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RELAXING THE REQUIREMENTS use none none implement toaccess none with none EAN128 The said list of problems includes SAT, Clique, and 3-Colorability. 10.2.1.

3. Pro babilistic Versions The de nitions in 10.2.

1.1 can be extended so as to account also for randomized computations. For example, extending De nition 10.

14, we have. De nition 10. none for none 20 (the class tpcBPP): For a probabilistic algorithm A, a Boolean function f , and a time-bound function t : N N, we say that the string x is t -bad for A with respect to f if with probability exceeding 1/3, on input x, either A(x) = f (x) or A runs more that t(. x. ) steps. We s none none ay that A typically solves (S, {X n }n N ) in probabilistic polynomial time if there exists a polynomial p such that the probability that X n is p-bad for A with respect to the characteristic function of S is negligible. We denote by tpcBPP the class of distributional problems that are typically solvable in probabilistic polynomial time.

The de nition of reductions can be similarly extended. This means that in De nition 10.16, both M T (x) and Q(x) (mentioned in Items 2 and 3, respectively) are random variables rather than xed objects.

Furthermore, validity is required to hold (for every input) only with probability 2/3, where the probability space refers only to the internal coin tosses of the reduction. Randomized reductions are closed under composition and preserve typical feasibility (see Exercise 10.24).

Randomized reductions allow the presentation of a distN P-complete problem that refers to the (perfectly) uniform ensemble. Recall that Theorem 10.17 establishes the distN P-completeness of (Su , U ), where U is a quasi-uniform ensemble (i.

e., Pr[Un = M, x, 1t ] = 2 (. M. +. x. ) / n , where n = M, x, 1t ). We rst no none for none te that (Su , U ) 2 can be randomly reduced to (Su , U ), where Su = { M, x, z : M, x, 1. z. Su } and P none for none r[Un = M, x, z ] = 2 (. M. +. x. +. z. ) / n for eve none none ry M, x, z {0, 1}n . The randomized 2 reduction consists of mapping M, x, 1t to M, x, z , where z is uniformly selected in {0, 1}t . Recalling that U = {Un }n N denotes the uniform probability ensemble (i.

e., Un is uniformly distributed on strings of length n) and using a suitable encoding we get Proposition 10.21: There exists S N P such that every (S , X ) distN P is randomly reducible to (S, U ).

Proof Sketch: By the foregoing discussion, every (S , X ) distN P is randomly reducible to (Su , U ), where the reduction goes through (Su , U ). Thus, we focus on reducing (Su , U ) to (Su , U ), where Su N P is de ned as follows. The string bin (.

u. ) bin (. v. ) u v w is in Su if and only if u, v, w Su and = log2 uvw + 1, where b none none in (i) denotes the -bit long binary encoding of the integer i [2 1 ] (i.e., the encoding is padded with zeros to a total length of ).

The reduction maps M, x, z to the string bin (. x. ) bin (. M. ) M x z, where = log2 (. M. + . x. + . z. ) + 1. Noting none for none that this reduction satis es all conditions of De nition 10.16, the proposition follows.

. 10.2.2. Rami cations In our opinio none none n, the most problematic aspect of the theory described in Section 10.2.1 is the choice to focus on simple probability ensembles, which in turn restricts.

10.2. AVERAGE-CASE COMPLEXITY distribution none none al versions of NP to the class distN P (De nition 10.15). As indicated in 10.

2.1.1, this restriction raises two opposite concerns (i.

e., that distN P is either too wide or too narrow).25 Here, we address the concern that the class of simple probability ensembles is too restricted, and consequently that the conjecture distN P tpcBPP is too strong (which would mean that distN P-completeness is a weak evidence for typicalcase hardness).

An appealing extension of the class of simple probability ensembles is presented in 10.2.2.

2, yielding a corresponding extension of distN P, and it is shown that if this extension of distN P is not contained in tpcBPP then distN P itself is not contained in tpcBPP. Consequently, distN P-complete problems enjoy the bene t of both being in the more restricted class (i.e.

, distN P) and being hard as long as some problem in the extended class is hard. Another extension appears in 10.2.

2.1, where we extend the treatment from decision problems to search problems. This extension is motivated by the realization that search problem are actually of greater importance to real-life applications (cf.

Section 2.1.1), and hence a theory motivated by real-life applications must address such problems, as we do next.

Prerequisites. For the technical development of 10.2.

2.1, we assume familiarity with the notion of a unique solution and results regarding it as presented in Section 6.2.

3. For the technical development of 10.2.

2.2, we assume familiarity with hashing functions as presented in Appendix D.2.

In addition, the technical development of 10.2.2.

2 relies on 10.2.2.

1.. 10.2.2.

1. Sea rch Versus Decision Indeed, as in the case of worst-case complexity, search problems are at least as important as decision problems. Thus, an average-case treatment of search problems is indeed called for.

We rst present distributional versions of PF and PC (cf. Section 2.1.

1), following the underlying principles of the de nitions of tpcP and distN P.. De nition 10. none none 22 (the classes tpcPF and distPC): As in Section 2.1.

1, we consider only polynomially bounded search problems, that is, binary relations R {0, 1} {0, 1} such that for some polynomial q it holds that (x, y) R implies . y. q(. x. ). def def Re none none call that R(x) = {y : (x, y) R} and S R = {x : R(x) = }. A distributional search problem consists of a polynomially bounded search problem coupled with a probability ensemble.

The class tpcPF consists of all distributional search problems that are typically solvable in polynomial time. That is, (R, {X n }n N ) tpcPF if there exists an algorithm A and a polynomial p such that the probability that on input X n algorithm A either errs or runs more that p(n) steps is negligible, where A errs on x S R if A(x) R(x) and errs on x S R if A(x) = . A distributional search problem (R, X ) is in distPC if R PC and X is simple (as in De nition 10.

15). Likewise, the class tpcBPPF consists of all distributional search problems that are typically solvable in probabilistic polynomial time (cf. De nition 10.

20). The de nitions of. On the one ha nd, if the de nition of distN P were too liberal, then membership in distN P would mean less than one may desire. On the other hand, if distN P were too restricted, then the conjecture that distN P contains hard problems would have been very questionable..

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