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....problem arises in private line data network design. In this setting the sets X and Y are disjoint. This problem is NP complete, and various heuristics have been developed. In previous works, this problem is reduced to the node weighted Steiner tree problem [12] or the directed Steiner tree problem [14]. Several heuristics and algorithms have been developed for the node weighted Steiner tree problem [16, 12, 13, 11, 7] for computing approximate solutions, as well as for computing optimal solutions. In this paper we focus on approximation algorithms. The node weighted Steiner tree problem ....

S. Raghavan, personal communication (1999).

....to analyze rigorously since the transition probabilities are themselves dependent on the current state. For example, 5; 6] implicitly assume that the probability that a node has a given degree is a continuous function. The authors of [10; 12] will o#er an improved analysis in an upcoming paper [16]. In [2] we derive a power law degree sequence for several graph evolution models for asymptotically large graphs by explicitly solving the recurrence relations given by the random evolution process for the expected degree sequence and showing tight concentration around the mean using Azuma s ....

P. Raghavan, personal communication.

....to analyze rigorously since the transition probabilities are themselves dependent on the current state. For example, 5; 6] implicitly assume that the probability that a node has a given degree is a continuous function. The authors of [10; 12] will offer an improved analysis in an upcoming paper [16]. In [2] we derive a power law degree sequence for several graph evolution models for asymptotically large graphs by explicitly solving the recurrence relations given by the random evolution process for the expected degree sequence and showing tight concentration around the mean using Azuma s ....

P. Raghavan, personal communication.

....other protocols using the queries in Table 1. It suffices to find the right combinatorial property, essentially saying a malicious adversary will succeed in forcing many queries on this class , then an adaptation of the proofs in [3, 15] will yield the analog of Theorem 17. For example, Raghavan [25] has identified the property that works for fMem,Equg. This is an interesting and potentially useful case, given the extensive use of this protocol and the important questions still open around it. 6 Circuits: A Complexity Theoretic Barrier When in the last section we listed some classes that are ....

V. Raghavan, personal communication.

....[Wig83] Wigderson gives an algorithm to color any n vertex 3 colorable graph with O( p n) colors, and more generally to color any k colorable graph with O(n 1 Gamma 1 k Gamma1 ) colors. More recently, several researchers: Berger and Rompel [BR88] Linial, Saks, and Wigderson [LSW] and Raghavan [Rag] independently improved this bound to O( n= log n) 1 Gamma 1 k Gamma1 ) colors, which for k = 3 results in a coloring of 3 colorable graphs with O( p n= p log n) colors. The result of Berger and Rompel, et al. was important because no progress had been made for some time and it ....

....Wigderson gives an algorithm to color any n vertex 3 colorable graph with O( p n) colors, and more generally to color any k colorable graph with O(n 1 Gamma 1 k Gamma1 ) colors. More recently, several researchers: Berger and Rompel [BR88] Linial, Saks, and Wigderson [LSW] and Raghavan [Rag] independently improved this bound to O( n= log n) 1 Gamma 1 k Gamma1 ) colors, which for k = 3 results in a coloring of 3 colorable graphs with O( p n= p log n) colors. The result of Berger and Rompel, et al. was important because no progress had been made for some time and it showed ....

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P. Raghavan. personal communication.

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P. Raghavan, personal communication.

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S. Raghavan, personal communication (1999).

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P. Raghavan, personal communication.

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