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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 cardi-nality. Both these problems are NP-hard. We prove that (1 ? o(1)) ln n is a threshold below which… (More)

It is well known that two prover proof systems are a convenient tool for establishing hardness of approximation results. In this paper, we show that two prover proof systems are also convenient starting points for establishing easiness of approximation results. Our approach combines the Feige-Lovv asz (STOC92) semideenite programming relaxation of one-round… (More)

Submodular maximization generalizes many important problems including Max Cut in directed/undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is NP-hard. In this paper, we design the first… (More)

In this paper we extend the notion of zero knowledge proofs of membership (which reveal one bit of information) to zero knowledge proofs of knowledge (which reveal no information whatsoever). After formally defining this notion, we show its relevance to identification schemes, in which parties prove their identity by demonstrating their knowledge rather… (More)

We investigate relations between average case complexity and the complexity of approximation. Our preliminary findings indicate that this is a research direction that leads to interesting insights. Under the assumption that refuting 3SAT is hard on average on a natural distribution, we derive hardness of approximation results for min bisection, dense… (More)

This paper considers the problem of computing the dense k-vertex subgraph of a given graph, namely, the subgraph with the most edges. An approximation algorithm is developed for the problem, with approximation ratio O(n), for some < 1=3.

1 Introduction A two party protocol in which party A uses one of several secret witnesses to an NP assertion is witness indistinguishable if party B cannot tell which witness A is actually using. The protocol is witness hiding if by the end of the protocol B cannot compute any new witness which he did not know before the protocol began. Witness hiding is a… (More)

We develop the algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into L<inf>1</inf> (and even Euclidean embeddings) are insufficient, but that the additional structure provided by many embedding theorems does suffice for our purposes.We obtain an… (More)

The contribution of this paper is two-fold. First, a connection is established between approximating the size of the largest clique in a graph and multi-prover interactive proofs. Second, an efficient multi-prover interactive proof for NP languages is constructed, where the verifier uses very few random bits and communication bits. Last, the connection… (More)

The domatic number problem is that of partitioning the ver-tices of a graph into the maximum number of disjoint dominating sets. Let n denote the number of vertices in a graph, 6 the minimum degree, and A the maximum degree. Trivially, the domatic number is at most (6 + 1). We show that every graph has a domatic partition with (1-o(1))(6 + 1)/lnn sets, and… (More)