Carsten Lund

Bio: Carsten Lund is an academic researcher from AT&T. The author has contributed to research in topics: Sampling (statistics) & Traffic generation model. The author has an hindex of 41, co-authored 95 publications receiving 10272 citations. Previous affiliations of Carsten Lund include Lynn University & AT&T Labs.

Proof verification and the hardness of approximation problems

Abstract: 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 [1998] 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 [1991] 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 et al. [1996] and Arora and Safra [1998] and show that there exists a positive e such that approximating the maximum clique size in an N-vertex graph to within a factor of Ne is NP-hard.

On the hardness of approximating minimization problems

Abstract: We prove results indicating that it is hard to compute efficiently good approximate solutions to the Graph Coloring, Set Covering and other related minimization problems. Specifically, there is an e > 0 such that Graph Coloring cannot be approximated with ratio n e unless P = NP. Set Covering cannot be approximated with ratio c log n for any c < 1/4 unless NP is contained in DTIME(n poly log n ). Similar results follow for related problems such as Clique Cover, Fractional Chromatic Number, Dominating Set, and others

Algebraic methods for interactive proof systems

Abstract: A new algebraic technique for the construction of interactive proof systems is presented. Our technique is used to prove that every language in the polynomial-time hierarchy has an interactive proof system. This technique played a pivotal role in the recent proofs that IP = PSPACE [28] and that MIP = NEXP [4].

Non-deterministic exponential time has two-prover interactive protocols

Abstract: We determine the exact power of two-prover interactive proof systems introduced by Ben-Or, Goldwasser, Kilian, and Wigderson (1988). In this system, two all-powerful noncommunicating provers convince a randomizing polynomial time verifier in polynomial time that the inputx belongs to the languageL. We show that the class of languages having tow-prover interactive proof systems is nondeterministic exponential time. We also show that to prove membership in languages inEXP, the honest provers need the power ofEXP only. The first part of the proof of the main result extends recent techniques of polynomial extrapolation used in the single prover case by Lund, Fortnow, Karloff, Nisan, and Shamir. The second part is averification scheme for multilinearity of a function in several variables held by an oracle and can be viewed as an independent result onprogram verification. Its proof rests on combinatorial techniques employing a simple isoperimetric inequality for certain graphs:

Deriving traffic demands for operational IP networks: methodology and experience

Abstract: Engineering a large IP backbone network without an accurate network-wide view of the traffic demands is challenging. Shifts in user behavior, changes in routing policies, and failures of network elements can result in significant (and sudden) fluctuations in load. In this paper, we present a model of traffic demands to support traffic engineering and performance debugging of large Internet Service Provider networks. By defining a traffic demand as a volume of load originating from an ingress link and destined to a set of egress links, we can capture and predict how routing affects the traffic traveling between domains. To infer the traffic demands, we propose a measurement methodology that combines flow-level measurements collected at all ingress links with reachability information about all egress links. We discuss how to cope with situations where practical considerations limit the amount and quality of the necessary data. Specifically, we show how to infer interdomain traffic demands using measurements collected at a smaller number of edge links-the peering links connecting the neighboring providers. We report on our experiences in deriving the traffic demands in the AT&T IP BAckbone, by collecting, validating, and joining very large and diverse sets of usage, configuration, and routing data over extended periods of time. The paper concludes with a preliminary analysis of the observed dynamics of the traffic demands and a discussion of the practical implications for traffic engineering.

A scalable, commodity data center network architecture

Abstract: Today's data centers may contain tens of thousands of computers with significant aggregate bandwidth requirements. The network architecture typically consists of a tree of routing and switching elements with progressively more specialized and expensive equipment moving up the network hierarchy. Unfortunately, even when deploying the highest-end IP switches/routers, resulting topologies may only support 50% of the aggregate bandwidth available at the edge of the network, while still incurring tremendous cost. Non-uniform bandwidth among data center nodes complicates application design and limits overall system performance.In this paper, we show how to leverage largely commodity Ethernet switches to support the full aggregate bandwidth of clusters consisting of tens of thousands of elements. Similar to how clusters of commodity computers have largely replaced more specialized SMPs and MPPs, we argue that appropriately architected and interconnected commodity switches may deliver more performance at less cost than available from today's higher-end solutions. Our approach requires no modifications to the end host network interface, operating system, or applications; critically, it is fully backward compatible with Ethernet, IP, and TCP.

Computational Complexity: A Modern Approach

Abstract: This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set.

Selection of relevant features and examples in machine

Abstract: In this survey, we review work in machine learning on methods for handling data sets containing large amounts of irrelevant information. We focus on two key issues: the problem of selecting relevant features, and the problem of selecting relevant examples. We describe the advances that have been made on these topics in both empirical and theoretical work in machine learning, and we present a general framework that we use to compare different methods. We close with some challenges for future work in this area. @ 1997 Elsevier Science B.V.

A threshold of ln n for approximating set cover

Abstract: Given a collection ℱ of subsets of S = {1,…,n}, 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 - 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 alogorithm (which is (1 - o(1)) ln n), and provious results of Lund and Yanakakis, that showed hardness of approximation within a ratio of (log2n) / 2 ≃0.72 ln n. For max k-cover, we show an approximation threshold of (1 - 1/e)(up to low-order terms), under assumption that P ≠ NP.