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Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than 2vx/-n vertices. We exhibit an algorithm which finds such a partition A, B, C in O(n) time.

We prove the existence of ε-Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payoffs to all players in any (exact) Nash equilibrium can be ε-approximated by the payoffs to the players in some such logarithmic support ε-Nash equilibrium. These strategies are also uniform on a multiset… (More)

- Richard A. DeMillo, Richard J. Lipton, Frederick G. Sayward
- Computer
- 1978

In many cases tests of a program that uncover simple errors are also effective in uncovering much more complex errors. This so-called coupling effect can be used to save work during the testing process.

- Richard M. Karp, Richard J. Lipton
- STOC
- 1980

It is well known that every set in P has small circuits [13]. Adleman [1] has recently proved the stronger result that every set accepted in polynomial time by a randomized Turing machine has small circuits. Both these results are typical of the known relationships between uniform and nonuniform complexity bounds. They obtain a nonuniform upper bound as a… (More)

- Romas Aleliunas, Richard M. Karp, Richard J. Lipton, László Lovász, Charles Rackoff
- 20th Annual Symposium on Foundations of Computer…
- 1979

1 I. INTRODUCTI ON It is well known that the reachability problem for directed graphs is logspace-complete for the complexity class NSPACE(log n) , and thus holds the key to the open question of whether DSPACE(logn)= NSPACE(logn) ([3,4,5,6]). Here as usual OSPACE(logn) is the class of languages that are accepted in logn space by deterministic Turing Ma chi… (More)

- Dan Boneh, Richard A. DeMillo, Richard J. Lipton
- EUROCRYPT
- 1997

We present a theoretical model for breaking various cryptographic schemes by taking advantage of random hardware faults. We show how to attack certain implementations of RSA and Rabin signatures. An implementation of RSA based on the Chinese Remainder Theorem can be broken using a single erroneous signature. Other implementations can be broken using a… (More)

- Richard J. Lipton, Robert E. Tarjan
- 18th Annual Symposium on Foundations of Computer…
- 1977

Any n-vertex planar graph has the property that it can be divided into components of roughly equal size by removing only O(√n) vertices. This separator theorem, in combination with a divide-and-conquer strategy, leads to many new complexity results for planar graph problems. This paper describes some of these results.

- Richard A. DeMillo, Richard J. Lipton, Alan J. Perlis
- Commun. ACM
- 1977

Language-based security is a protection mechanism that allows software components to interact in a shared address space, such that each component is guaranteed to respect its interfaces and not steal or corrupt internal data of other components. This protection mechanism is complicated to implement correctly, so we might want a formal verification of it.But… (More)

- Richard J. Lipton
- Commun. ACM
- 1975

When proving that a parallel program has a given property it is often convenient to assume that a statement is indivisible, i.e. that the statement cannot be interleaved with the rest of the program. Here sufficient conditions are obtained to show that the assumption that a statement is indivisible can be relaxed and still preserve properties such as… (More)

- Avrim Blum, Merrick L. Furst, Michael Kearns, Richard J. Lipton
- CRYPTO
- 1993

Modern cryptography has had considerable impact on the development of computational learning theory. T ools from cryptography h a ve been used in proving nearly all of the strong negative results for learning. In this paper, we give results in the reverse direction by showing how to construct several cryptographic primitives based on certain assumptions on… (More)