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 study the question of determining whether an unknown function has a particular property or is ffl-far from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the function on instances of its choice. First, we establish some connections between property testing and problems in learning theory. Next, we focus on testing graph properties, and devise algorithms to test whether a graph has properties such as being k-colorable or having a ae-clique (clique of density ae w.r.t the vertex set). Our graph property testing algorithms are probabilistic and make assertions which are correct with high probability, utilizing only poly(1=ffl) edge-queries into the graph, where ffl is the distance parameter. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph which corre...

The study of self-testing and self-correcting programs leads to the search for robust characterizations of functions. Here we make this notion precise and show such a characterization for polynomials. From this characterization, we get the following applications.

Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c � 1 and given any n nodes in � 2, a randomized version of the scheme finds a (1 � 1/c)-approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes are in � d, the running time increases to O(n(log n) (O(�dc))d�1). For every fixed c, d the running time is n � poly(log n), that is nearly linear in n. The algorithm can be derandomized, but this increases the running time by a factor O(n d). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-approximation in polynomial time. We also give similar approximation schemes for some other NP-hard Euclidean problems: Minimum Steiner Tree, k-TSP, and k-MST. (The running times of the algorithm for k-TSP and k-MST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constant-factor approximation. We also give efficient approximation schemes for Euclidean Min-Cost Matching, a problem that can be solved exactly in polynomial time. All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as �p for p � 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.

We show that a parallel repetition of any two-prover one-round proof system (MIP(2, 1)) decreases the probability of error at an exponential rate. No constructive bound was previously known. The constant in the exponent (in our analysis) depends only on the original probability of error and on the total number of possible answers of the two provers. The dependency on the total number of possible answers is logarithmic, which was recently proved to be almost the best possible [U. Feige and O. Verbitsky, Proc. 11th Annual IEEE Conference on Computational Complexity, IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 70--76].

We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with link-costs and, for each pair fi; jg of nodes, an edge-connectivity requirement r ij . The goal is to find a minimum-cost network using the available links and satisfying the requirements. Our algorithm outputs a solution whose cost is within 2dlog 2 (r + 1)e of optimal, where r is the highest requirement value. In the course of proving the performance guarantee, we prove a combinatorial min-max approximate equality relating minimum-cost networks to maximum packings of certain kinds of cuts. As a consequence of the proof of this theorem, we obtain an approximation algorithm for optimally packing these cuts; we show that this algorithm has application to estimating the reliability of a probabilistic network.

This paper continues the investigation of the connection between proof systems and approximation. The emphasis is on proving tight non-approximability results via consideration of measures like the "free bit complexity" and the "amortized free bit complexity" of proof systems.

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a coloring is called the chromatic number of�, and is usually denoted by��.Determining the chromatic number of a graph is known to be NP-hard (cf. [19]). Besides its theoretical significance as a canonical NPhard problem, graph coloring arises naturally in a variety of applications such as register allocation [11, 12, 13] is the maximum degree of any vertex. Be-and timetable/examination scheduling [8, 40]. In many We consider the problem of coloring�-colorable graphs with the fewest possible colors. We give a randomized polynomial time algorithm which colors a 3-colorable graph on vertices with� � ���� colors where sides giving the best known approximation ratio in terms of, this marks the first non-trivial approximation result as a function of the maximum degree. This result can be generalized to�-colorable graphs to obtain a coloring using�� � ��� � � � �colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovász�-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the�-function. 1

We prove the following about the Nearest Lattice Vector Problem (in any `p norm), the Nearest Codeword Problem for binary codes, the problem of learning a halfspace in the presence of errors, and some other problems. 1. Approximating the optimum within any constant factor is NP-hard. 2. If for some ffl ? 0 there exists a polynomial-time algorithm that approximates the optimum within a factor of 2 log 0:5\Gammaffl n , then every NP language can be decided in quasi-polynomial deterministic time, i.e., NP ` DTIME(n poly(log n) ). Moreover, we show that result 2 also holds for the Shortest Lattice Vector Problem in the `1 norm. Also, for some of these problems we can prove the same result as above, but for a larger factor such as 2 log 1\Gammaffl n or n ffl . Improving the factor 2 log 0:5\Gammaffl n to p dimension for either of the lattice problems would imply the hardness of the Shortest Vector Problem in `2 norm; an old open problem. Our proofs use reductions from few-pr...