It is not intended to be an exact definition, but should help you to understand the concept. If y is npcomplete and x 2npsuch that y p x, then x is npcomplete. Introduction to theory of computation p, np, and npcompleteness sungjin im university of california, merced 04232015. If an nphard problem belongs to set np, then it is npcomplete. X is npcomplete x is nphard x is in np, but not necessarily npcomplete question 18 both npcomplete both in p npcomplete and in p, respectively undecidable and npcomplete, r espectively c d analysis of algorithms np complete gate it 2006 discuss it for problems x and y, y is npcomplete and x r educes to y in polynomial time.
Npcomplete article about npcomplete by the free dictionary. Strategy 3sat sequencing problemspartitioning problemsother problems proving other problems npcomplete i claim. To do this, we use the polynomialtime reduction from 3sat to indset that we just gave. We want some way to classify problems that are hard to solve, i. A language in l is called npcomplete iff l is nphard and l. Note that nphard problems do not have to be in np, and they do not have to be decision problems. This discussion is deliberately fuzzy, since it is supposed to be addressed to a child as per the question. Nphard problems a problem x is nphard if there is an npcomplete problem y that can be polynomially turing reduced to it i. So all we need to show is that every problem in np is polynomialtime reducible to indset.
We can show that problems are npcomplete via the following steps. Npcomplete the group of problems which are both in np and nphard are known as npcomplete problem. Nphardness a language l is called nphard iff for every l. All i know is that np is a subset of npcomplete, which is a subset of nphard, but i have no idea what they actually mean. 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 length, and. Npcomplete problems can provably be solved in polynomial time, but only in a nonblackbox setting. In computational complexity theory, np is the class of decision problems whose solution can be veri ed in polytime. 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.
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. 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. It means that we can verify a solution quickly np, but its at least as hard as the hardest problem in np nphard. Algorithm cs, t is a certifier for problem x if for every string s, s.
It was set up this way because its easier to compare the difficulty of decision problems than that of optimization problems. Usually we focus on length of the output from the transducer, because. The precise definition here is that a problem x is nphard, if there is an npcomplete problem y, such that y is reducible to x in polynomial time. The second part is giving a reduction from a known npcomplete problem. A problem that is npcomplete can be solved in polynomial time iff all other npcomplete problems can also be solved in polynomial time nphard. This is a rough guide to the meaning of npcomplete. Np hard and np complete an algorithm a is of polynomial complexity is there exist a polynomial p such that the computing time of a is opn. To keep things simple, lets just talk about problems with yesno answers. To answer the rest of question, you first need to understand which nphard problems are also npcomplete. Np perhaps you have heard of some of the following terms.
Under the assumption of p np, no nphard problem can be solved in. Thus a solution for one npcomplete problem would solve all problems in. Npcomplete means that a problem is both np and nphard. A trivial example of np, but presumably not npcomplete is finding the bitwise and of two strings of n boolean bits.
Np is the set of problems for which there exists a. I dont really know what it means for it to be nondeterministic. Chapter 34 as an engineer or computer scientist, it is important not only to be able to solve problems, but also to know which problems one can expect to solve ef. Nphard and npcomplete problems 2 the problems in class npcan be veri. I given a new problem x, a general strategy for proving it npcomplete is 1. P is a set of all decision problems solvable by a deterministic algorithm in polynomial time. Demainey alan guoyz march 9, 2012 abstract we prove nphardness results for ve of nintendos largest video game franchises. Nphard and npcomplete an algorithm a is of polynomial complexity is there exist a polynomial p such that the computing time of a is opn. The first part of an npcompleteness proof is showing the problem is in np. Thus, finding an efficient algorithm for any npcomplete problem implies that an efficient algorithm can be found for all np problems, since a solution for any problem belonging to this class can be recast into a solution for any other member of the class. A simple example of an nphard problem is the subset sum problem. The challenge of the problem is that the traveling salesman wants to minimize the total length of the trip. A problem is said to be nphard if everything in np can be transformed in polynomial time into it, and a problem is npcomplete if it is both in np and nphard. P and npcomplete class of problems are subsets of the np class of problems.
Np or p np nphardproblems are at least as hard as an npcomplete problem, but npcomplete technically refers only to decision problems,whereas. Informally, a language lis in np if there is a \guessandcheck algorithm for l. Wikipedia isnt much help either, as the explanations are still a bit too high level. What is the definition of p, np, npcomplete and nphard. Actually thats not hard to believe, its not like this site is a trove of content but its a goal for 2015 to get content more consistently up here. They are the hardest problems in np definition of npcomplete q is an npcomplete problem if. Given n jobs with processing times p j, schedule them on m machines so as to minimize the makespan. That is, there has to be an e cient veri cation algorithm with the. Therefore, npcomplete set is also a subset of nphard set.
Nphardness nondeterministic polynomialtime hardness is, in computational complexity theory, the defining property of a class of problems that are informally at least as hard as the hardest problems in np. Tractability polynomial time ptime onk, where n is the input size and k is a constant problems solvable in ptime are considered tractable np complete problems have no known ptime. Np, because we constructed a polynomialtime verifier for it. If an nphard problem can be solved in polynomial time then all npcomplete problems can also be solved in polynomial time. Npcompleteness applies to the realm of decision problems. If both are satisfied then it is an np complete problem. Np is the class of decision problems for which an affirmative answer can be verified in polynomial time, and np complete problems are the hardest of this class. Np l there is a polynomialtime verifier for l the name np comes from another way of characterizing np. I would like to add to the existing answers and also focus strictly on nphard vs npcomplete class of problems. A common method of demonstrating a problem to be intrinsically hard is to show the problem to be npcomplete. If any npcomplete problem has a polynomial time algorithm, then p np. If any npcomplete problem has a polynomial time algorithm, all problems in np do. Complexity and npcompleteness supplemental reading in clrs.
In reality, though, being able to solve a decision. The npcomplete problems represent the hardest problems in np. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or. A problem is said to be in complexity class p if there ex. N verify that the answer is correct, but knowing how to and two bit strings doesnt help one quickly find, say, a hamiltonian cycle or tour.
The traveling salesman problem consists of a salesman and a set of cities. Npcomplete partitioning problems columbia university. 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. More npcomplete problems nphard problems tautology problem node cover knapsack. If you introduce nondeterministic turing machines and appropriately define polynomial time, then np is the set of problems that an ntm can solve in polynomial time. The point to be noted here, the output is already given, and you can verify the outputsolution within the polynomial time but. Tractability polynomial time ptime onk, where n is the input size and k is a constant problems solvable in ptime are considered tractable npcomplete problems have no known ptime. P and np many of us know the difference between them. Intuitively, these are the problems that are at least as hard as the npcomplete problems. If we know a single problem in npcomplete that helps when we are asked to prove some other problem is npcomplete.
Npcomplete complexity npc, nondeterministic polynomial time complete a set or property of computational decision problems which is a subset of np i. How to explain np complete and nphard to a child quora. Classic nintendo games are nphard greg aloupis erik d. The salesman has to visit each one of the cities starting from a certain one and returning to the same city. Np problems have their own significance in programming, but the discussion becomes quite hot when we deal with differences between np, p, npcomplete and nphard. These are just my personal ideas and are not meant to be rigorous. What are the differences between np, npcomplete and nphard. Trying to understand p vs np vs np complete vs np hard.