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149
A Tight Bound on Approximating Arbitrary Metrics by Tree Metrics
 In Proceedings of the 35th Annual ACM Symposium on Theory of Computing
, 2003
"... In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; t ..."
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Cited by 317 (8 self)
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In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion#sto n)distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buyatbulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.
On Approximating Arbitrary Metrics by Tree Metrics
 In Proceedings of the 30th Annual ACM Symposium on Theory of Computing
, 1998
"... This paper is concerned with probabilistic approximation of metric spaces. In previous work we introduced the method of ecient approximation of metrics by more simple families of metrics in a probabilistic fashion. In particular we study probabilistic approximations of arbitrary metric spaces by \hi ..."
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Cited by 281 (16 self)
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This paper is concerned with probabilistic approximation of metric spaces. In previous work we introduced the method of ecient approximation of metrics by more simple families of metrics in a probabilistic fashion. In particular we study probabilistic approximations of arbitrary metric spaces by \hierarchically wellseparated tree" metric spaces. This has proved as a useful technique for simplifying the solutions to various problems.
Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to ..."
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Cited by 194 (15 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
Approximation Algorithms for Directed Steiner Problems
 Journal of Algorithms
, 1998
"... We give the first nontrivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both problems, the best ratios known before our work we ..."
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Cited by 177 (8 self)
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We give the first nontrivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both problems, the best ratios known before our work were the trivial O(k)approximations. For the directed Steiner tree problem, we design a family of algorithms that achieves an approximation ratio of i(i \Gamma 1)k 1=i in time O(n i k 2i ) for any fixed i ? 1, where k is the number of terminals. Thus, an O(k ffl ) approximation ratio can be achieved in polynomial time for any fixed ffl ? 0. Setting i = log k, we obtain an O(log 2 k) approximation ratio in quasipolynomial time. For the directed generalized Steiner network problem, we give an algorithm that achieves an approximation ratio of O(k 2=3 log 1=3 k), where k is the number of pairs of vertices that are to be connected. Related problems including the group Steiner...
Approximating a Finite Metric by a Small Number of Tree Metrics
 In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science
, 1998
"... Bartal [4, 5] gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O(log n log log n). His result has found several applications and in particular has resulted in approximation algorithms f ..."
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Cited by 92 (9 self)
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Bartal [4, 5] gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O(log n log log n). His result has found several applications and in particular has resulted in approximation algorithms for many graph optimization problems. However approximation algorithms based on his
LowerStretch Spanning Trees
, 2005
"... ... as a subgraph a spanning tree into which the edges of G can be embedded with average stretch exp (O ( √ log n log log n)), and that there exists an nvertex graph G such that all its spanning trees have average stretch Ω(log n). Closing the exponential gap between these upper and lower bounds i ..."
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Cited by 84 (11 self)
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... as a subgraph a spanning tree into which the edges of G can be embedded with average stretch exp (O ( √ log n log log n)), and that there exists an nvertex graph G such that all its spanning trees have average stretch Ω(log n). Closing the exponential gap between these upper and lower bounds is listed as one of the longstanding open questions in the area of lowdistortion embeddings of metrics (Matousek 2002). We significantly reduce this gap by constructing a spanning tree in G of average stretch O((log n log log n) 2). Moreover, we show that this tree can be constructed in time O(m log 2 n) in general, and in time O(m log n) if the input graph is unweighted. The main ingredient in our construction is a novel graph decomposition technique. Our new algorithm can be immediately used to improve the running time of the recent solver for diagonally dominant linear systems of Spielman and Teng from to m2 (O( √ log n log log n)) log(1/ɛ) m log O(1) n log(1/ɛ), and to O(n(log n log log n) 2 log(1/ɛ)) when the system is planar. Applying a recent reduction of Boman, Hendrickson and Vavasis, this provides an O(n(log n log log n) 2 log(1/ɛ)) time algorithm for solving the linear systems that arise when applying the finite element method to solve twodimensional elliptic partial differential equations. Our result can also be used to improve several earlier approximation algorithms that use lowstretch spanning trees.
Hedging uncertainty: Approximation algorithms for stochastic optimization problems
 In Proceedings of the 10th International Conference on Integer Programming and Combinatorial Optimization
, 2004
"... We initiate the design of approximation algorithms for stochastic combinatorial optimization problems; we formulate the problems in the framework of twostage stochastic optimization, and provide nearly tight approximation algorithms. Our problems range from the simple (shortest path, vertex cover, ..."
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Cited by 81 (13 self)
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We initiate the design of approximation algorithms for stochastic combinatorial optimization problems; we formulate the problems in the framework of twostage stochastic optimization, and provide nearly tight approximation algorithms. Our problems range from the simple (shortest path, vertex cover, bin packing) to complex (facility location, set cover), and contain representatives with different approximation ratios. The approximation ratio of the stochastic variant of a typical problem is of the same order of magnitude as its deterministic counterpart. Furthermore, common techniques for designing approximation algorithms such as LP rounding, the primaldual method, and the greedy algorithm, can be carefully adapted to obtain these results. 1
Rounding via Trees: Deterministic Approximation Algorithms for Group Steiner Trees and kmedian
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Finding and Approximating Topk Answers in Keyword Proximity Search
 In Proceedings of the Twenty Fourth ACM SIGACTSIGMODSIGART Symposium on Principles of Database Systems
, 2005
"... Various approaches for keyword proximity search have been implemented in relational databases, XML and the Web. Yet, in all of them, an answer is a Qfragment, namely, a subtree T of the given data graph G, such that T contains all the keywords of the query Q and has no proper subtree with this prop ..."
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Cited by 56 (9 self)
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Various approaches for keyword proximity search have been implemented in relational databases, XML and the Web. Yet, in all of them, an answer is a Qfragment, namely, a subtree T of the given data graph G, such that T contains all the keywords of the query Q and has no proper subtree with this property. The rank of an answer is inversely proportional to its weight. Three problems are of interest: finding an optimal (i.e., topranked) answer, computing the topk answers and enumerating all the answers in ranked order. It is shown that, under data complexity, an efficient algorithm for solving the first problem is sufficient for solving the other two problems with polynomial delay. Similarly, an efficient algorithm for finding a θapproximation of the optimal answer suffices for carrying out the following two tasks with polynomial delay, under queryanddata complexity. First, enumerating in a (θ + 1)approximate order. Second, computing a (θ + 1)approximation of the topk answers. As a corollary, this paper gives the first efficient algorithms, under data complexity, for enumerating all the answers in ranked order and for computing the topk answers. It also gives the first efficient algorithms, under queryanddata complexity, for enumerating in a provably approximate order and for computing an approximation of the topk answers.
A Recursive Greedy Algorithm for Walks in Directed Graphs
 PROC. OF IEEE FOCS
, 2005
"... Given an arcweighted directed graph G = (V, A, ℓ) and a pair of nodes s, t, we seek to find an st walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the ori ..."
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Cited by 52 (3 self)
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Given an arcweighted directed graph G = (V, A, ℓ) and a pair of nodes s, t, we seek to find an st walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasipolynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [4] is O(log 2 OPT). The second application is an O(log 2 k) approximation for the kTSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first nontrivial approximation algorithm for this problem. The third application is an O(log 2 k) approximation (in quasipoly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios [15, 19, 8] by a logarithmic factor and almost matches the inapproximability threshold on trees [20]. This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than Ω(log 1−ɛ OPT), even in undirected graphs. Even though our algorithm runs in quasipoly time, we believe that the implications for the approximability of several basic optimization problems are interesting.