Complexity and Real Computation
Citations
6,340 citations
Cites background from "Complexity and Real Computation"
...Other models of computation also exist, such as integer RAM and real RAM (see [120]); there are debates as to which model is most appropriate, especially when performing geometric computations with real numbers....
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...Computation models over the reals are covered in [120]....
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4,200 citations
Cites background from "Complexity and Real Computation"
...[17] focuses on the development of the theory in numerical analysis....
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1,651 citations
Cites background from "Complexity and Real Computation"
...Lemma 7 of Chapter 14 of [8]; this reference gives more background to this discussion)....
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1,330 citations
1,310 citations
References
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"Complexity and Real Computation" refers background or methods in this paper
...Karmarkar [1984] gave a proof employing interior point methods. Gonzaga [1989] proves the theorem using the barrier method. Our analysis of the barrier method using the robust ex theorem is from [Renegar and Shub 1992]. For the reduction of the feasibility problem to the barrier method and the bit analysis we have largely followed Vavasis [1991]. Renegar [1995a, 1995b] proves that LPF~ is in P when the condition of the problem is taken into account....
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...See [Griffiths and Harris 1978] for background on projective space and complex algebraic geometry....
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...Karmarkar [1984] gave a proof employing interior point methods. Gonzaga [1989] proves the theorem using the barrier method....
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...Karmarkar [1984] gave a proof employing interior point methods....
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7,422 citations
7,005 citations
6,675 citations
"Complexity and Real Computation" refers background or result in this paper
...At the beginning of the 1970s Cook [1971] defined a reduction for a class of decision problems corresponding to these search problems and proved the existence of complete problems. Concretely, he proved the completeness of the Satisfiability Problem of propositional logic. Independently Levin [1973] obtained similar results for a class of search problems and proved the existence of six complete problems, including the one of finding a satisfying truth assignment for a given propositional formula. This is the search version of Cook's completeness result for decision problems. Shortly afterwards, Karp [1972] considered the class of decision problems dealt with by Cook and coined the name NP for it....
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...At the beginning of the 1970s Cook [1971] defined a reduction for a class of decision problems corresponding to these search problems and proved the existence of complete problems. Concretely, he proved the completeness of the Satisfiability Problem of propositional logic. Independently Levin [1973] obtained similar results for a class of search problems and proved the existence of six complete problems, including the one of finding a satisfying truth assignment for a given propositional formula. This is the search version of Cook's completeness result for decision problems. Shortly afterwards, Karp [1972] considered the class of decision problems dealt with by Cook and coined the name NP for it. Moreover he showed that a series of familiar decision problems from different areas of discrete mathematics were also complete, coining the name NP-complete. This gave strong impetus to the subject that was reflected in work exhibiting hundreds of NP-complete problems and, on the other hand, in attempts to prove the inequality P =f=. NP leading to results on the structure of the class NP. A lively exposition on the P versus NP question (containing a large list of NP-complete problems) can be found in the already classic book by Garey and Johnson [1979]. A survey of the state of the art of this question is given in [Sipser 1992]....
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...At the beginning of the 1970s Cook [1971] defined a reduction for a class of decision problems corresponding to these search problems and proved the existence of complete problems. Concretely, he proved the completeness of the Satisfiability Problem of propositional logic. Independently Levin [1973] obtained similar results for a class of search problems and proved the existence of six complete problems, including the one of finding a satisfying truth assignment for a given propositional formula....
[...]
...At the beginning of the 1970s Cook [1971] defined a reduction for a class of decision problems corresponding to these search problems and proved the existence of complete problems. Concretely, he proved the completeness of the Satisfiability Problem of propositional logic. Independently Levin [1973] obtained similar results for a class of search problems and proved the existence of six complete problems, including the one of finding a satisfying truth assignment for a given propositional formula. This is the search version of Cook's completeness result for decision problems. Shortly afterwards, Karp [1972] considered the class of decision problems dealt with by Cook and coined the name NP for it. Moreover he showed that a series of familiar decision problems from different areas of discrete mathematics were also complete, coining the name NP-complete. This gave strong impetus to the subject that was reflected in work exhibiting hundreds of NP-complete problems and, on the other hand, in attempts to prove the inequality P =f=. NP leading to results on the structure of the class NP. A lively exposition on the P versus NP question (containing a large list of NP-complete problems) can be found in the already classic book by Garey and Johnson [1979]. A survey of the state of the art of this question is given in [Sipser 1992]. In this latter article, a recently discovered letter of G6del to von Neumann dated 1956 is reproduced in which G6del stated the P versus NP question in the form of the time required by a Turing machine to test whether a formula of the predicate calculus has a proof of a given length. The rise of complexity issues in the numerical tradition is less attached to the advent of the digital computer. Early in 1937, in a short note of Scholz [1937], complexity questions arose under the form of the number of additions needed to produce a given integer starting from 1....
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...At the beginning of the 1970s Cook [1971] defined a reduction for a class of decision problems corresponding to these search problems and proved the existence of complete problems. Concretely, he proved the completeness of the Satisfiability Problem of propositional logic. Independently Levin [1973] obtained similar results for a class of search problems and proved the existence of six complete problems, including the one of finding a satisfying truth assignment for a given propositional formula. This is the search version of Cook's completeness result for decision problems. Shortly afterwards, Karp [1972] considered the class of decision problems dealt with by Cook and coined the name NP for it. Moreover he showed that a series of familiar decision problems from different areas of discrete mathematics were also complete, coining the name NP-complete. This gave strong impetus to the subject that was reflected in work exhibiting hundreds of NP-complete problems and, on the other hand, in attempts to prove the inequality P =f=. NP leading to results on the structure of the class NP. A lively exposition on the P versus NP question (containing a large list of NP-complete problems) can be found in the already classic book by Garey and Johnson [1979]. A survey of the state of the art of this question is given in [Sipser 1992]. In this latter article, a recently discovered letter of G6del to von Neumann dated 1956 is reproduced in which G6del stated the P versus NP question in the form of the time required by a Turing machine to test whether a formula of the predicate calculus has a proof of a given length. The rise of complexity issues in the numerical tradition is less attached to the advent of the digital computer. Early in 1937, in a short note of Scholz [1937], complexity questions arose under the form of the number of additions needed to produce a given integer starting from 1. Seventeen years later Ostrowski [1954] conjectured the optimality of Homer's rule for evaluating univariate polynomials....
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