Leonid Levin. Average Case Complete Problems. SIAM Journal on Computing 15, 285--286, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....problems that are NP complete, their average case complexity might not be intractable, meaning that such an algorithm could have a good performance in practice. The average level of intractability differs per NP complete problem. The theory of average case complexity was first advocated by Levin [14]. We will now give a calculation that suggests that the average and even amortizeel i complexity of AMCRA is polynomial in time for fixed m and all weights wi independent random variables. Lemma 1: The expected number of non dominated vectors in a set of T i.i.d. vectors in m dimensions is upper ....

Levin, L.A., Average Case Complete Problems, SIAM J. Cornput., 1986, 15(1):285- 286.

....framework providing a language and tools for comparing and classifying the compilability of formalisms on average. Our framework is built on notions and insights from the theory of compilability clases due to Cadoli et al. 2000a; 2000b] and the theory of average case time complexity due to Levin [1986] . 1.1 Background Compilability. Informally, a formalism A is compilable to a formalism B if for every knowledge base x in formalism A, there is a knowledge base in formalism B representing the same information as x with size polynomial in the length In previous work, compilability has also ....

....L does not imply hardness of L on typical or real world instances. Our objection is really this old observation, masquerading in the new context of compilability. One theory that was developed in response to this old observation is the theory of average case time complexity (ACTC) initiated by Levin [1986] . Our theory of averagecase compilability will be built on a key notion of ACTC that of polynomial on average. Moreover, there are useful analogies between our theory and the theory of ACTC. Consequently, we provide a brief overview of ACTC. Average case time complexity. By definition, a ....

Leonid Levin. Average case complete problems. SIAM Journal on Computing, 15:285-286, 1986.

....distributions and sophisticated algorithms. It is also frequently unclear what constitutes a reasonable assumption about the distribution of problem instances, apart from the analytical difficulties of the program. A completeness notion for average case complexity has been introduced by Levin [Lev86]. This also seems to be difficult to apply, and has only been demonstrated for a few problems. The main weaknesses of average case analysis as a mathematical program for coping with intractability seem to be: ffl In general, it seems to be too difficult to prove the mathematical results that the ....

L. Levin, "Average Case Complete Problems," SIAM J. Computing 15 (1986), 285--286.

....occur in problems that are not NPcomplete. Monasson et al. 17] report an analytic solution and experimental investigation of the phase transition in Ksatis ability (the rst problem shown to be NP complete) Gent and Walsh [8] show that phase transitions occur in the ######### problem. Levin [14] advocated a different study of NP complete problems by introducing the concept of average case complexity. He indicated that some NP complete problems are easy on average, while other (average case NP complete) problems may not be. There exists also some work in the literature revealing ....

L.A. Levin, Average Case Complete Problems, SIAM J. Comput., 15(1):285-286, 1986.

....hidden in the big oh term (and in further asymptotic terms) is sometimes a real challenge. From a more theoretical viewpoint, it is natural to raise the question Is there any relevant notion of smoothed complexity completeness , quite similarly to the DistNP complete class de ned by Levin [15, 11]. While classical worst case complete problems are rather well known, this average complete problems were introduced quite recently. A smoothed complexity completeness would give a valuable criterion for problems on which one could use cryptographic schemes on a rather wide region of instances. ....

Leonid A. Levin. Average case complete problems. SIAM J. Comput., 15(1):285-286, 1986. 10

....the simplest of algorithms and distributions. As a consequence, the methods are difficult to apply, and moreover, it is frequently unclear what constitutes a reasonable assumption about the distribution of problem instances. The completeness notion for average case complexity introduced by Levin [Lev86] also seems to have limited applicability. Another drawback of the program is that it lacks a positive toolkit. If I want to design an algorithm with good average case performance (however this is evaluated) how do I do that The real strength of average case analysis as a means of coping is ....

L. Levin, "Average Case Complete Problems," SIAM J. Computing 15 (1986), 285--286.

....distributions and sophisticated algorithms. It is also frequently unclear what constitutes a reasonable assumption about the distribution of problem instances, apart from the analytical difficulties of the program. A completeness notion for average case complexity has been introduced by Levin [Lev86]. This also seems to be difficult to apply, and has only been demonstrated for a few problems. The main weaknesses of averagecase analysis as a mathematical program for coping with intractability seem to be: ffl In general, it seems to be too difficult to prove the mathematical results that the ....

L. Levin, "Average Case Complete Problems," SIAM J. Computing 15 (1986), 285--286.

....in each possible run of the machine, regardless of the outcome of its coin tosses. 2. Expected probabilistic polynomial time. The standard approach is to look at the running time as a random variable and bound its expectation (by a polynomial in the length of the input) As observed by Levin [73] (cf. 46] this definitional approach is quite problematic (e.g. it is not model independent and is not closed under algorithmic composition) and an alternative treatment of this random variable is preferable. Consequently, the notion of expected polynomial time raises a variety of conceptual ....

L.A. Levin. Average Case Complete Problems. SIAM Jour. of Computing, Vol. 15, pages 285--286, 1986.

....be an NP complete problem to determine whether initial conditions exist that lead to particular behavior. p. 769) In computer science, the complexity of typical instances of NP complete problems has been investigated for decades. Highlights include Levin s theory of average case completeness [9] and studies of phase transitions in randomly generated combinatorial problems [10] It remains open to show that some NP complete problem is hard on average, under a simple distribution, so long as P NP. However, worst case average case equivalence has been shown for several cryptographic ....

L. A. Levin (1986), Average case complete problems, SIAM Journal on Computing 15(1), pp. 285--286.

....particular any one of the valid inputs may be generated: If there exists a valid input that cannot be generated, then this input may be the only one for which the algorithm fails. The uniform distribution also plays an important role in the theoretical study of average case complexity classes (see [13,17]) and algorithms that sample structures uniformly at random are of independent interest. We would like to mention another interesting and related random structure, that recently attracted attention: Unlike to the random graph, still little is known about the random planar graph. Random planar ....

L. Levin. Average case complete problems. SIAM J. on Comp., 15:285-286, 1986.

....implications for which we cannot prove whether or not their converse holds. In nearly all such cases we provide oracles relative to which the converse fails. We use the techniques of Kolmogorov complexity to describe our oracles and to simplify the technical arguments. 1 Introduction Levin [Lev86] was the first to advocate the general study of average case complexity and he provided the central notions for its study. More recently, Cai and Selman [CS99] observed that Levin s definition of Average P has limitations when applied to distributional problems with unreasonable distributions and ....

....P = NP [Gur91] All distributions are to have infinite support we explicitly exclude from consideration distributions for which (x) 0 for all but a finite number of strings x. Consideration of such distributions would allow every problem to be an essentially finite problem. Levin [Lev86] defines a function f from S to nonnegative reals to be polynomial on average if there is an integer k 0 such that : 1) Average P is the class of distributional problems (L; where L is a language and is a polynomial time computable distribution, such that L can be decided by ....

L. Levin. Average case complete problems. SIAM Journal on Computing, 15:285--286, 1986.

....which is loosely connected to the present paper is the question whether there are problems in NP which cannot be solved in expected polynomial time with respect to a probability measure on the input space. Our results have no direct relation to this problem, and hence we refer the reader to [Gur91, Lev86] and their references for a discussion of these questions. 2 Bounds by diagonalization We use the alphabet f00; 01; 10; 11g to encode Turing machines. We will assume that a description of a Turing machine contains a straightforward encoding of its next step function. We will also assume that all ....

L. Levin. Average case complete problems. SIAM Journal of Computing, 15:285--286, 1986.

....simulation) is less than satisfactory for several reasons: Philosophical considerations: 1. Ecient computation: Equating ecient computation with expected polynomial time is more controversial than equating ecient computation with (strict) probabilistic polynomial time. For example, Levin ([21], see also [15] 16, Sec. 4.3.1.6] has shown that when expected polynomial time is de ned as above, the de nition is too machine dependent, and is not closed under reductions. He proposed a stronger de nition that is closed under reductions and is less machine dependent. However, it is still ....

L. A. Levin. Average Case Complete Problems. SIAM Journal on Computing, 15(1):285-286, 1986.

....is less than satisfactory for several reasons: ffl Philosophical considerations: 1. Efficient computation: Equating efficient computation with expected polynomial time is more controversial than equating efficient computation with (strict) probabilistic polynomial time. For example, Levin ([20], see also [14] 15, Sec. 4.3.1.6] has shown that when expected polynomial time is defined as above, the definition is too machine dependent, and is not closed under reductions. He proposed a stronger definition that is closed under reductions and is less machine dependent. However, it is still ....

L. A. Levin. Average Case Complete Problems. SIAM Journal on Computing, 15(1):285--286, 1986. 31

....which is loosely connected to the present paper is the question whether there are problems in NP which cannot be solved in expected polynomial time with respect to a probability measure on the input space. Our results have no direct relation to this problem, and hence we refer the reader to [Gur91, Lev86] and their references for a discussion of these questions. 2 Bounds by diagonalization We use the alphabet f00; 01; 10; 11g to encode Turing machines. We will assume that a description of a Turing machine contains a straightforward encoding of its next step function. We will also assume that all ....

L. Levin. Average case complete problems. SIAM Journal of Computing, 15:285--286, 1986.

....is a pair (L; where L is a language over a nite alphabet and is a distribution de ned on . Given a distributional problem, it is an important issue either to nd an expected polynomial time algorithm that solves the problem or to prove that such an algorithm does not exist. Levin [Lev86] provided two central notions for studying this issue. One is analogous to the class P, and provides an easiness notion; the other is analogous to the class of NP complete sets, and provides a hardness notion. For the rst, Levin de ned a robust notion of what it means for an algorithm that ....

....polynomial time. The converse is false unless P = NP [Gur91] Also, we explicitly exclude from consideration distributions for which 0 (x) 0 for all but a nite number of strings x. Consideration of such distributions would allow every problem to be an essentially nite problem. Levin [Lev86] de nes a function f from to nonnegative reals to be polynomial on average if there is an integer k 0 such that X jxj 1 0 (x) f(x) 1=k jxj 1: 1) Average P is the class of distributional problems (L; where L is a language and is a polynomial time computable ....

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L. Levin. Average case complete problems. SIAM Journal on Computing, 15:285-286, 1986.

....Note that NP hard problems are not necessarily easy on the average. Distributions on which a problem is hard on the average case are necessary for cryptography. It is therefore important to identify problems and corresponding distributions, on which the problem is hard on the average. Levin [Lev86] put a basis for a theory of average NP completeness. The emphasis in Levin s theory appears to be to identify distributions on which the underlying problem is hard. In contrast, the emphasis in our work is to provide algorithms that perform well on 8 average with respect to distributions that ....

L. A. Levin. Average case complete problems. SIAM J. Comput., 15(1):285--286, 1986.

....access to A. Most theorems in computational complexity relativize. 2.1 Average Case Complexity Even if P 6= NP, we still could have that NP is easy for all practical purposes, i.e. given any reasonable distribution, NP is easy on average on that distribution. To formalize this problem, Levin [Lev86] developed a notion of distributional problems. Given a language L and a distribution on , we say L is polynomial time on average with respect to if there exists a Turing machine M computing L and an 0 such that X x2 T M (x) jxj 0 (x) 1 Here (x) represents the ....

L. Levin. Average case complete problems. SIAM Journal on Computing, 15:285-286, 1986.

....The second notion is the notion of domination. Given two probability distributions and , say that dominates if is equal to or greater than modulo a polynomial factor. In other words, dominates if there exists a polynomial p such that Pr [fxg] Pr [fxg] Delta p(jxj) It can be checked [Le1,Gu1] that (X; is Ptime on average if there exists such that dominates and (X; is Ptime on average. There seems to be a consensus that natural probability distributions are dominated by Ptime computable ones. Thus, Challenger may draw instances with respect to a Ptime computable probability ....

....many attempts in order to force Solver to work hard. Roughly speaking, Challenger needs t = 2 i attempts to draw a string with i leading zeroes and to force Solver to work t 2 = 4 i steps. ffl A: Very good. You force me to disclose the official definition of polynomiality on average [Le1,Gu1]. The definition is slightly different from what you (most justifiably) assumed it to be. Let n be a function that assigns probabilities to strings of length n. The connection between this case and the case of one distribution over all strings is discussed at length in [Gu1] Further, let T be a ....

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Leonid Levin. "Average Case Complete Problems". SIAM Journal of Computing, 1986.

....problem is, in many cases, a more signi cant measure than the worst case complexity. It is known that some NP complete problems, such as k coloring and Hamiltonian cycle, can be solved quickly on average. This has motivated the development of the study of average case analysis of algorithms. Levin [56] initiated a general study of average case complexity by de ning a robust notion of average polynomial time and the notion of distributional NP completeness. Cai and Selman [19] gave a general de nition of T on average for arbitrary time bounds T . They observed diculties with Levin s de nition of ....

....Gurevich and Shelah [33] for the Hamiltonian problem, both under the commonly used distribution on graphs. Because of these examples, as well as ordinary computational practice, the average case complexity of a problem seems to be a more signi cant measure than the worst case complexity. Levin [56] was the rst to advocate a general study of average case complexity. An average case complexity class consists of pairs (L; called distributional problems, where L is a language and is a distribution over the input domain. Given a distributional problem, it is an important issue to either ....

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L. Levin. Average case complete problems. SIAM Journal of Computing, 15:285-286, 1986. BIBLIOGRAPHY 85

....Do Not Preserve Fast Convergence Rates in Average Time Jay Belanger A. Pavan y Jie Wang z Abstract Cai and Selman [CS96] proposed a general de nition of average computation time that, when applied to polynomials, results in a modi cation of Levin s [Lev86] notion of average polynomial time. The e ect of the modi cation is to control the rate of convergence of the expressions that de ne average computation time. With this modi cation, they proved a hierarchy theorem for average time complexity that is as tight as the Hartmanis Stearns [HS65] ....

....under a fairly reasonable condition on distributions, called condition W, a distributional problem is solvable in average polynomial time under the modi cation exactly when it is solvable in average polynomial time under Levin s de nition. Various notions of reductions, as de ned by Levin [Lev86] and others, play a central role in the study of average case complexity. However, the class of distributional problems that are solvable in average polynomial time under the modi cation is not closed under Division of Mathematics and Computer Science, Truman State University, Kirksville, ....

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L. Levin. Average case complete problems. SIAM Journal on Computing, 15:285-286,

.... of the Davis Putnam propositional decision procedure [DP60, Gol79, Fra86, Fra91, PT92] Several approaches have arisen which give pessimistic results: One approach is to show that if the decision problem for a given L is NP Complete, then it is also average case hard in the sense of Levin [Lev86, BDCGL89, BG91, Gur91, VR92, SY92, RS93]. We define the general terms and some of the results of this approach, and prove that MLS is NP average complete. Secondly, it is clear that if we tailor a logic to a certain NP Complete problem, in terms of our choice of axioms and inference rules (for example, we can build a strong inference ....

....[DP60, Gol79, Fra86, Fra91, PT92] We explore item 2 more fully in this chapter; the utility of item 1 is self evident. Several approaches have arisen which give pessimistic results: 1) One approach is to prove that a given NP Complete decision problem is averagecase hard in the sense of Levin ([Lev86], BDCGL89] BG91] Gur91] VR92] SY92] RS93] So far, it has fairly di#cult to prove average case NP hardness under the various definitions. We should note that the definitions in the literature haven t yet been standardized. We define the general terms and some of the results of this ....

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L. Levin. Average case complete problems. SIAM J. Comput., 15:285--286, 1986.

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Leonid Levin. Average Case Complete Problems. SIAM Journal on Computing 15, 285--286, 1986.

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Leonid A. Levin, Average Case Complete Problems, SIAM Journal of Computing 15 (1986), 285--286.

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Leonid A. Levin, "Average Case Complete Problems", SIAM Journal of Computing, No. 15, 1986, 285--286.

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Leonid Levin, Average Case Complete Problems, SIAM Journal of Computing 15 1986, 285--286.

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Leonid A Levin. Average case complete problems. SIAM J. Comput., 15(1):285--286, 1986. Online available at http://www.cs.bu.edu/fac/lnd/research/dvi/rp.dvi.

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Leonid A. Levin. Average case complete problems. SIAM J. Comput., 15(1):285-- 286, 1986.

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L. Levin "Average Case Complete Problems", SIAM Journal of Computing 15: 285-286, 1986.

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L. A. Levin. Average case complete problems. SIAM Journal of Computing, 15(1):285--286, 1986. Preliminary version in STOC'84.

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L. Levin "Average Case Complete Problems", SIAM Journal of Computing 15: 285286, 1986. 12

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Leonid A. Levin. Average case complete problems. SIAM Journal on Computing, 15(1):285--286, February 1986.

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Leonid Levin. Average case complete problems. SIAM Journal on Computing, 15(1):285-- 286, 1986.

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L. Levin. Average case complete problems. SIAM Journal on Computing, 15(1):285--286, 1986.

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L. A. Levin. Average case complete problems. SIAM J. Comput. 15:285--286 (1986).

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L. Levin. Average case complete problems. SIAM Journal on Computing, 15(1):285--286, 1986.

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Levin, L.A., Average case complete problems, SIAM J. on Computing, 15(1986), 285-286.

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Levin, L.A., Average case complete problems, 16th Symposium on Theory of Computing, 1984, 465.

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Leonid A. Levin. Average case complete problems. SIAM Journal on Computing, 15(1):285-286, 1986.

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L. Levin. Average case complete problems. SIAM Journal on Computing, 15:285--286, 1986.

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L. Levin. "Average case complete problems". SICOMP 15, 1986, 285--286.

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L.A. Levin. Average Case Complete Problems. SICOMP, Vol. 15, pages 285--286, 1986.

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L.A. Levin. Average Case Complete Problems. SICOMP, Vol. 15, pages 285--286, 1986.

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L.A. Levin. Average Case Complete Problems. SICOMP, Vol. 15, pages 285--286, 1986.

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Leonid Levin. Average case complete problems. SIAM Journal on Computing, 15(1):285--286, 1986. 167

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L.A. Levin. Average Case Complete Problems. SICOMP, Vol. 15, pages 285-286, 1986.

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L. Levin. Average case complete problems. SIAM J. Computing, 15:285-286, 1986.

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L. Levin. Average case complete problems. SIAM Journal on Computing, 15:285--286, 1986.

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L. A. Levin, Average Case Complete Problems, SIAM J.Comput., 15:1(1986), 285-286.

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L.A. Levin, "Average Case Complete Problems", SICOMP, Vol. 15, 1986, pp. 285-- 286.

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