R. Venkatesan. Average-Case Intractability. Ph.D. Thesis, Boston University, 1991. Home/Search Document Not in Database Summary Related Articles Check |
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Reductions Do Not Preserve Fast Convergence Rates in Average.. - Jay Belanger Pavan (2 citations) (Correct)
.... , and let t : IN. Then t is polynomial on average if there exists a positive integer k such that P x t 1=k (x)jxj 1 (x) 1. Several authors have explained the motivations and justi cations of this de nition, and have explained why seemingly more obvious formulations fail [Lev86, Gur89, Gur91, Ven91, Imp95, Wan97]. Let AP denote the class of all distributional decision problems (A; such that A can be solved by a deterministic algorithm whose running time is polynomial on average. Let and be two distributions. Then is dominated by , denoted by , if there is a polynomial p such that for ....
R. Venkatesan. Average-Case Intractability. Ph.D. Thesis, Boston University, 1991.
Randomized Reductions and Isomorphisms - Wang (1999) (Correct)
....#(1) the number of blank edges k with probability #(n 2 ) Hence, the probability distribution is proportional to n 4 2 n 2 , which is flat. Let (E, E ) denote the distributional graph spot coloring problem. 5. 2 12 Venkatesan and Levin [VL88] a slightly di#erent proof was given in [Ven91]) showed that (E, E ) is average case NP complete under a randomized reduction. 4 Theorem 4 (T , T ) # r (E, E ) Proof of Theorem 4 For a large part, we follow a proof given in [VL88] Proof of Theorem 4 1 showing that distributional graph spot coloring is average case NP complete. To ....
....part, we follow a proof given in [VL88] Proof of Theorem 4 1 showing that distributional graph spot coloring is average case NP complete. To arrive quickly to the point showing why a randomized isomorphism exists, we will omit proofs to certain lemmas; all of the missing proofs can be found in [VL88, Ven91]. A tournament is an acyclic (except for self loops) complete digraph. Any Proof of Theorem 4 2 tournament contains a Hamiltonian path; there is a deterministic p time algorithm that starts from the node with smallest label and finds a Hamiltonian path uniquely (see, e.g. Liu68] Let k = T ....
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R. Venkatesan. Average-Case Intractability. PhD thesis, Boston University, 1991.
Average-Case Intractable NP Problems - Wang (1997) (2 citations) (Correct)
....a notion for measuring efficiency on the average case and a notion of completeness for measuring intractability on the average case. Expected polynomial time is a natural notion to measure average efficiency of an algorithm. But it is, among other things, machine dependent (see, for example, [Gur91, Ven91, Wan96]) and so it cannot be used to build a general theory on average case intractability for NP problems. To overcome this obstacle, Levin defined a robust notion on what it means for the running time of a deterministic algorithm to be polynomial on average with respect to the input distribution. A ....
....f : Sigma IN is polynomial on average if there exists an 0 such that P x f (x)jxj Gamma1 (x) 1. This definition is robust and machine independent. Also, if a function is an expected polynomial over distribution , then it is polynomial on average. The reader is referred to [Gur89, Gur91, Ven91, Wan96] for motivation and justification of this definition. A problem is solvable in average polynomial time with respect to distribution if it can be solved by a deterministic algorithm whose running time is polynomial on average. A problem with an associated probability distribution is called a ....
[Article contains additional citation context not shown here]
R. Venkatesan. Average-Case Intractability. Ph.D. Thesis (Advisor: L. Levin), Boston University, 1991.
Average-case Computational Complexity Theory - Wang (1996) (28 citations) (Correct)
....issues regarding defining a robust notion of average polynomial time, and we discuss them below. These issues were either mentioned explicitly or hinted at by Levin [Lev86] and various aspects of the issues have been elaborated on by Johnson [Joh84] Gurevich [Gur89, Gur91a, Gur91b] Venkatesan [Ven91], and Impagliazzo [Imp95] From this, Levin s definition of average polynomial time (given here as Definition 2.1) can be derived naturally and can be well justified. Model Independence. Let Sigma n = fx : jxj = ng. Let A be a subset of Sigma n and jAj proportional to 2 n (1 Gamma 2 ....
R. Venkatesan. Average-Case Intractability. Ph.D. Thesis (Advisor: L. Levin), Boston University, 1991.
Reductions Do Not Preserve Fast Convergence Rates in Average .. - Belanger, Pavan, Wang (2 citations) (Correct)
....let t : Sigma IN. Then t is polynomial on average if there exists a positive integer k such that P x t 1=k (x)jxj Gamma1 (x) 1. Several authors have explained the motivations and justifications of this definition, and have explained why seemingly more obvious formulations fail [Lev86, Gur89, Gur91, Ven91, Imp95, Wan96]. Let AP denote the class of all distributional decision problems (A; such that A can be solved by a deterministic algorithm whose running time is polynomial on average. Let and be two distributions. Then is dominated by , denoted by , if there is a polynomial p such that for all ....
R. Venkatesan. Average-Case Intractability. Ph.D. Thesis, Boston University, 1991.
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