We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probability 1 (i.e., for every choice of its random string). For strings not in the language, the verifier rejects every provided “proof " with probability at least 1/2. Our result builds upon and improves a recent result of Arora and Safra [6] whose verifiers examine a nonconstant number of bits in the proof (though this number is a very slowly growing function of the input length). As a consequence we prove that no MAX SNP-hard problem has a polynomial time approximation scheme, unless NP=P. The class MAX SNP was defined by Papadimitriou and Yannakakis [82] and hard problems for this class include vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige, Goldwasser, Lovász, Safra and Szegedy [42], and Arora and Safra [6] and shows that there exists a positive ɛ such that approximating the maximum clique size in an N-vertex graph to within a factor of N ɛ is NP-hard. 1

We attempt to reconcile the two distinct views of approximation classes: syntactic and computational. Syntactic classes such as MAX SNP permit structural results and have natural complete problems, while computational classes such as APX allow us to work with classes of problems whose approximability is well-understood. Our results provide a syntactic characterization of computational classes, and give a computational framework for syntactic classes. We compare the syntactically defined class MAX SNP with the computationally defined class APX, and show that every problem in APX can be “placed ” (i.e. has approximation preserving reduction to a problem) in MAX SNP. Our methods introduce a general technique for creating approximation-preserving reductions which show that any “well ” approximable problem can be reduced in an approximation-preserving manner to a problem which is hard to approximate to corresponding factors. We demonstrate this technique by applying it to the classes RMAX(2) and MIN F+Π2(1)which have the clique problem and the set cover problem, respectively, as complete problems. We use the syntactic nature of MAX SNP to define a general paradigm, non-oblivious local search, useful for developing simple yet efficient approximation algorithms. We show that such algorithms can find good approximations for all MAX SNP problems, yielding approximation ratios comparable to the best-known for a variety of specific MAX SNP-hard problems. Non-oblivious local search provably out-performs standard local search in both the degree of approximation achieved and the efficiency of the resulting algorithms.

This chapter is a self-contained survey of recent results about the hardness of approximating NP-hard optimization problems.

Introduction The Maximum Traveling Salesman Problem (MAX TSP), also known informally as the "taxicab ripoff problem", is stated as follows: Given an n \Theta n real matrix c = (c ij ), called a weight matrix, find a hamiltonian cycle i 1 7! i 2 7! : : : 7! i n 7! i 1 , for which the maximum value of c i 1 i 2 + c i 2 i 3 + : : : + c i n\Gamma1 i n + c i n i 1 is attained. Here (i 1 ; : : : ; i n ) is a permutation of the set f1; : : : ; ng. Of course, in this general setting, the Maximum Traveling Salesman Problem is equivalent to the Minimum Traveling Salesman Problem, Partially supported by NSF Grant DMS 9734138 since the maximum weight hamiltonian cycle with the weight matrix c corresponds to the minimum weight hamiltonian cycle with the weight matrix \Gammac. What makes the MAX TSP special is that there are some interesting and natural special cases of weights c ij , not preserved by the sign reversal, where much more can be said about the problem than in the general case. Be

We are given a set of points pl,...,p. and a symmetric distance matrix (o!ij) giving the distance between pi and pj. We wish to construct a tour that minimizes ~~=1 1(z), where l(i) is

We consider the following problem: given a collection of strings s 1 ; . . . ; s m , find the shortest string s such that each s i appears as a substring (a consecutive block) of s. Although this problem is known to be NP-hard, a simple greedy procedure appears to do quite well and is routinely used in DNA sequencing and data compression practice, namely: repeatedly merge the pair of distinct strings with maximum overlap until only one string remains. Let n denote the length of the optimal superstring. A common conjecture states that the above greedy procedure produces a superstring of length O(n) (in fact, 2n), yet the only previous nontrivial bound known for any polynomial-time algorithm is a recent O(n log n) result. We show that the greedy algorithm does in fact achieve a constant factor approximation, proving an upper bound of 4n. Furthermore, we present a simple modified version of the greedy algorithm that we show produces a superstring of length at most 3n. We also show the sup...

The definition of the class NP [Coo71, Lev73] highlights the problem of verification of proofs as one of central interest to theoretical computer science. Recent efforts have shown that the efficiency of the verification can be greatly improved by allowing the verifier access to random bits and accepting probabilistic guarantees from the verifier [BFL91, BFLS91, FGL + 91, AS92]. We improve upon the efficiency of the proof systems developed above and obtain proofs which can be verified probabilistically by examining only a constant number of (randomly chosen) bits of the proof. The efficiently verifiable proofs constructed here rely on the structural properties of low-degree polynomials. We explore the properties of these functions by examining some simple and basic questi...

We consider the problem of approximating the longest path in undirected graphs. In an attempt to pin down the best achievable performance ratio of an approximation algorithm for this problem, we present both positive and negative results. First, a simple greedy algorithm is shown to find long paths in dense graphs. We then consider the problem of finding paths in graphs that are guaranteed to have extremely long paths. We devise an algorithm that finds paths of a logarithmic length in Hamiltonian graphs. This algorithm works for a much larger class of graphs (weakly Hamiltonian), where the result is the best possible. Since the hard case appears to be that of sparse graphs, we also consider sparse random graphs. Here we show that a relatively long path can be obtained, thereby partially answering an open problem of Broder, Frieze and Shamir. To explain the difficulty of obtaining better approximations, we also prove hardness results. We show that, for any ffl ! 1, the problem of findin...

Given a planar Rraph on n nodes with costs (weights) on its edges,- define;he distance between nodes i &d 2 as ’ the length of the shortest path between i and i. Consider this as &I instance of me & TSP. For any E> 6, our algorithm finds a salesman tour of total cost at most (1 + E) times optimal in time n”(llea). We also present a quasi-polynomial time algorithm for the Steiner version of this problem.

We consider the special case of the traveling salesman problem (TSP) in which the distance metric is the shortest-path metric of a planar unweighted graph. We present a polynomial-time approximation scheme (PTAS) for this problem.