All np complete problems are np hard, but all np hard problems are not np complete. To answer the rest of question, you first need to understand which nphard problems are also npcomplete. Abstract in 1955, john nash sent a remarkable letter to the national security agency, in which seeking to build theoretical foundations for cryptographyhe all but formulated what today. Computing i recently watched the travelling salesman movie that got me thinking about computational complexity, but i cant get my head around these three things np complete, np incomplete, np hard problems. More npcomplete problems np hard problems tautology problem node cover knapsack. Another essential part of an npcompleteness proof is showing the problem is in np. Decision vs optimization problems npcompleteness applies to the realm of decision problems. Minesweeper and complexity theory, the np completeness of minesweeper. That is, there has to be an e cient veri cation algorithm with the. So, nondeterminism is still part of the definition of np and is historically important, but the current status is that it is not mandatory. Nphard and npcomplete problems 2 the problems in class npcan be veri. 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. The reason is that if a is in np, and b is a np complete problem, then a can be reduced to b. In your case, problem a is complete for np, or npcomplete, if every problem in np reduces to a, and a is in np.
In computational complexity theory, a problem is npcomplete when it can be solved by a. Np hard and np complete classes a problem is in the class npc if it is in np and is as hard as any problem in np. It is not intended to be an exact definition, but should help you to understand the concept. That is a yesno question and so is a decision problem.
It was set up this way because its easier to compare the difficulty of decision problems. Abstract in 1955, john nash sent a remarkable letter to the national security agency, in which seeking to build theoretical foundations for cryptographyhe all. Np hardness 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. Np complete have the property that it can be solved in polynomial time if all other np complete problems can be solved in polynomial time.
Home theory of computation p, np, npcomplete, nphard p, np, npcomplete, nphard. 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. Euler diagram for p, np, npcomplete, and nphard set of problems. Under the assumption of p np, no np hard problem can be solved in. Roughly speaking, an npcomplete problem is one that is as hard as any problem in a large class of problems. A decision problem d is npcomplete if its as hard as any problem in np, i. A problem is nphard if it follows property 2 mentioned above, doesnt need to follow property 1. A language in l is called npcomplete iff l is nphard and. Problem x polynomial cook reduces to problem y if arbitrary. 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.
Minesweeper and np completeness minesweeper is np complete. Np or p np nphardproblems are at least as hard as an npcomplete problem, but npcomplete technically refers only to decision problems,whereas. Aug 17, 2017 npcompleteness a problem x is npcomplete if x. 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. 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 hard and np complete problems if an nphard problem can be solved in polynomial time, then all npcomplete problems can be solved in polynomial time. Informally, a search problem b is nphard if there exists some npcomplete problem a that turing reduces to b. Now, this includes all ridiculously hard problems exptime, undecidable, or worse, so we just look at the set of np hard problems that are also np. Therefore, if any np complete problem has a polynomial time algorithm, then p np. The np complete problems represent the hardest problems in np. Therefore, np complete set is also a subset of np hard set.
Informally, a language lis in np if there is a \guessandcheck algorithm for l. Therefore, npcomplete set is also a subset of nphard set. We want some way to classify problems that are hard to solve, i. We can see that np complete problems are the hardest problem in np. 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. The girl who proved p np one of my all time favorite blog entries is a truly epic tale of dating gone wrong that culminates in the strangest reference to p np youll probably ever encounter. This leads us to conclude that npcomplete is a class of problems that fall under both np and nphard hardest of the np problems is npcomplete and the hardest of npcomplete is nphard. My original paper appeared under this title in the spring 2000 issue of the mathematical intelligencer volume 22 number 2, pages 915 it was discussed by ian stewart in the mathematical recreations column in the scientific american, in october 2000, and has been discussed in newspapers in the usa including the boston globe on. Other npcomplete problems obtained through polynomialtime reductions from a known npcomplete problem. The second part is giving a reduction from a known npcomplete problem.
All npcomplete problems are nphard, but all nphard problems are not npcomplete. 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 left side is valid under the assumption that p. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Np hardness a language l is called np hard iff for every l. There are algorithms for which there is no known solution, for example.
In your case, problem a is complete for np, or np complete, if every problem in np reduces to a, and a is in np. Equivalently, npcomplete is the class of decision problems in np to which all other problems in np can be reduced to. Np hard if it can be solved in polynomial time then. A problem is np hard if it follows property 2 mentioned above, doesnt need to follow property 1. If language a is known to be nphard, then showing that a. What is the definition of p, np, npcomplete and nphard. Let us consider the classical version of tsp travelling sales person problem find the shortest distance between two cities on a map such that the sales person visits each city only once. 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. Id like to read your explanations, and the reason is they might be different from whats out there, or there is something that im not aware of.
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. 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. Sometimes weve claimed a problem is nphard as evidence. The current hot topic has been the use of pcp theorem to prove various np hardness results for approximation versions of np hard problems. This is an example of a problem that is thought to be hard, but is not thought to be np complete. The p vs np problem has an unusual status in that people have thought of rigorous reasons that its hard. If a problem is proved to be npc, there is no need to waste time on trying to find an efficient algorithm for it. A language in l is called np complete iff l is np hard and l. 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. Equivalently, npcomplete is the class of decision problems in np to which all other problems in np can be reduced to in polynomial time by a deterministic turing machine. Np complete search problem this is a list of some of the. A language in l is called npcomplete iff l is nphard and l. The class of nphard problems is very rich in the sense that it contain many problems from a wide.
This is a rough guide to the meaning of np complete. Np hard and np complete problems if an np hard problem can be solved in polynomial time, then all np complete problems can be solved in polynomial time. Np set of decision problems for which there exists a polytime certifier. Algorithm cs, t is a certifier for problem x if for every string s, s. Np, the existence of problems within np but outside both p and np complete was established by ladner. Decision vs optimization problems np completeness applies to the realm of decision problems. To obfuscate this already confusing topic by using n to denote the number of operations is more than a little laughable. To describe sat, a very important problem in complexity theory to describe two more classes of problems. My original paper appeared under this title in the spring 2000 issue of the mathematical intelligencer volume 22 number 2, pages 915. Complexity and np completeness supplemental reading in clrs. If both are satisfied then it is an np complete problem. If an nphard problem belongs to set np, then it is npcomplete.
What if, instead of requiring v v, we require v s for some given subset. If we know a single problem in np complete that helps when we are asked to prove some other problem is np complete. P vs np satisfiability reduction nphard vs npcomplete pnp patreon. Np hard and npcomplete problems 2 the problems in class npcan be veri. Finally, a problem is complete for a class c if it is in c and hard for c. Npcomplete problems are often addressed by using heuristic methods and approximation. The graph isomorphism problem is suspected to be neither in p nor np complete, though it is in np. The problem in nphard cannot be solved in polynomial time, until p np.
I assume that you are looking for intuitive definitions, since the technical definitions require quite some time to understand. The easiest way to prove that some new problem is np complete is first to prove that it is in np, and then to reduce some known np complete problem. Second, when people prove a barrier result meaning, a negative result about how not to prove a conjecture, obviously the community will take it as a challenge to. If any np complete problem has a polynomial time algorithm, all problems in np do. If problem a is hard for np, or nphard, if every problem in np reduces to a.
A simple example of an np hard problem is the subset sum problem a more precise specification is. If we know a single problem in npcomplete that helps when we are asked to prove some other problem is npcomplete. If any np complete problem has a polynomial time algorithm, then p np. P is set of problems that can be solved by a deterministic turing machine in polynomial time. 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. Np complete the group of problems which are both in np and np hard are known as np complete problem. These are just my personal ideas and are not meant to be rigorous.
That is the problem which asks given a program and its input, will it run forever. For more advanced reading, i highly recommend scott a. So you really did graduate from computer engineering. Np or p np np hardproblems are at least as hard as an np complete problem, but np complete technically refers only to decision problems,whereas. Note that a problem satisfying condition 2 is said to be nphard, whether or not it. Once that one npcomplete language was known, it was relatively simple to show the npcompleteness of other languages via reduction. Nphardness a language l is called nphard iff for every l. In computational complexity theory, np is the class of decision problems whose solution can be veri ed in polytime. A problem is np hard if all problems in np are polynomial time reducible to it. Karp 3 if np complete is karpcompleteness, i can conclude that all of np can be solved in time onfn, where fn is. Euler diagram for p, np, np complete, and np hard set of problems. A problem is said to be np hard if everything in np can be transformed in polynomial time into it, and a problem is np complete if it is both in np and np hard. P is a set of all decision problems solvable by a deterministic algorithm in polynomial time.
Np, while the right side is valid under the assumption that p np. It has the neat property that every np complete problem is polynomial reducible to every other np complete problem simply because all np problems are. The problem in np hard cannot be solved in polynomial time, until p np. If problem a is hard for np, or np hard, if every problem in np reduces to a.
What are the differences between np, np complete and np hard i am aware of many resources all over the web. Trying to understand p vs np vs np complete vs np hard. Since y is np complete, this would imply that p np. Npcomplete the group of problems which are both in np and nphard are known as npcomplete problem. Np hard problems a problem x is np hard if there is an np complete problem y that can be polynomially turing reduced to it i. Np is the set of problems for which there exists a. It is easy to prove that the halting problem is np hard but not np complete. What are the differences between np, npcomplete and nphard. What is the definition of p, np, np complete and np hard. If a problem is proved to be npc, there is no need to waste time on trying to find an efficient algorithm for p vs np satisfiability reduction np hard vs np complete p np. This is a rough guide to the meaning of npcomplete. For example, the traveling salesman problem tsp, integer programming ip, the longest cycle, and satis. The first part of an npcompleteness proof is showing the problem is in np. A trivial example of np, but presumably not npcomplete is finding the bitwise and of two strings of n boolean bits.