## Average-Case Intractability vs. Worst-Case Intractability (1998)

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Venue: | IN THE 23RD INTERNATIONAL SYMPOSIUM ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE |

Citations: | 3 - 1 self |

### BibTeX

@INPROCEEDINGS{Köbler98average-caseintractability,

author = {Johannes Köbler and Rainer Schuler},

title = {Average-Case Intractability vs. Worst-Case Intractability},

booktitle = {IN THE 23RD INTERNATIONAL SYMPOSIUM ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE},

year = {1998},

pages = {493--502},

publisher = {Springer}

}

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### Abstract

We use the assumption that all sets in NP (or other levels of the polynomial-time hierarchy) have efficient average-case algorithms to derive collapse consequences for MA, AM, and various subclasses of P/poly. As a further consequence we show for C 2 fP(PP);PSPACEg that C is not tractable in the average-case unless C = P.

### Citations

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Some connections between nonuniform and uniform complexity classes
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(Show Context)
Citation Context ... A set A is P ;honest (C)-printable if A is P(C)-printable and the respective Turing transducer is honest and non-adaptive. Next we review the notion of advice functions introduced by Karp and Lipton =-=[KL80]-=- to characterize non-uniform complexity classes. A function h : 0 ! \Sigma is called a polynomiallength function if for some polynomial p and for all ns0, jh(0 n )j = p(n). For a class C of sets, let ... |

160 |
Hiding instances in multioracle queries
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(Show Context)
Citation Context ...nder the distribution induced by the random self-reduction) can be decided by a randomized algorithm in (worst-case) polynomial time. For example, Lipton [Lip91] used an idea of Beaver and Feigenbaum =-=[BF90]-=- to show that multivariate polynomials of low degree are (functionally) random self-reducible. In particular, it follows from Lipton's result that if there is an algorithm computing the permanent effi... |

130 | On Hiding Information from an Oracle - Abadi, Feigenbaum, et al. - 1989 |

114 | BPP has subexponential time simulations unless EXPTIME has publishable proofs
- Babai, Fortnow, et al.
- 1993
(Show Context)
Citation Context ...from [BS92, AKM96, Kob94, Gav95], we can state similar collapse consequences as in Theorem 3.1 for several subclasses of P/poly. We note that by using a different proof technique it has been shown in =-=[BFNW93] that BPP = P f-=-ollows from the assumption that every tally set in \Sigma p 4 is contained in P. Corollary 3.3 1. If NP " TALLY ` AP FP then NP " P/log = P. 2. If \Delta p 2 " TALLY ` AP FP then \Delta... |

107 | On the theory of average case complexity
- Ben-David, Chor, et al.
- 1992
(Show Context)
Citation Context ...ble efficiently on average (i.e., in time polynomial on -average) with respect to certain natural distributionsson the instances. However, this is not true for all NP-complete problems, unless E = NE =-=[BCGL92]-=-. In fact, some natural NP problems A are under a particular distributionscomplete for NP in the sense that A is not efficiently solvable on -average unless any NP problem is efficiently solvable with... |

83 | Oracles and queries that are sufficient for exact learning - Bshouty, Cleve, et al. - 1996 |

75 |
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- Impagliazzo
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(Show Context)
Citation Context ...for C = \Sigma p k . In contrast to worst-case complexity, where NP ` P implies that PH ` P, it is not known whether NP ` AP FP implies that all sets in \Delta p 2 = P(NP) are contained in AP FP (see =-=[Imp95]-=- for an exposition). Consider for example an NP optimization problem. It is not known whether an efficient average-case algorithm for the corresponding decision problem can be used to compute efficien... |

74 | On the random-self-reducibility of complete sets
- Feigenbaum, Fortnow
- 1993
(Show Context)
Citation Context ...ty and worst-case complexity. Namely, if all NP problems 1 can be decided in time polynomial on average, then all sets in NE can be decided in (worst-case) exponential time. Similarly, as observed in =-=[FF93]-=-, any random self-reducible set which can be decided in time polynomial on average (under the distribution induced by the random self-reduction) can be decided by a randomized algorithm in (worst-case... |

71 | Average case completeness
- Gurevich
- 1991
(Show Context)
Citation Context ... ) computes the number p=q. As the following remark shows, requiring that the 5 density functions0 of a distribution is P-computable is a strictly weaker condition unless P 6= NP. Remark. As shown in =-=[Gur91]-=-, if P 6= NP then there exists a distribution whose density functions0 is P-computable but whose distribution functionsis not P-computable. As usual let FP denote the set of polynomial-time computable... |

61 | Counting classes: Thresholds, parity, mods, and fewness - Beigel, Gill - 1992 |

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50 |
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(Show Context)
Citation Context ...If \Sigma p 3 " TALLY ` AP FP then \Sigma p 3 " \Pi p 3 " P/poly = P. Proof. 1. As shown in [BS92], every set A in NP " P/log is contained in P(S) for some sparse set S 2 NP. Furth=-=ermore, as shown in [Har83], P(NP " SPARSE) = P-=-(NP " TALLY), implying that NP " P/log ` P(NP " TALLY). Assuming NP " TALLY ` AP FP , the collapse now follows by Theorem 2.3. 2. As shown in [AKM96], every set A in IC[log; poly] ... |

46 | On Pseudorandomness and Resource-Bounded Measure
- Arvind, Köbler
- 1997
(Show Context)
Citation Context ... P/poly via an abundant advice set H ∈ co-NP (implicit in [NW94]). More recently, it has been shown in [AK01,GZ97] that the Arthur-Merlin class MA is contained in ZPP(NP). In fact, the proof given in =-=[AK01]-=- actually shows that any set in MA (AM) 15sbelongs to NP/poly via an abundant advice set in co-NP (respectively, Π p 2). By using these results we get as an extension of Impagliazzo’s result that unde... |

45 |
Tally languages and complexity classes
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- 1974
(Show Context)
Citation Context ... FP was given by Ben-David et.al. [BCGL92]. Theorem 3.1 [BCGL92] If NP ` AP FP , then E = NE (or, equivalently, NP " TALLY ` P). Proof. Recall that E = NE if and only if every tally set in NP is =-=in P [Boo74]-=-. Since, by Theorem 2.3, every tally set in AP FP is already in P it follows that NP ` AP FP implies E = NE. Put in other words, if NP problems have efficient average-case decision algorithms, then P(... |

24 | Reductions to sets of low information content - Arvind, Han, et al. - 1993 |

19 |
On resource-bounded measure and pseudorandomness
- Arvind, Köbler
- 1997
(Show Context)
Citation Context ...ies stronger collapse consequences for the polynomial hierarchy. For example, if \Delta p 2 ` AP FP then NP ` P/poly implies PH = ZPP . Finally, by applying a technique used to show that MA ` ZPP(NP) =-=[AK97]-=- we extend a result in [Imp95] showing that NP ` AP FP implies BPP = ZPP . More specifically, we derive under the same assumption NP ` AP FP that MA can be derandomized, i.e., MA = NP, whereas under t... |

19 |
Provable Security of Cryptosystems: A Survey
- Angluin, Lichtenstein
(Show Context)
Citation Context ...h is equivalent to NE = E). A different structural property that can be used to relate the average case and worst case complexities of a computational problem is random self-reducibility. As noted in =-=[AL83]-=-, a random self-reducible problem is intractable on average (under the distribution induced by the random self-reduction) unless it is easy in the worst case (see also, e.g., [AFK89,Fei93,FF93]). In f... |

17 | Upper bounds for the complexity of sparse and tally descriptions
- Arvind, Köbler, et al.
- 1996
(Show Context)
Citation Context ...2 NP. Furthermore, as shown in [Har83], P(NP " SPARSE) = P(NP " TALLY), implying that NP " P/log ` P(NP " TALLY). Assuming NP " TALLY ` AP FP , the collapse now follows by The=-=orem 2.3. 2. As shown in [AKM96]-=-, every set A in IC[log; poly] is in P(T ) for some tally set T 2 P(NP \Phi A). Therefore, if additionally A 2 \Delta p 2 , then A 2 P(T ) for some tally set T 2 \Delta p 2 . Since by Theorem 2.3 ever... |

15 | Relations among Mod-classes - Hertrampf - 1990 |

13 |
Relative to a random oracle A, PA 6= NPA 6= co-NPA with probability 1
- Bennett, Gill
- 1981
(Show Context)
Citation Context ...\Delta p 3 ` AP FP then \Sigma p 2 " \Pi p 2 " P/poly = P. ffl If \Sigma p 3 ` AP FP then \Sigma p 3 " \Pi p 3 " P/poly = P. Since BPP is contained in \Sigma p 2 "\Pi p 2 [Sip=-=83, Lau83] and in P/poly [BG81]-=- we get in particular: ffl If \Delta p 3 ` AP FP then BPP = P. It is interesting to note that Corollary 4.5 implies stronger collapse consequences for the polynomial hierarchy. For example, if \Delta ... |

13 | Fine separation of average time complexity classes - Cai, Selman - 1999 |

9 | On helping and interactive proof systems
- Arvind, Köbler, et al.
- 2003
(Show Context)
Citation Context .... An advice string can be checked by a co-NP(IP[P/poly]) computation as follows. On input z 2 \Sigma p(n) , verify for all x 2 \Sigma n that (x; z) 2 I , x 2 L. Since IP[P/poly] is low for \Sigma p 2 =-=[AKS95]-=-, i.e., \Sigma p 2 (IP[P/poly]) = \Sigma p 2 , it follows that the correctness of an advice string can be checked by a \Sigma p 2 computation, and hence L has a multivalued advice function in NPMV hon... |

6 |
Bounding the complexity of advice functions
- a
- 1992
(Show Context)
Citation Context ...tion \Delta p 3 " TALLY ` AP FP and part one of Theorem 3.2, in FP. The consequence that BPP ` P is immediate since BPP ` \Sigma p 2 " \Pi p 2 [Sip83, Lau83] and BPP ` P/poly [BG81]. 9 5. As=-= shown in [Gav95], every se-=-t A in P/poly is in P(T ) for some tally set T in NP(A \Phi \Sigma p 2 ). Therefore, if additionally A 2 \Sigma p 3 " \Pi p 3 , then A 2 P(T ) for some tally set T 2 NP (\Sigma p 3 " \Pi p 3... |

6 | New lowness results for ZPPNP and other complexity classes
- Arvind, Köbler
(Show Context)
Citation Context ...ulfill the property that for all n, all strings x of length n, and all strings w of length p(n), either 〈x, w〉 ∈ I or 〈x, w〉 ∈ I ′ . Clearly, (NP ∩ co-NP)/poly ⊆ NPMVt/poly ⊆ NP/poly ∩ co-NP/poly. In =-=[AK02]-=- it is shown that any self-reducible set A ∈ NPMVt/poly is low for Σ p 2. In fact, the proof shows that A (as well as its complement) is in NP/poly via an advice function in NPMV(NP), implying the fol... |

5 | On the power of generalized MOD-classes
- Köbler, Toda
- 1993
(Show Context)
Citation Context ...contained in AP FP unless K ` ZPP where the middle bit class MP, the classes Mod k P , ks2, and the generalized Mod class ModP have been introduced and studied in [GKR + 95], [CH90, Her90, BG92], and =-=[KT96]-=-, respectively. In Theorem 4.4 below we show a similar collapse for the subclass of P/poly consisting of all sets L for which a multivalued advice function can be computed by a randomized algorithm un... |

4 |
Locating P/poly optimally in the extended low hierarchy
- obler
- 1994
(Show Context)
Citation Context ...tion \Sigma p 2 " TALLY ` AP FP and part two of Theorem 3.2, in FP. This implies that S 2 \Sigma p 2 and using an analogous reasoning as in the proof of part one it follows that A 2 P. 4. As show=-=n in [Kob94], eve-=-ry set A in P/poly has an advice function h computable in FP(NP(A) \Phi \Sigma p 2 ). Therefore, if additionally A 2 \Sigma p 2 " \Pi p 2 , then h 2 FP (\Sigma p 2 ) and thus, using the assumptio... |

3 |
Sch oning. Logarithmic advice classes
- azar, U
- 1992
(Show Context)
Citation Context ...set are in P. 4. If \Delta p 3 " TALLY ` AP FP then \Sigma p 2 " \Pi p 2 " P/poly = P and hence BPP = P. 5. If \Sigma p 3 " TALLY ` AP FP then \Sigma p 3 " \Pi p 3 " P/po=-=ly = P. Proof. 1. As shown in [BS92], every set A in NP " P/l-=-og is contained in P(S) for some sparse set S 2 NP. Furthermore, as shown in [Har83], P(NP " SPARSE) = P(NP " TALLY), implying that NP " P/log ` P(NP " TALLY). Assuming NP " T... |

1 |
Using efficient average-case algorithms to collapse worstcase complexity classes
- obler, Schuler
- 1997
(Show Context)
Citation Context ... FP is contradictory to Lutz' hypothesis that NP is not small in EXP, as follows directly from the fact that AP FP is small in EXP [SY95, CS96]. In this extended abstract most proofs are omitted; see =-=[KS97]-=- for a complete version. 2 Preliminaries All languages are over the binary alphabet \Sigma = f0; 1g. The length of a string x 2 \Sigma is denoted by jxj. For a language A, let A =n denote the set of a... |