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

Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max k-cover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NP-hard. We prove that (1 \Gamma o(1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low order terms) between the ratio of approximation achievable by the greedy algorithm (which is (1 \Gamma o(1)) ln n), and previous results of Lund and Yannakakis, that showed hardness of approximation within a ratio of (log 2 n)=2 ' 0:72 lnn. For max k-cover we show an approximation threshold of (1 \Gamma 1=e) (up to low order terms), under the assumption that P != NP .

We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c . To obtain this result, we show that the treewidth of a planar graph with domination number (G) is O( (G)), and that such a tree decomposition can be found in O( (G)n) time. The same technique can be used to show that the k-face cover problem ( find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c n) time, where c 1 = 3 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k-dominating set, e.g., k-independent dominating set and k-weighted dominating set.

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...

A set of vertices in a graph is a dominating set if every vertex outside the set has aneighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number ofvertices, ffi the minimum degree, and \Delta the maximum degree.We show that every graph has a domatic partition with (1-o(1))(ffi + 1) / ln n dominatingsets, and moreover, that such a domatic partition can be found in polynomial time. This implies a (1 + o(1)) ln n approximation algorithm for domatic number, since the domaticnumber is always at most ffi + 1. We also show this to be essentially best possible. Namely,extending the approximation hardness of set cover by combining multi-prover protocols with zero-knowledge techniques, we show that for every ffl> 0, a (1- ffl) ln n-approximation impliesthat N P ` DT IM E(nO(log log n)). This makes domatic number the first natural maximiza-tion problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better.We also show that every graph has a domatic partition with (1-o(1))(ffi + 1) / ln \Delta dominating sets, where the " o(1) " term goes to zero as \Delta increases. This can be turned intoan efficient algorithm that produces a domatic partition of \Omega ( ffi / ln \Delta) sets.

: We investigate here NP optimization problems from a logical definability standpoint. We show that the class of optimization problems whose optimum is definable using first-order formulae coincides with the class of polynomially bounded NP optimization problems on finite structures. After this, we analyze the relative expressive power of various classes of optimization problems that arise in this framework. Some of our results show that logical definability has different implications for NP maximization problems than it has for NP minimization problems, in terms of both expressive power and approximation properties. To appear in Information and Computation. Research partially supported by NSF Grants CCR8905038 and CCR-9108631. y e-mail addresses: kolaitis@cse.ucsc.edu, thakur@cse.ucsc.edu z supersedes Technical report UCSC-CRL-90-48 1 Introduction and Summary of Results It is well known that optimization problems had a major influence on the development of the theory of NP-co...

We investigate the well-known anomalous differences in the approximability properties of NP-complete optimization problems. We define a notion of polynomial time reduction between optimization problems, and introduce conditions guaranteeing that such reductions preserve various types of approximate solutions. We then prove that a weighted version of the satisfiability problem, the traveling salesperson problem, and the zero-one integer programming problem are in a strong sense approximation complete for the class of NP minimization problems. Finally, we discuss the reasons that cause the standard relative error approximation quality measure to break down in computationally simple problem transformations, and give a general construction for producing quality measures that are more robust with respect to an arbitrary given class of invertible transformations. 1

. In this paper we generalize the notion of polynomial-time approximation scheme preserving reducibility, called PTAS-reducibility, introduced in [4]. As a first application of this generalization, we prove the APX-completeness of a polynomially bounded optimization problem, that is, an APX problem whose measure function is bounded by a polynomial in the length of the instance and such that any APX problem is reducible to it. As far as we know, no such problem was known before. This result has been recently used in [10] to show that several natural optimization problem are APX-complete, such as Max Cut, Max Sat, Min Node Cover, and Min \Delta-TSP. Successively, we apply the notion of APX-completeness to the study of the relative complexity of evaluating an ffl-approximate value and computing an ffl-approximate solution for any ffl. We first show that if P 6= NP " coNP then the former question can be easier than the latter even if the optimization problem is NP-hard. We therefore give ...

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Hierarchies have long been used for organization, summarization, and access to information. In this dissertation we define summarization in terms of a probabilistic language model and use this definition to explore a new technique for automatically generating topic hierarchies. We use the language model to characterize the documents that will be summarized and then apply a graph-theoretic algorithm to determine the best topic words for the hierarchical summary. This work is very different from previous attempts to generate topic hierarchies because it relies on statistical analysis and language modeling to identify descriptive words for a document and organize the words in a hierarchical structure. We compare