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38
Optimal Hierarchical Decompositions for Congestion Minimization in Networks
, 2008
"... Hierarchical graph decompositions play an important role in the design of approximation and online algorithms for graph problems. This is mainly due to the fact that the results concerning the approximation of metric spaces by tree metrics (e.g. [10, 11, 14, 16]) depend on hierarchical graph decompo ..."
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Cited by 42 (1 self)
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Hierarchical graph decompositions play an important role in the design of approximation and online algorithms for graph problems. This is mainly due to the fact that the results concerning the approximation of metric spaces by tree metrics (e.g. [10, 11, 14, 16]) depend on hierarchical graph decompositions. In this line of work a probability distribution over tree graphs is constructed from a given input graph, in such a way that the tree distances closely resemble the distances in the original graph. This allows it, to solve many problems with a distancebased cost function on trees, and then transfer the tree solution to general undirected graphs with only a logarithmic loss in the performance guarantee. The results about oblivious routing [30, 22] in general undirected graphs are based on hierarchical decompositions of a different type in the sense that they are aiming to approximate the bottlenecks in the network (instead of the pointtopoint distances). We call such decompositions cutbased decompositions. It has been shown that they also can be used to design approximation and online algorithms for a wide variety of different problems, but at the current state of the art the performance guarantee goes down by an O(log 2 n log log n)factor when making the transition from tree networks to general graphs. In this paper we show how to construct cutbased decompositions that only result in a logarithmic loss in performance, which is asymptotically optimal. Remarkably, one major ingredient of our proof is a distancebased decomposition scheme due to Fakcharoenphol, Rao and Talwar [16]. This shows an interesting relationship between these seemingly different decomposition techniques. The main applications of the new decomposition are an optimal O(log n)competitive algorithm for oblivious routing in general undirected graphs, and an O(log n)approximation for Minimum Bisection, which improves the O(log 1.5 n) approximation
Oblivious network design
 In Proceedings of the 17th Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2006
"... Consider the following network design problem: given a network G = (V, E), sourcesink pairs {si, ti} arrive and desire to send a unit of flow between themselves. The cost of the routing is this: if edge e carries a total of fe flow (from all the terminal pairs), the cost is given by ∑ e ℓ(fe), wher ..."
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Cited by 32 (8 self)
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Consider the following network design problem: given a network G = (V, E), sourcesink pairs {si, ti} arrive and desire to send a unit of flow between themselves. The cost of the routing is this: if edge e carries a total of fe flow (from all the terminal pairs), the cost is given by ∑ e ℓ(fe), where ℓ is some concave cost function; the goal is to minimize the total cost incurred. However, we want the routing to be oblivious: when terminal pair {si, ti} makes its routing decisions, it does not know the current flow on the edges of the network, nor the identity of the other pairs in the system. Moreover, it does not even know the identity of the function ℓ, merely knowing that ℓ is a concave function of the total flow on the edge. How should it (obliviously) route its one unit of flow?
Approximation algorithms for multicommoditytype problems with guarantees independent of the graph size
 IN: PROCEEDINGS, IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS
, 2009
"... Linial, London and Rabani [3] proved that the mincut maxflow ratio for general maximum concurrent flow problems (when there are k commodities) is O(log k). Here we attempt to derive a more general theory of Steiner cut and flow problems, and we prove bounds that are polylogarithmic in k for a muc ..."
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Cited by 18 (4 self)
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Linial, London and Rabani [3] proved that the mincut maxflow ratio for general maximum concurrent flow problems (when there are k commodities) is O(log k). Here we attempt to derive a more general theory of Steiner cut and flow problems, and we prove bounds that are polylogarithmic in k for a much broader class of multicommodity flow and cut problems. Our structural results are motivated by the meta question: Suppose we are given a poly(log n) approximation algorithm for a flow or cut problem when can we give a poly(log k) approximation algorithm for a generalization of this problem to a Steiner cut or flow problem? Thus we require that these approximation guarantees be independent of the size of the graph, and only depend on the number of commodities (or the number of terminal nodes in a Steiner cut problem). For many natural applications (when k = n o(1) ) this yields much stronger guarantees. We construct vertexsparsifiers that approximately preserve the value of all terminal mincuts. We prove such sparsifiers exist through zerosum games and metric geometry, and we construct such sparsifiers through oblivious routing guarantees. These results let us reduce a broad class of multicommoditytype problems to a uniform case (on k nodes) at the cost of a loss of a poly(log k) in the approximation guarantee. We then give poly(log k) approximation algorithms for a number of problems for which such results were previously unknown, such as requirement cut, lmulticut, oblivious 0extension, and natural Steiner generalizations of oblivious routing, mincut linear arrangement and minimum linear arrangement.
Oblivious routing on nodecapacitated and directed graphs
 IN PROCEEDINGS OF THE 16TH ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA), 2005
, 2005
"... Oblivious routing algorithms for general undirected networks were introduced by Räcke [17], and this work has led to many subsequent improvements and applications. Comparatively little is known about oblivious routing in general directed networks, or even in undirected networks with node capacities. ..."
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Cited by 16 (8 self)
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Oblivious routing algorithms for general undirected networks were introduced by Räcke [17], and this work has led to many subsequent improvements and applications. Comparatively little is known about oblivious routing in general directed networks, or even in undirected networks with node capacities. We present the first nontrivial upper bounds for both these cases, providing algorithms for kcommodity oblivious routing problems with competitive ratio O (√ k log(n)) for undirected nodecapacitated graphs and O (√ k n 1/4 log(n)) for directed graphs. In the special case that all commodities have a common source or sink, our upper bound becomes O ( √ n log(n)) in both cases, matching the lower bound up to a factor of log(n). The lower bound (which first appeared in [6]) is obtained on a graph with very high degree. We show that in fact the degree of a graph is a crucial parameter for nodecapacitated oblivious routing in undirected graphs, by providing an O(∆ polylog(n))competitive oblivious routing scheme for graphs of degree ∆. For the directed case, however, we show that the lower bound of Ω (√ n) still holds in lowdegree graphs. Finally, we settle an open question about routing problems in which all commodities share a common source or sink. We show that even in this simplified scenario there are networks in which no oblivious routing algorithm can achieve a competitive ratio better than Ω(log n).
Approximating the kMulticut Problem
"... We study the kmulticut problem: Given an edgeweighted undirected graph, a set of l pairs of vertices, and a target k ≤ l, find the minimum cost set of edges whose removal disconnects at least k pairs. This generalizes the well known multicut problem, where k = l. We show that the kmulticut problem ..."
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Cited by 16 (0 self)
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We study the kmulticut problem: Given an edgeweighted undirected graph, a set of l pairs of vertices, and a target k ≤ l, find the minimum cost set of edges whose removal disconnects at least k pairs. This generalizes the well known multicut problem, where k = l. We show that the kmulticut problem on trees can be approximated within a factor of 8 3 + ɛ, for any fixed ɛ> 0, and within O(log 2 n log log n) on general graphs, where n is the number of vertices in the graph. For any fixed ɛ> 0, we also obtain a polynomial time algorithm for kmulticut on trees which returns a solution of cost at most (2 + ɛ) · OP T, that separates at least (1 − ɛ) · k pairs, where OP T is the cost of the optimal solution separating k pairs. Our techniques also give a simple 2approximation algorithm for the multicut problem on trees using total unimodularity, matching the best known algorithm [8].
A unified approach to approximating partial covering problems
 IN PROCEEDINGS OF THE 14TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS
, 2006
"... An instance of the generalized partial cover problem consists of a ground set U and a family of subsets S ⊆ 2 U. Each element e ∈ U is associated with a profit p(e), whereas each subset S ∈ S has a cost c(S). The objective is to find a minimum cost subcollection S ′ ⊆ S such that the combined prof ..."
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Cited by 15 (2 self)
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An instance of the generalized partial cover problem consists of a ground set U and a family of subsets S ⊆ 2 U. Each element e ∈ U is associated with a profit p(e), whereas each subset S ∈ S has a cost c(S). The objective is to find a minimum cost subcollection S ′ ⊆ S such that the combined profit of the elements covered by S ′ is at least P, a specified profit bound. In the prizecollecting version of this problem, there is no strict requirement to cover any element; however, if the subsets we pick leave an element e ∈ U uncovered, we incur a penalty of π(e). The goal is to identify a subcollection S ′ ⊆ S that minimizes the cost of S ′ plus the penalties of uncovered elements. Although problemspecific connections between the partial cover and the prizecollecting variants of a given covering problem have been explored and exploited, a more general connection remained open. The main contribution of this paper is to establish a formal relationship between these two variants. As a result, we present a unified framework for approximating problems that can be formulated or interpreted as special cases of generalized partial cover. We demonstrate the applicability of our method on a diverse collection of covering problems, for some of which we obtain the first nontrivial approximability results.
Oblivious routing in directed graphs with random demands
 IN PROCEEDINGS OF THE 37TH ACM SYMPOSIUM ON THEORY OF COMPUTING (STOC
, 2005
"... Oblivious routing algorithms for general undirected networks were introduced by Räcke, and this work has led to many subsequent improvements and applications. More precisely, Räcke showed that there is an oblivious routing algorithm with polylogarithmic competitive ratio (w.r.t. edge congestion) for ..."
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Cited by 13 (4 self)
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Oblivious routing algorithms for general undirected networks were introduced by Räcke, and this work has led to many subsequent improvements and applications. More precisely, Räcke showed that there is an oblivious routing algorithm with polylogarithmic competitive ratio (w.r.t. edge congestion) for any undirected graph. Comparatively little positive results are known about oblivious routing in general directed networks. Using a novel approach, we present the first oblivious routing algorithm which is O(log 2 n)competitive with high probability in directed graphs given that the demands are chosen randomly from a known demanddistribution. On the other hand, we show that no oblivious routing algorithm can be o( log n log log n) competitive even with constant probability in general directed graphs. Our routing algorithms are not oblivious in the traditional definition, but we add the concept of demanddependence, i.e., the path chosen for an st pair may depend on the demand between s and t. This concept that still preserves that routing decisions are only based on local information proves very powerful in our randomized demand model. Finally, we show that our approach for designing competitive oblivious routing algorithms is quite general and has applications in other contexts like stochastic scheduling.
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 ⊆ G connects a demand (Si, Ti) when it contains a path with o ..."
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Cited by 13 (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 ⊆ 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, nonmetric facility location, tree multicast, and group Steiner tree. Finding a nontrivial approximation ratio for generalized connectivity was left as an open problem. Our starting point is the first polylogarithmic approximation for generalized connectivity, attaining 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 directed Steiner network problem, we obtain an O(k 1/2+ɛ) approximation, which improves on the currently best performance guarantee of Õ(k2/3) due to Charikar et al. (SODA ’98). For the set connector problem, recently introduced by Fukunaga and Nagamochi (IPCO ’07), we present a polylogarithmic approximation; this result improves on the previously known ratio which can be Ω(n) in the worst case.
Robust combinatorial optimization with exponential scenarios
 In IPCO
, 2007
"... Abstract. Following the wellstudied twostage optimization framework for stochastic optimization [15, 18], we study approximation algorithms for robust twostage optimization problems with an exponential number of scenarios. Prior to this work, Dhamdhere et al. [8] introduced approximation algorith ..."
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Cited by 13 (3 self)
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Abstract. Following the wellstudied twostage optimization framework for stochastic optimization [15, 18], we study approximation algorithms for robust twostage optimization problems with an exponential number of scenarios. Prior to this work, Dhamdhere et al. [8] introduced approximation algorithms for twostage robust optimization problems with explicitly given scenarios. In this paper, we assume the set of possible scenarios is given implicitly, for example by an upper bound on the number of active clients. In twostage robust optimization, we need to prepurchase some resources in the first stage before the adversary’s action. In the second stage, after the adversary chooses the clients that need to be covered, we need to complement our solution by purchasing additional resources at an inflated price. The goal is to minimize the cost in the worstcase scenario. We give a general approach for solving such problems using LP rounding. Our approach uncovers an interesting connection between robust optimization and online competitive algorithms. We use this approach, together with known online algorithms, to develop approximation algorithms for several robust covering problems, such as set cover, vertex cover, and edge cover. We also study a simple buyatonce algorithm that either covers all items in the first stage or does nothing in the first stage and waits to build the complete solution in the second stage. We show that this algorithm gives tight approximation factors for unweighted variants of these covering problems, but performs poorly for general weighted problems. 1
J.S.: Improved bounds for online routing and packing via a primaldual approach
 In: Proc. IEEE FOCS (2006
"... ..."