Circuitsat given a combinational circuit with n gates, m. To get the theory going, we need to establish at least one np complete problem. Steve cook, leonid levin and richard karp 11, 28, 25 developed the initial theory of npcompleteness that generated multiple acm turing awards.
That is, for each language m in np, we can take an input x for m, transform it in polynomial time to an input x. We show how to prove that a problem is npcomplete, and give some help for coping with npcompleteness. Exp since we can enumerate all the exponential number of possible solutions to the problem and check each of them in polynomial time. Introduction to theory of computation p, np, and np. Most proofs of npcompleteness dont look like the one above. It is important to computer science because it has been proven that any problem in np can be transformed into another problem in npcomplete. Sometimes, we can only show a problem nphard if the problem is in p, then p np, but the problem may not be in np. Still faster than any exponential, and faster than we have a right to expect. P6 np famous open problem in computer science since 1971 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. Now we have shown what npcomplete is, lets show how we prove a problem is npcomplete. Npc np complete is a subset of np, not the other way around. Using the notion of npcompleteness, we can make an analogy between nphardness and bigo notation. 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. Following are some npcomplete problems, for which no polynomial time algorithm.
Np is the set of problems for which there exists a. This means that we provide a method running in poly nomial time that converts every. N in np really stands for nondeterministic if we can guess the solution we can solve the problem in polynomial time. Murali april 28, may 3, 2016 coping with np completeness.
Np complete problems are the hardest problems in np set. Cooklevin theorem 1971 sat is np complete 3sat 3sat f f is in conjunctive normal form, each clause has exactly 3 literals and f is satisfiable 3sat is np complete 2sat is in p np complete to prove a problem is np complete show a polynomial time reduction from 3sat other np complete problems. Np complete problems problem a is np complete ifa is in np polytime to verify proposed solution any problem in np reduces to a second condition says. That is, there has to be an e cient veri cation algorithm with the. Things that you will nd here but not in this textbook include. Solving np complete problemssmall vertex coverstreesload balancingset cover coping with np completeness t. This is the problem that given a program p and input i, will it halt. To get the theory going, we need to establish at least one npcomplete problem. Completeness always includes being an element of the class the problem is complete for. Gave rise to new and novel mathematical approaches, proofs, and analyses. Npcompleteness theory as we have already mentioned, the theory is based considering decision problems.
A reduction from problem a to problem b is a polynomialtime algorithm that converts inputs to problem a into equivalent inputs to problem b. P np if every np problem has a deterministic polynomial algorithm. We want some way to classify problems that are hard to solve, i. Introduction to npcompleteness these notesslides are intended as an introduction to the theory of npcompleteness, as a supplementary material to the rst sections in chapter 34 npcompleteness of the textbook. Eppsteins longer, but very good introduction to npcompleteness, with sections like why should we care. Many significant computerscience problems belong to this classe. Lecture npcompleteness spring 2015 a problem x is nphard if every problem y. Each input is a positive integer common way to encode an integer is in binary primes decision problem is 10,11,101,111, since 10 encodes 2, 11 encodes 3, 101 encodes. The problem in np hard cannot be solved in polynomial time, until p np. Reductions reduce language l 1 to l 2 via function f. That is, for any problem a in np, we need to find a reduction from a to sat.
Algorithm cs, t is a certifier for problem x if for every string s, s. A problem is nphard if all problems in np are polynomial time reducible to it. In 1971 stephen cook proved that the boolean satisfiability sat problem was npcomplete using an argument from how computers work not via. Pdf keywords some known npcomplete problems methodology for np completeness proofs example proofs conclusion see also references find, read. Cormen, leiserson and rivest, introduction to algorithms, 2nd ed, 2001. If a polynomial time algorithm exists for any of these problems, all problems in np would be polynomial time solvable. In reality, though, being able to solve a decision problem in polynomial time will often permit us to solve the corresponding optimization problem in. Said another way, if there is any np complete problem that admits an efficient solution then every np problem does so. Given this formal definition, the complexity classes are. Npcomplete problems are in np, the set of all decision problems whose solutions can be verified in polynomial time.
But if i use cookcompleteness, i cannot say anything of this type. A problem is npcomplete if it is both nphard and in np. A problem a is np complete if a 2np and every problem x 2np is reducible to a. Solving npcomplete problemssmall vertex coverstreestreewidthapproximation algorithms coping with npcompleteness t. Equivalent means that both problem a and problem b must output the. As another example, any np complete problem is np hard. It is especially useful to understand npcompleteness.
Np, then no polynomial time approximation algorithm for tsp wo triangle inequality can have a constant ratio bound. A language b is npcomplete if it satisfies two conditions. This can be done by showing that given the witness w, the correctness of w can be veri. We show how to prove that a problem is np complete, and give some help for coping with np completeness. A problem is in the class npc if it is in np and is as hard as any problem in np. An answer either way will give fundamental insights into the nature of computation. Definitive guide to npcompleteness lists hundreds of npcomplete problems gives reduction types and refs michael garey david johnson. Np hard and npcomplete classes a problem is in the class npc if it is in np and is as hard as any problem in np.