PDF Data Structures and Algorithms 2: Graph Algorithms and NP-Completeness

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If you still have questions after meeting with the TA, then come and see me. The minute quizzes will be given during our regular lecture times on the following dates. Quiz 1 will have three sections: The students with the last name starting with a letter between A to L inclusive will write the quiz in AQ ; the students whose last name begins with a letter between M to T inclusive will write the quiz in AQ ; the students whose last name begins with a letter between W and Z inclusive will write the quiz in AQ Read ahead: Sec.

Read: KT, Secs. Read ahead: KT, Secs.


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Slides for Lectures Read: KT, Sec. Read ahead: KT, Sec. Randomized Algorithms: Contention resolution. Slides for lectures Aug 2: The final exam on Aug 9 will be on all topics of the course, with equal emphasis. You'll have 3 hours for an exam that will be roughly two quizzes in size. To prepare, go over all lecture slides and notes, HWs, quizzes, and the textbook.

The questions on the final exam will be similar to those on the quizzes we had.

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NP-completeness will not be part of Quiz 4, but rather appear on the final exam. A constructive and efficient solution [Note 2] to an NP -complete problem such as 3-SAT would break most existing cryptosystems including:. These would need to be modified or replaced by information-theoretically secure solutions not inherently based on P - NP equivalence. On the other hand, there are enormous positive consequences that would follow from rendering tractable many currently mathematically intractable problems. For instance, many problems in operations research are NP -complete, such as some types of integer programming and the travelling salesman problem.

Efficient solutions to these problems would have enormous implications for logistics. Many other important problems, such as some problems in protein structure prediction , are also NP -complete; [33] if these problems were efficiently solvable it could spur considerable advances in life sciences and biotechnology. But such changes may pale in significance compared to the revolution an efficient method for solving NP -complete problems would cause in mathematics itself.

Namely, it would obviously mean that in spite of the undecidability of the Entscheidungsproblem , the mental work of a mathematician concerning Yes-or-No questions could be completely replaced by a machine. After all, one would simply have to choose the natural number n so large that when the machine does not deliver a result, it makes no sense to think more about the problem. Similarly, Stephen Cook says [36]. Example problems may well include all of the CMI prize problems. Research mathematicians spend their careers trying to prove theorems, and some proofs have taken decades or even centuries to find after problems have been stated—for instance, Fermat's Last Theorem took over three centuries to prove.

A method that is guaranteed to find proofs to theorems, should one exist of a "reasonable" size, would essentially end this struggle. It would allow one to show in a formal way that many common problems cannot be solved efficiently, so that the attention of researchers can be focused on partial solutions or solutions to other problems. For example, it is possible that SAT requires exponential time in the worst case, but that almost all randomly selected instances of it are efficiently solvable.

Russell Impagliazzo has described five hypothetical "worlds" that could result from different possible resolutions to the average-case complexity question. A Princeton University workshop in studied the status of the five worlds. These barriers have also led some computer scientists to suggest that the P versus NP problem may be independent of standard axiom systems like ZFC cannot be proved or disproved within them. The interpretation of an independence result could be that either no polynomial-time algorithm exists for any NP -complete problem, and such a proof cannot be constructed in e.

ZFC, or that polynomial-time algorithms for NP -complete problems may exist, but it is impossible to prove in ZFC that such algorithms are correct.

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Additionally, this result implies that proving independence from PA or ZFC using currently known techniques is no easier than proving the existence of efficient algorithms for all problems in NP. While the P versus NP problem is generally considered unsolved, [46] many amateur and some professional researchers have claimed solutions. Gerhard J. Consider all languages of finite structures with a fixed signature including a linear order relation. Then, all such languages in P can be expressed in first-order logic with the addition of a suitable least fixed-point combinator.

Effectively, this, in combination with the order, allows the definition of recursive functions. As long as the signature contains at least one predicate or function in addition to the distinguished order relation, so that the amount of space taken to store such finite structures is actually polynomial in the number of elements in the structure, this precisely characterizes P. Similarly, NP is the set of languages expressible in existential second-order logic —that is, second-order logic restricted to exclude universal quantification over relations, functions, and subsets.

The languages in the polynomial hierarchy , PH , correspond to all of second-order logic. Thus, the question "is P a proper subset of NP " can be reformulated as "is existential second-order logic able to describe languages of finite linearly ordered structures with nontrivial signature that first-order logic with least fixed point cannot?

No algorithm for any NP -complete problem is known to run in polynomial time. However, these algorithms do not qualify as polynomial time because their running time on rejecting instances are not polynomial. The following algorithm, due to Levin without any citation , is such an example below. If there is an algorithm say a Turing machine , or a computer program with unbounded memory that can produce the correct answer for any input string of length n in at most cn k steps, where k and c are constants independent of the input string, then we say that the problem can be solved in polynomial time and we place it in the class P.

NP-Complete Problems

Formally, P is defined as the set of all languages that can be decided by a deterministic polynomial-time Turing machine. That is,. NP can be defined similarly using nondeterministic Turing machines the traditional way. However, a modern approach to define NP is to use the concept of certificate and verifier. Formally, NP is defined as the set of languages over a finite alphabet that have a verifier that runs in polynomial time, where the notion of "verifier" is defined as follows.

In general, a verifier does not have to be polynomial-time. However, for L to be in NP , there must be a verifier that runs in polynomial time. This is a common way of proving some new problem is NP -complete. In the second episode of season 2 of Elementary , "Solve for X" revolves around Sherlock and Watson investigating the murders of mathematicians who were attempting to solve P versus NP.

P vs. NP and the Computational Complexity Zoo

From Wikipedia, the free encyclopedia. Unsolved problem in computer science. If the solution to a problem is easy to check for correctness, must the problem be easy to solve?

Algorithms and Data Structures | AIT-Budapest

Main article: NP-completeness. See also: Complexity class. Main article: NP -intermediate. Vardi , Rice University. Such a machine could solve an NP problem in polynomial time by falling into the correct answer state by luck , then conventionally verifying it. Such machines are not practical for solving realistic problems but can be used as theoretical models. Corollary 1. ACM site. Levin Communications of the ACM.

Thomson Course Technology, Definition 7. Gasarch June NP poll" PDF. Retrieved 26 September Retrieved 27 August Discrete Applied Mathematics. Holyer SIAM J. Lichtenstein Journal of Combinatorial Theory, Series A.

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