5.2 Application to Nonapproximability of Optimization Functions

{ 26 { makes expander codes inappropriate for on-line communications, but they may be useful for storage on write-once media, because of their eecient decoding algorithms. Both of these recent developments were inspired by the innuential role that codes play in the theory of probabilistically checkable proof systems. In a forthcoming paper, Arora 2] explores this role in depth. His work formalizes the notion of a \code-like reduction," observes that the nonapproximability results discussed in Section 5.2 use such reductions, and shows that these reductions have certain limitations. { 25 { on input (x; L) is thus the maximum, over 2 f0; 1g s(n) , of the number of simultaneously satissed constraints C x;r. Now we argue that (1=10)-approximating MAX-PCP is NP-hard. This follows directly from the existence of a \gap" in acceptance probabilities in the deenition of a probabilistically checkable proof system. If x 2 L, then MAX-PCP(x; L) = n c , i.e., there is some proof string for which the veriier accepts on all coin-toss sequences r, and hence all constraints can be satissed simultaneously. On the other hand, if x 6 2 L, then MAX-PCP(x; L) :5n c , because, for all proof strings , the veriier accepts on at most 1=2 the coin-toss sequences r, and hence at most 1=2 of the constraints are simultaneously satissed. A (1=10)-approximation algorithm for MAX-PCP would thus always return a value that was at least (1=1:1)n c :9n c or at most :55n c , thus allowing us to distinguish between the cases x 2 L and x 6 2 L. Because L is an arbitrary NP language, this implies that (1=10)-approximating MAX-PCP is NP-hard. Finally, note that MAX-PCP is in the class MAX-SNP, because it is a constant-arity constraint-satisfaction problem. This means that there is a linear reduction from MAX-PCP to any optimization problem that is MAX-SNP-hard. Thus, any such problem is NP-hard to-approximate, for some constant. This means that no such problem has a PTAS, unless P = NP. The PCP Theorem has been used to derive nonapproximability results for many natural optimization functions, including chromatic number, clique number, MAX-3SAT, shortest vector, nearest vector, and halfspace learning. See 1, 57] for a thorough discussion of these applications. Similarly, the results of Condon et al. 19, 20] are used to derive nonap-proximability results for PSPACE-hard optimization functions, including nite-automaton intersection, MAX-Quatiied-3SAT, dynamic graph reliability, and games such as …