## Average-case computational complexity theory (1997)

Venue: | Complexity Theory Retrospective II |

Citations: | 32 - 2 self |

### BibTeX

@INPROCEEDINGS{Wang97average-casecomputational,

author = {Jie Wang},

title = {Average-case computational complexity theory},

booktitle = {Complexity Theory Retrospective II},

year = {1997},

pages = {295--328},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

ABSTRACT Being NP-complete has been widely interpreted as being computationally intractable. But NP-completeness is a worst-case concept. Some NP-complete problems are \easy on average", but some may not be. How is one to know whether an NP-complete problem is \di cult on average"? The theory of average-case computational complexity, initiated by Levin about ten years ago, is devoted to studying this problem. This paper is an attempt to provide an overview of the main ideas and results in this important new sub-area of complexity theory. 1

### Citations

725 | Proof verification and the hardness of approximation problems - Arora, Lund, et al. - 1998 |

290 |
Elements of the Theory of Computation
- Lewis, Papadimitriou
- 1997
(Show Context)
Citation Context ...ding on whether or not M accepts x), and there exists a polynomial p such that the length of every computation path of M 0 on x is strictly less than p(jxj). Following a standard procedure (e.g., see =-=[LP81]-=-), we can construct tiles to represent the transition function of M 0 , and tiles that can duplicate tape symbols a and pairs (q� a) in the vertical direction, where q is the accepting state. We then ... |

203 |
Average Case Complete Problems
- Levin
- 1986
(Show Context)
Citation Context ...which have resisted so far such \average-case" attacks. Are these problems di - cult on average? What does it mean for a problem to be di cult on average, and how is one to know? In his seminal paper =-=[Lev86]-=-, Levin initiated the study of answering these questions. Two fundamental and robust notions were de ned along similar lines to the NP-completeness theory. Namely, the notion of average polynomial tim... |

201 |
On the computational complexity of algorithms
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- 1965
(Show Context)
Citation Context .... Tocheck the domination condition, it su ces to note that ((x� k� h� )�i) appears as a query with probability jHjxj�kj ;1jxj ;12 ;k (x). 5 Hierarchies of Average-Case Complexity Hartmanis and Sterns =-=[HS65]-=- showed that, for multi-tape Turing machines, if t(n) log t(n) = o(T (n)) and both t and T are time-constructible, then DTIME(t(n)) is a proper subset of DTIME(T (n)). This is the best hierarchy known... |

201 | The NP-completeness column: an ongoing guide
- Johnson
- 1985
(Show Context)
Citation Context ...notion of average polynomial time, which are discussed below. These issues were either mentioned explicitly or hinted at by Levin [Lev86] and they have been elaborated from various aspects by Johnson =-=[Joh84]-=-, Gurevich [Gur89, Gur91a, Gur91b], Venkatesan [Ven91], and Impagliazzo [Imp95], from which Levin's de nition of average polynomial time (given in De nition 2.1) can be derived naturally and be well j... |

160 |
On isomorphism and density of NP and other complete sets
- Berman, Hartmanis
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(Show Context)
Citation Context ...d in various other aspects and a brief survey of these results is presented in this section. 1. NP-Isomorphism Problem with Respect to Random Instances. The Berman-Hartmanis NP-isomorphism conjecture =-=[BH77]-=- has provided much of the impetus for research in structural complexity theory during the last two decades. It conjectures that all many{one NP-complete prob1. Average-Case Computational Complexity T... |

112 | On the theory of average case complexity
- Ben-David, Chor, et al.
- 1992
(Show Context)
Citation Context ...lled a distributional problem. Other names such as\random problems" [Lev86] and \randomized problems" [Gur91a] have also been used in the literature. The term \distributional problem" was rst used in =-=[BCGL92]-=-. We adopt this terminology in this paper.s1. Average-Case Computational Complexity Theory 5 3 Average-Case Completeness Given two distributional problems, we wishtoknow which one is computationally m... |

94 | On the symmetry of algorithmic information - Gács - 1974 |

77 |
A Personal View of Average-Case Complexity
- Impagliazzo
- 1995
(Show Context)
Citation Context ... either mentioned explicitly or hinted at by Levin [Lev86] and they have been elaborated from various aspects by Johnson [Joh84], Gurevich [Gur89, Gur91a, Gur91b], Venkatesan [Ven91], and Impagliazzo =-=[Imp95]-=-, from which Levin's de nition of average polynomial time (given in De nition 2.1) can be derived naturally and be well justi ed. 2 Levin [Lev86] used to denote a distribution function and used 0 to d... |

73 | Average Case Completeness
- Gurevich
- 1991
(Show Context)
Citation Context ...st all of our statements can be stated directly in terms of probability distributions, denoting a probability distribution by without a prime seems more convenient. The notations and were rst used in =-=[Gur91a]-=-.s1. Average-Case Computational Complexity Theory 3 Model Independence. Assume an algorithm runs in polynomial time ona1; 2 ;0:1n fraction of instances of length n, and runs in 2 0:09n time on instanc... |

65 |
No better ways to generate hard NP instances than picking uniformly at random
- Impagliazzo, Levin
- 1990
(Show Context)
Citation Context ... overcome this obstacle, Venkatesan and Levin [VL88] introduced a new type of reduction using randomized algorithms. An important application of such reductions is the result of Impagliazzo and Levin =-=[IL90]-=- who showed that polynomial-time sampling cannot generate harder instances than picking instances uniformly at random. 4.1 Flat Distributions and Incompleteness A distribution is at if (9" >0)(8x)[ (x... |

46 |
Tally languages and complexity classes
- Book
- 1974
(Show Context)
Citation Context ...\DistNP AP if ...", or something of this nature unless there is a major breakthrough. The following two results are of this type and they are related to worst-case complexity classes. It was shown in =-=[Boo74]-=- thatDTIME(2 O(n) ) 6= NTIME(2 O(n) ) implies the existence of a tally language D in NP ; P. Let (1 n )= 1 n(n+1) . If DistNP AP, then D 2 P, a contradiction. Hence, if DTIME(2 O(n) ) 6= NTIME(2 O(n) ... |

41 | Expected computation time for Hamiltonian path problem
- Gurevich, Shelah
- 1987
(Show Context)
Citation Context ...obability distributions on instances. Algorithms that are fast on average have been found for several NP-complete problems, suchasthevertex k-coloring problem [Wil84] and the Hamiltonian path problem =-=[GS87]-=- under commonly used distributions on graphs. Yet there are NP-complete problems which have resisted so far such \average-case" attacks. Are these problems di - cult on average? What does it mean for ... |

41 |
Random instances of a graph coloring problem are hard. In: Proceedings of the twentieth annual ACM symposium on Theory of computing, 1988, 217–222. Raghav Bhaskar Satyanarayana V
- Levin, Venkatesan
- 2009
(Show Context)
Citation Context ... solving the search problem R in polynomial time on -average. Reductions on distributional search problems, which have the desired properties, can be similarly de ned following the framework given in =-=[VL88]-=-. De nition 3.7 Let (Ri� i) (i 2f1� 2g) betwo distributional search problems and Yi = fx : i(x) > 0 &(9w)Ri(x� w)g. (R1� 1) is p-time reducible to (R2� 2) if there are p-time computable functions f an... |

31 |
Properties of logarithmico-exponential functions
- Hardy
- 1911
(Show Context)
Citation Context ...and Selman [CS95] obtained such a result when time functions are restricted to be logarithmico-exponential. The class L of logarithmico-exponential (log-exp, in short) functions, rst studied by Hardy =-=[Har11]-=-, is the smallest class of functions f : IR ! IR con1. Average-Case Computational Complexity Theory 26 taining every constant function f(x) =c and the identity function f(x) =x such thatiff(x) andg(x... |

25 |
Proof Veri cation and Hardness of Approximation Problems
- Arora, Lund, et al.
- 1992
(Show Context)
Citation Context ...y help to obtain results for actual optimization problems. Although certain NP optimization problems cannot have good approximation solutions in deterministic polynomial time unless P = NP (see, e.g. =-=[ALMSS92]-=-), exact solutions, or good approximations could still be obtained in average polynomial time, or some average-case \completeness" results could be shown. 10 To see this, let K0 be K1 restricted to de... |

25 | Backtrack: an O(1) expected time algorithm for the graph coloring problem
- Wilf
- 1984
(Show Context)
Citation Context ...average with respect to the underlying probability distributions on instances. Algorithms that are fast on average have been found for several NP-complete problems, suchasthevertex k-coloring problem =-=[Wil84]-=- and the Hamiltonian path problem [GS87] under commonly used distributions on graphs. Yet there are NP-complete problems which have resisted so far such \average-case" attacks. Are these problems di -... |

23 |
Constant time factors do matter
- Jones
- 1993
(Show Context)
Citation Context ...g machine that accepts L requires 9 Note that such results are machine dependent. For example, the linear speedup theorem that holds for multi-tape Turing machines does not hold for some other models =-=[Jon93]-=-.s1. Average-Case Computational Complexity Theory 25 more than t(x) steps for all but nitely many inputx. It follows that there is an expected time hierarchy which is independent of distributions and ... |

20 | Matrix transformation is complete for the average case - Blass, Gurevich - 1995 |

19 | Average Case Complexity - Gurevich - 1991 |

14 |
Complete and Incomplete Randomized NP Problems
- Gurevich
- 1987
(Show Context)
Citation Context ...ing more average-case NP-complete problems has been a central issue in research. The notions of p-time reductions and ap-time reductions have played an important role in this study. However, Gurevich =-=[Gur87]-=- observed that NP-complete problems with \ at" distributions (see below for a de nition) cannot be complete for DistNP under p-time or ap-time reductions unless EXP = NEXP, where EXP = DTIME(2 poly ) ... |

13 |
On the NP-isomorphism problem with respect to random instances
- Wang, Belanger
- 1994
(Show Context)
Citation Context ...y using a \perfect rounding" technique, which isinteresting in its own right (see [Gur91a] for an exposition). Gurevich [Gur91a] provided a di erent and easier proof, based on which Wang and Belanger =-=[WB95]-=- further showed that if the distribution is not too small, it will also dominate the same uniform distribution within a constant factor. 6 Wemayalsoweaken the second requirement in De nition 3.5 to on... |

11 | Randomizing Reductions of Search Problems
- Blass, Gurevich
- 1991
(Show Context)
Citation Context ...ty. They showed that a graph edge coloring problem with a at distribution is complete under a randomized reduction by allowing algorithms to toss coins to determine the next moves. Blass and Gurevich =-=[BG93]-=- conducted a thorough study on such randomized reductions. We assume that a randomized algorithm does not ip a coin unless the computation requires a random bit. For simplicity, coins are assumed to b... |

10 |
Average-case completeness of a word problem for groups
- Wang
- 1995
(Show Context)
Citation Context ... state is reached. Several more natural NP-complete problems such asthePost correspondence problem [Gur91a], the word problem for Thue systems, [WB95] and the word problem for nitely presented groups =-=[Wan95a]-=- have also been shown to be complete for DistNP using p-time reductions in which each parameter of the instances is selected uniformly at random. Proofs of these results are more involved and are not ... |

9 |
Structural average case complexity
- Schuler, Yamakami
- 1992
(Show Context)
Citation Context ... provides an a rmative answer [Sch95a], but it is not known whether there are such natural problems. If a problem is in AP under every exponential-time computable distribution, then it has to be in P =-=[SY92]-=-. Ass1. Average-Case Computational Complexity Theory 30 a remark, we note that no AP problems under p-time computable distributions are harder on average than simply a P problem. 10 More results on st... |

9 |
Precise Average Case Complexity
- Reischuk, Schindelhauer
- 1993
(Show Context)
Citation Context ...al time under De nition 5.2, but (A� ) is not. There is yet another di erent approach to study average-case hierarchies under an entirely di erent average-case measurement. Reischuk and Schindelhauer =-=[RS93]-=- suggested measuring average-case complexity with respect to the ranking of the input distribution by decreasing weights in1. Average-Case Computational Complexity Theory 28 stead of the individual v... |

8 | Sets computable in polynomial time on the average - Schuler, Yamakami - 1995 |

8 | Average-Case Intractable NP Problems
- Wang
- 1997
(Show Context)
Citation Context ...ome more distributional problems are shown to be average-case NP-complete under randomized reductions in [VR92]. Due to page limitations, exposition of these results will be given in a separate paper =-=[Wan]-=-. To demonstrate the idea of using randomized reductions, we show that a halting problem with a at distribution is complete for DistNP under p-time randomized reductions. We then show, in the next sec... |

7 | Isomorphisms of NP-complete problems on random instances - Belanger, Wang - 1993 |

7 | The Complexity of Malign Ensembles
- Miltersen
- 1991
(Show Context)
Citation Context ...der the universal distribution, any hierarchy results for DTIME(t(n)), however tight, will also apply to the expectedtime complexity classes of time t. This line of research was extended by Miltersen =-=[Mil91]-=-. However, the distributions used in these results are either not computable (e.g., m), or require super-polynomial time to compute. To study average-case hierarchies in the framework of De nition 2.1... |

6 |
Rankable distributions do not provide harder instances than uniform distributions
- Belanger, Wang
- 1995
(Show Context)
Citation Context ...wn whether there are natural NP-complete problems under commonly used distributions that are solvable in average polynomial time with respect to the rank of the distributions. Also, Belanger and Wang =-=[BW95]-=- showed that no NP problems averaging with respect to p-time computable ranking functions are harder than NP problems averaging under uniform distributions. In particular, they showed that there is an... |

6 | Average case intractability of diophantine and matrix problems - Venkatesan, Rajagopalan - 1992 |

5 | Truth-table closure and Turing closure of Average Polynomial Time have different measures in EXP
- Schuler
- 1996
(Show Context)
Citation Context ...structural properties of the class of sets that are in AP under every p-time computable distribution can be found in [SY92, SY95, Sch95b, KS95], and a connection to the measure theory can be found in =-=[Sch96]-=-. A similar approach has also been applied to studying the class of optimization problems. Under the worst-case complexity, P = NP implies that all NP optimization problems are polynomial-time solvabl... |

5 |
Towards average-case complexity analysis of NP optimization problems
- Schuler, Watanabe
- 1995
(Show Context)
Citation Context ... also been applied to studying the class of optimization problems. Under the worst-case complexity, P = NP implies that all NP optimization problems are polynomial-time solvable. Schuler and Watanabe =-=[SW95]-=- investigated how much of this can be carried out in the average-case setting. They showed that if every NP decision problem is in AP under every P NP -samplable distribution, then every NP optimizati... |

4 |
Some properties of sets tractable under every polynomial-time computable distribution
- Schuler
- 1995
(Show Context)
Citation Context ...er a Set of Distributions. One may wonder whether there is a set which is not in P but is in AP under every p-time computable distribution. A diagonalization construction provides an a rmative answer =-=[Sch95a]-=-, but it is not known whether there are such natural problems. If a problem is in AP under every exponential-time computable distribution, then it has to be in P [SY92]. Ass1. Average-Case Computation... |

4 |
Average-Case Intractability
- Venkatesan
- 1991
(Show Context)
Citation Context ... below. These issues were either mentioned explicitly or hinted at by Levin [Lev86] and they have been elaborated from various aspects by Johnson [Joh84], Gurevich [Gur89, Gur91a, Gur91b], Venkatesan =-=[Ven91]-=-, and Impagliazzo [Imp95], from which Levin's de nition of average polynomial time (given in De nition 2.1) can be derived naturally and be well justi ed. 2 Levin [Lev86] used to denote a distribution... |

4 | Random Instances of Bounded String Rewriting Are - Wang - 1995 |

3 |
Reductions and convergence rates of average time
- Belanger, Wang
- 1996
(Show Context)
Citation Context ...verage polynomial time under De nition 5.2 when restricted to distributions that satisfy condition W. If condition W is not satis ed, one may get some unwanted results. For example, Belanger and Wang =-=[BW]-=- showed that p-time reductions are not closed for average polynomial time under De nition 5.2. In particular, they constructed two distributional decision problems (A� ) and (B� ), with p-time computa... |

3 |
On the Reduction Theory for Average-Case Complexity
- Blass, Gurevich
- 1991
(Show Context)
Citation Context ...tribution if for all x, (x) g(x) (x), for some function g which is polynomial on -average. The following de nition was shown to be the most general in the context of deterministic many{one reductions =-=[BG91]-=-.s1. Average-Case Computational Complexity Theory 12 De nition 3.6 (A� ) is ap-time reducible to (B� ) if there is a many{ one reduction f that reduces A to B such that f is computable in time polynom... |

3 | Average polynomial time is hard for exponential time under sn-reductions - Schuler - 1995 |

3 | Test instance generation for promised NP search problems
- Watanabe
- 1994
(Show Context)
Citation Context ...he input is guaranteed to be a satis able Boolean formula. The correctness of the algorithm is often justi ed by testing the algorithm on some randomly generated satis able Boolean formulas. Watanabe =-=[Wat94]-=- provided a framework for investigating the difculty of generating test instances for promise NP search problems and some \completeness" results were shown based on the known average-case NP-complete ... |

2 | Average time complexity classes
- Cai, Selman
- 1996
(Show Context)
Citation Context ...s not exist for the worst-case deterministic time because of the linear speedup theorem. It is therefore possible to prove anaverage-case hierarchy result as tight asin the worst case. Cai and Selman =-=[CS95]-=- obtained such a result when time functions are restricted to be logarithmico-exponential. The class L of logarithmico-exponential (log-exp, in short) functions, rst studied by Hardy [Har11], is the s... |

2 |
A note on almost-everywhere complex sets with application to polynomial complexity degrees
- Geske, Huynh, et al.
- 1991
(Show Context)
Citation Context ...h thatavery large portion of instances x will require more than t(x) to compute. This approach has a long history (e.g., see [GGH94]). Among others, the following result of Geske, Huynh, and Seiferas =-=[GHS91]-=- says that for every fully time-constructible function T , there is a set L 2 DTIME(O(T (n)) such that for every function t, if t(x)logt(x) =o(T (x)), then every Turing machine that accepts L requires... |

2 | On average time hierarchies
- Goldmann, Grape, et al.
- 1994
(Show Context)
Citation Context ...functions T and t with t<T, whether there is a language in DTIME(T (n)) such thatavery large portion of instances x will require more than t(x) to compute. This approach has a long history (e.g., see =-=[GGH94]-=-). Among others, the following result of Geske, Huynh, and Seiferas [GHS91] says that for every fully time-constructible function T , there is a set L 2 DTIME(O(T (n)) such that for every function t, ... |

2 | The challenger-solver game: variations on the theme of P =? NP - Gurevich - 1989 |

2 |
On average-P vs. average-NP
- Wang, Belanger
- 1993
(Show Context)
Citation Context ...lowable distributions is needed. If arbitrary distributions are allowed, then no complete distributional problems can exist with respect to one{one, ptime reductions. In particular, Wang and Belanger =-=[WB93a]-=- showed that for any distribution , there is a distribution such that for any one{one, p-time reduction f, cannot be dominated by with respect to f. Reductions that are one{one are a reasonable assump... |

2 | No NP problems averaging over ranking of distributions are harder - Belanger, Wang |

1 |
Orders of In nity, The 'in nitarcalcul' of Paul du Bois-Reymond
- Hardy
- 1924
(Show Context)
Citation Context ...iff(x) andg(x) are in ;, then so are f(x);g(x), exp(f(x)) (i.e., e f (x) ), and ln f(x)(iff(x)iseventually positive). It follows that f(x)+g(x), f(x) g(x), and f(x)=g(x) are also in L. Itwas shown in =-=[Har24]-=- thatevery function in L is either eventually positive oreventually negative or identically zero. It follows that every function in L is eventually monotonic by the fact that L is closed under di eren... |

1 |
On the de nition of some complexity classes of real numbers
- Ko
- 1983
(Show Context)
Citation Context ...p-time computable if there is a deterministic algorithm which on input x outputs f(x) in time bounded by a polynomial in jxj. Gurevich [Gur91a] extended this notion to real-valued functions following =-=[Ko83]-=-. De nition 3.4 Let f : + ! [0� 1] be a real-valued function. Then f is p-time computable if there is a deterministic algorithm which, on every string x and every positive integer k, outputs a nite bi... |

1 |
Average case complexity underthe universal distribution equals worst-case complexity
- Li, Vitanyi
- 1992
(Show Context)
Citation Context ...at there is an expected time hierarchy which is independent of distributions and is as tight as the Hartmanis-Sterns hierarchy forworst-case deterministic time. In a di erent approach, Li and Vitanyi =-=[LV92]-=- showed that under the universal distribution with probability m(x) =2 ;K(x) , where K(x) isthe pre x variant of Kolmogorov complexity [Gac74], the expected time of any problem on strings of the same ... |