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30
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.
Asymmetric kcenter is log ∗ nhard to Approximate
 In Proc. SODA
, 2005
"... In the Asymmetric kCenter problem, the input is an integer k and a complete digraph over n points together with a distance function obeying the directed triangle inequality. The goal is to choose a set of k points to serve as centers and to assign all the points to the centers, so that the maximum ..."
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Cited by 34 (4 self)
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In the Asymmetric kCenter problem, the input is an integer k and a complete digraph over n points together with a distance function obeying the directed triangle inequality. The goal is to choose a set of k points to serve as centers and to assign all the points to the centers, so that the maximum distance of any point to its center is as small as possible. We show that the Asymmetric kCenter problem is hard to approximate up to a factor of log ∗ n − Θ(1) unless NP ⊆ DTIME(n log log n). Since an O(log ∗ n)approximation algorithm is known for this problem, this essentially resolves the approximability of this problem. This is the first natural problem whose approximability threshold does not polynomially relate to the known approximation classes. We also resolve the approximability threshold of the metric kCenter problem with costs.
On Average Throughput and Alphabet Size in Network Coding
 IEEE TRANSACTION ON INFORMATION THEORY (TO APPEAR)
, 2005
"... We examine the throughput benefits that network coding offers with respect to the average throughput achievable by routing, where the average throughput refers to the average of the rates that the individual receivers experience. We relate these benefits to the integrality gap of a standard LP for ..."
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Cited by 22 (2 self)
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We examine the throughput benefits that network coding offers with respect to the average throughput achievable by routing, where the average throughput refers to the average of the rates that the individual receivers experience. We relate these benefits to the integrality gap of a standard LP formulation for the directed Steiner tree problem. We describe families of configurations over which network coding at most doubles the average throughput, and analyze a class of directed graph configurations with N receivers where network coding offers benefits proportional to √ N. We also discuss other throughput measures in networks, and show how in certain classes of networks, average throughput bounds can be translated into minimum throughput bounds, by employing vector routing and channel coding. Finally, we show configurations where use of randomized coding may require an alphabet size exponentially larger than the minimum alphabet size required.
Set Connectivity Problems in Undirected Graphs and the Directed Steiner Network Problem
"... In the generalized connectivity problem, we are given an edgeweighted graph G = (V, E) and a collection D = {(S1, T1),..., (Sk, Tk)} of distinct demands; each demand (Si, Ti) is a pair of disjoint vertex subsets. We say that a subgraph F of G connects a demand (Si, Ti) when it contains a path with ..."
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Cited by 19 (3 self)
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In the generalized connectivity problem, we are given an edgeweighted graph G = (V, E) and a collection D = {(S1, T1),..., (Sk, Tk)} of distinct demands; each demand (Si, Ti) is a pair of disjoint vertex subsets. We say that a subgraph F of G connects a demand (Si, Ti) when it contains a path with one endpoint in Si and the other in Ti. The goal is to identify a minimum weight subgraph that connects all demands in D. Alon et al. (SODA ’04) introduced this problem to study online network formation settings and showed that it captures some wellstudied problems such as Steiner forest, facility location with nonmetric costs, tree multicast, and group Steiner tree. Finding a nontrivial approximation ratio for generalized connectivity was left as an open problem. We describe the first polylogarithmic approximation algorithm for generalized connectivity that has a performance guarantee of O(log 2 n log 2 k). Here, n is the number of vertices in G and k is the number of demands. We also prove that the cutcovering relaxation of this problem has an O(log 3 n log 2 k) integrality gap. Building upon the results for generalized connectivity, we obtain improved approximation algorithms for two problems that contain generalized connectivity as a special case. For the
The polymatroid Steiner problems
 J. Comb. Optim
, 2005
"... Abstract. The Steiner tree problem asks for a minimum cost tree spanning a given set of terminals S ⊆ V in a weighted graph G = (V, E, c), c: E → R +. In this paper we consider a generalization of the Steiner tree problem, so called Polymatroid Steiner Problem, in which a polymatroid P = P (V) is de ..."
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Cited by 15 (0 self)
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Abstract. The Steiner tree problem asks for a minimum cost tree spanning a given set of terminals S ⊆ V in a weighted graph G = (V, E, c), c: E → R +. In this paper we consider a generalization of the Steiner tree problem, so called Polymatroid Steiner Problem, in which a polymatroid P = P (V) is defined on V and the Steiner tree is required to span at least one base of P (in particular, there may be a single base S ⊆ V). This formulation is motivated by the following application in sensor networks – given a set of sensors S = {s1,..., sk}, each sensor si can choose to monitor only a single target from a subset of targets Xi, find minimum cost tree spanning a set of sensors capable of monitoring the set of all targets X = X1 ∪... ∪ Xk. The Polymatroid Steiner Problem generalizes many known Steiner tree problem formulations including the group and covering Steiner tree problems. We show that this problem can be solved with the polylogarithmic approximation ratio by a generalization of the combinatorial algorithm of Chekuri et. al. [7]. We also define the Polymatroid directed Steiner problem which asks for a minimum cost arborescence connecting a given root to a base of a polymatroid P defined on the terminal set S. We show that this problem can be approximately solved by algorithms generalizing methods of Charikar et al [6].
MultiEmbedding and Path Approximation of Metric Spaces
 IN 14TH ANN. ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 2003
"... Metric embeddings have become a frequent tool in the design of algorithms. The applicability is often dependent on how high the embedding's distortion is. For example embedding into ultrametrics (or arbitrary trees) requires linear distortion. Using probabilistic metric embeddings, the bound re ..."
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Cited by 14 (6 self)
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Metric embeddings have become a frequent tool in the design of algorithms. The applicability is often dependent on how high the embedding's distortion is. For example embedding into ultrametrics (or arbitrary trees) requires linear distortion. Using probabilistic metric embeddings, the bound reduces to O(log n log log n). Yet, the lower bound is still logarithmic. We make
A Combinatorial Approximation Algorithm for the Group Steiner Problem
 Discrete Applied Mathematics
, 2002
"... In the group Steiner problem we are given a graph with edge weights w(e) and m subsets of vertices fg i g i=1 . Each subset g i is called a group and the vertices in S g i are called terminals. It is required to find a minimum weight tree that contains at least one terminal from every group. ..."
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Cited by 12 (1 self)
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In the group Steiner problem we are given a graph with edge weights w(e) and m subsets of vertices fg i g i=1 . Each subset g i is called a group and the vertices in S g i are called terminals. It is required to find a minimum weight tree that contains at least one terminal from every group.
An Improved Approximation Ratio for the Covering Steiner Problem. On the Covering Steiner problem
 Theory of Computing
, 2006
"... Abstract: In the Covering Steiner problem, we are given an undirected graph with edgecosts, and some subsets of vertices called groups, with each group being equipped with a nonnegative integer value (called its requirement); the problem is to find a minimumcost tree which spans at least the requi ..."
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Cited by 5 (1 self)
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Abstract: In the Covering Steiner problem, we are given an undirected graph with edgecosts, and some subsets of vertices called groups, with each group being equipped with a nonnegative integer value (called its requirement); the problem is to find a minimumcost tree which spans at least the required number of vertices from every group. The Covering Steiner problem is a common generalization of the kMST and Group Steiner problems; indeed, when all the vertices of the graph lie in one group with a requirement of k, we get the kMST problem, and when there are multiple groups with unit requirements, we obtain the Group Steiner problem. While many covering problems (e.g., the covering integer programs such as set cover) become easier to approximate as the requirements increase, the Covering Steiner problem