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72
Proof verification and hardness of approximation problems
 IN PROC. 33RD ANN. IEEE SYMP. ON FOUND. OF COMP. SCI
, 1992
"... 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 probabilit ..."
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Cited by 797 (39 self)
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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 SNPhard 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 Nvertex graph to within a factor of N ɛ is NPhard.
The Impact of Internet Policy and Topology on Delayed Routing Convergence
 In Proc. IEEE INFOCOM
, 2001
"... Although recent advances in the IETF's Differentiated Services workinggroup promise to improve the performance of applicationlevel services within some networks, across the widearea Internet these QoS algorithms are usuallypredicated on the existence of a stable underlying forwarding infrastr ..."
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Cited by 154 (2 self)
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Although recent advances in the IETF's Differentiated Services workinggroup promise to improve the performance of applicationlevel services within some networks, across the widearea Internet these QoS algorithms are usuallypredicated on the existence of a stable underlying forwarding infrastructure. In recent work, we showed that the Internet lacks effective interdomain pathfailover [1]. Specifically, we found that multihomed Internet sites may experience periods of degraded performance as well as complete loss of connectivitypersisting fifteen minutes or more after a single fault.
On Syntactic versus Computational Views of Approximability
, 1994
"... We attempt to reconcile the two distinct views of approximation classes: syntactic and computational. Syntactic classes such as MAX SNP permit structural results and have natural complete problems, while computational classes such as APX allow us to work with classes of problems whose approximabilit ..."
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Cited by 126 (10 self)
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We attempt to reconcile the two distinct views of approximation classes: syntactic and computational. Syntactic classes such as MAX SNP permit structural results and have natural complete problems, while computational classes such as APX allow us to work with classes of problems whose approximability is wellunderstood. Our results provide a syntactic characterization of computational classes, and give a computational framework for syntactic classes. We compare the syntactically defined class MAX SNP with the computationally defined class APX, and show that every problem in APX can be “placed" (i.e. has approximation preserving reduction to a problem) in MAX SNP. Our methods introduce a general technique for creating approximationpreserving reductions which show that any “well ” approximable problem can be reduced in an approximationpreserving manner to a problem which is hard to approximate to corresponding factors. We demonstrate this technique by applying it to the classes RMAX(2) and MIN F+n2 (1) which have the clique problem and the set cover problem, respectively, as complete problems. We use the syntactic nature of MAX SNP to define a general paradigm, nonoblivious local search, useful for developing simple yet efficient approximation algorithms. We show that such algorithms can find good approximations for all MAX SNP problems, yielding approximution ratios comparable to the bestknown for a variety of specific MAX SNPhard problem. Nonoblivious local search provably outperforms standard local search in both the degree of approximation achieved and the efficiency of the resulting algorithms.
Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems
, 1992
"... 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 acce ..."
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Cited by 65 (8 self)
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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 lowdegree polynomials. We explore the properties of these functions by examining some simple and basic questions about them. We consider questions of the form: • (testing) Given an oracle for a function f, is f close to a lowdegree polynomial? • (correcting) Let f be close to a lowdegree polynomial g, is it possible to efficiently reconstruct the value of g on any given input using an oracle for f? 2 The questions described above have been raised before in the context of coding theory as the problems of errordetecting and errorcorrecting of codes. More recently
Stochastic shortest paths via quasiconvex maximization
 PROCEEDINGS OF EUROPEAN SYMPOSIUM OF ALGORITHMS
, 2006
"... We consider the problem of finding shortest paths in a graph with independent randomly distributed edge lengths. Our goal is to maximize the probability that the path length does not exceed a given threshold value (deadline). We give a surprising exact n Θ(log n) algorithm for the case of normally ..."
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Cited by 31 (9 self)
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We consider the problem of finding shortest paths in a graph with independent randomly distributed edge lengths. Our goal is to maximize the probability that the path length does not exceed a given threshold value (deadline). We give a surprising exact n Θ(log n) algorithm for the case of normally distributed edge lengths, which is based on quasiconvex maximization. We then prove average and smoothed polynomial bounds for this algorithm, which also translate to average and smoothed bounds for the parametric shortest path problem, and extend to a more general nonconvex optimization setting. We also consider a number other edge length distributions, giving a range of exact and approximation schemes.
Optimal route planning under uncertainty
 In Proc. of International Conference on Automated Planning and Scheduling
, 2006
"... We present new complexity results and efficient algorithms for optimal route planning in the presence of uncertainty. We employ a decision theoretic framework for defining the optimal route: for a given source S and destination T in the graph, we seek an STpath of lowest expected cost where the edg ..."
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Cited by 27 (8 self)
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We present new complexity results and efficient algorithms for optimal route planning in the presence of uncertainty. We employ a decision theoretic framework for defining the optimal route: for a given source S and destination T in the graph, we seek an STpath of lowest expected cost where the edge travel trimes are eandom variable and the cost is a nonlinear function of total travel time. Although this is a natural model for routeplanning on realworld road networks, results are sparse due to the analytic difficulty of finding closed form expressions for the exptected cost (Fan, Kalaba and Moore), as well as the computational/combinatorial difficulty of efficiently finding an optimal path which minimizes the exptected cost. We identify a family of appropriate cost models and travel time distributions that are closed under convolution and physically valid. We obtain hardness results for routing problems with a given start time and cost functions with a global minimum, in a variety of deterministic and stochastic settings. In general the global cost is not separable into edge costs, precluding classic shortestpath approaches. However, using partial minimization techniques, we exhibit an efficient solution via dynamic programming with low polynomial complexity.
The Power of Local Optimization: Approximation Algorithms for Maximumleaf Spanning Tree
 In Proceedings, Thirtieth Annual Allerton Conference on Communication, Control and Computing
, 1996
"... Given an undirected graph G, finding a spanning tree of G with maximum number of leaves is NPcomplete. We use the simple technique of local optimization to provide the first approximation algorithms for this problem. Our algorithms run in polynomial time to produce locally optimal solutions. We pro ..."
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Cited by 24 (3 self)
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Given an undirected graph G, finding a spanning tree of G with maximum number of leaves is NPcomplete. We use the simple technique of local optimization to provide the first approximation algorithms for this problem. Our algorithms run in polynomial time to produce locally optimal solutions. We prove that locally optimal solutions to this problem are globally nearoptimal. In particular, we prove that two such algorithms have performance ratios of 5 and 3. The latter algorithm employs more powerful localimprovement steps than the former and hence has higher running time. This may indicate an interesting tradeoff between the performance ratios and the running times of the series of algorithms we describe. Keywords: Approximation algorithms, NPcomplete problems, Performance ratio, Local optimization, Communication network design, Combinatorial algorithms. 1 Introduction Given an undirected graph G = (V; E), the Maximum Leaf Spanning Tree problem is to find a spanning tree of G with ...
Colorcoding: a new method for finding simple paths, cycles and other small subgraphs within large graphs (Extended Abstract)
"... We describe a novel randomized method, the method of colorcoding for finding simple paths and cycles of a specified length k, and other small subgraphs, within a given graph G = (V, E). The randomized algorithms obtained using this method can be derandomized using families of perfect hash functions ..."
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Cited by 22 (1 self)
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We describe a novel randomized method, the method of colorcoding for finding simple paths and cycles of a specified length k, and other small subgraphs, within a given graph G = (V, E). The randomized algorithms obtained using this method can be derandomized using families of perfect hash functions. Using the colorcoding method we obtain, among others, the following new results: • For every fixed k, if a graph G = (V, E) contains a simple cycle of size exactly k, then such a cycle can be found in either O(V ω) expected time or O(V ω log V) worstcase time, where ω < 2.376 is the exponent of matrix multiplication. (Here and in what follows we use V and E instead of V  and E  whenever no confusion may arise.) • For every fixed k, if a planar graph G = (V, E) contains a simple cycle of size exactly k, then
Approximation algorithms and hardness results for labeled connectivity problems
 In 31st MFCS
, 2006
"... Abstract. Let G = (V, E) be a connected multigraph, whose edges are associated with labels specified by an integervalued function L: E → N. In addition, each label ℓ ∈ N to which at least one edge is mapped has a nonnegative cost c(ℓ). The minimum label spanning tree problem (MinLST) asks to find ..."
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Cited by 20 (5 self)
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Abstract. Let G = (V, E) be a connected multigraph, whose edges are associated with labels specified by an integervalued function L: E → N. In addition, each label ℓ ∈ N to which at least one edge is mapped has a nonnegative cost c(ℓ). The minimum label spanning tree problem (MinLST) asks to find a spanning tree in G that minimizes the overall cost of the labels used by its edges. Equivalently, we aim at finding a minimum cost subset of labels I ⊆ N such that the edge set {e ∈ E: L(e) ∈ I} forms a connected subgraph spanning all vertices. Similarly, in the minimum label st path problem (MinLP) the goal is to identify an st path minimizing the combined cost of its labels, where s and t are provided as part of the input. The main contributions of this paper are improved approximation algorithms and hardness results for MinLST and MinLP. As a secondary objective, we make a concentrated effort to relate the algorithmic methods utilized in approximating these problems to a number of wellknown techniques, originally studied in the context of integer covering. 1