We introduce the problem of finding a spanning tree along with a partition of the tree edges into fewest number of feasible sets, where constraints on the edges define feasibility. The motivation comes from wireless networking, where we... more
We introduce the problem of finding a spanning tree along with a partition of the tree edges into fewest number of feasible sets, where constraints on the edges define feasibility. The motivation comes from wireless networking, where we seek to model the irregularities seen in actual wireless environments. Not all node pairs may be able to communicate, even if geographically close --- thus, the available pairs are modeled with a link graph $\mathcal{L}=(V,E)$. Also, signal attenuation need not follow a nice geometric formulas --- hence, interference is modeled by a conflict (hyper)graph $\mathcal{C}=(E,F)$ on the links. The objective is to maximize the efficiency of the communication, or equivalently minimizing the length of a schedule of the tree edges in the form of a coloring. We find that in spite of all this generality, the problem can be approximated linearly in terms of a versatile parameter, the inductive independence of the interference graph. Specifically, we give a simple...
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Scheduling jobs with pairwise conicts is modeled by the graph multicoloring problem. It occurs in two versions: in the preemptive case, each vertex may get any set of colors, while in the non-preemptive case, the set of colors assigned to... more
Scheduling jobs with pairwise conicts is modeled by the graph multicoloring problem. It occurs in two versions: in the preemptive case, each vertex may get any set of colors, while in the non-preemptive case, the set of colors assigned to each vertex has to be contiguous. We study these versions of the multicoloring problem on trees, under the sum-of-completion- times
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The data migration problem is to compute an efficient plan for moving data stored on devices in a network from one configuration to another. We consider this problem with the objective of minimizing the sum of completion times of all... more
The data migration problem is to compute an efficient plan for moving data stored on devices in a network from one configuration to another. We consider this problem with the objective of minimizing the sum of completion times of all storage devices. It is modeled by a transfer graph, where vertices represent the storage devices, and the edges indicate the data transfers required between pairs of devices. Each vertex has a nonnegative weight, and each edge has a release time and a processing time. A vertex completes when all the edges incident on it complete; the constraint is that two edges incident on the same vertex cannot be processed simultaneously. The objective is to minimize the sum of weighted completion times of all vertices. Kim ( Journal of Algorithms, 55:42--57, 2005 ) gave a 9-approximation algorithm for the problem when edges have arbitrary processing times and are released at time zero. We improve Kim's result by giving a 5.06-approximation algorithm. We also add...
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This paper studies two network design problems whose goals are to find a tree that minimizes the maximum degree. First is the Minimum Degree Group Steiner Tree problem (MD-GST), where we are given an n-vertex undirected graph G(V,E), and... more
This paper studies two network design problems whose goals are to find a tree that minimizes the maximum degree. First is the Minimum Degree Group Steiner Tree problem (MD-GST), where we are given an n-vertex undirected graph G(V,E), and a collection of p subsets of vertices called groups {gi}i∈[p], and the goal is find a tree that contains at least one vertex from every group gi while minimizing the maximum degree. Second is the Minimum Degree k-Steiner Tree problem (MDkT), where we are given an n-vertex undirected graph G(V,E), a set of p terminals and a number k < p, and the goal is to find a tree that spans at least k terminals while minimizing the maximum degree. We study these two problems when an input graph has bounded treewidth. We present an O(log n/ log log n) approximation algorithm for MDGST on bounded treewidth graphs and showed that the latter problem, MDkT, can be reduced to MD-GST via a blackbox reduction that loses only an extra O(logn) factor in the approximati...
We consider the non-uniform multicommodity buy-at-bulk network design problem. In this problem we are given a graph G(V;E) with two cost functions on the edges, a buy cost b : E ! R + and a rent costr : E ! R + , and a set of source-sink... more
We consider the non-uniform multicommodity buy-at-bulk network design problem. In this problem we are given a graph G(V;E) with two cost functions on the edges, a buy cost b : E ! R + and a rent costr : E ! R + , and a set of source-sink pairssi;ti2 V (1 i ) with each pair i having a positive demand i. Our goal is to design a minimum cost network G(V;E 0 ) such that for every 1 i , si and ti are in the same connected component in G(V;E 0 ). The total cost of G(V;E 0 ) is the sum of buy costs of the edges in E 0 plus sum of total demand going through every edge inE 0 times the rent cost of that edge. Since the costs of dieren t edges can be dieren t, we say that the problem is non-uniform. The rst non-trivial approximation algorithm for this problem is due to Charikar and Karagiozova (STOC’ 05) whose algorithm has an approximation guarantee of exp(O( p logn log logn)), when all i = 1 and exp(O( p logN log logN)) for the general demand case where N is the sum of all demands. We improv...
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Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph $G=(V, E)$ with edge costs $c \in \mathbb{R}_{\geq 0}^E$, a root $r \in V$ and $k$ terminals... more
Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph $G=(V, E)$ with edge costs $c \in \mathbb{R}_{\geq 0}^E$, a root $r \in V$ and $k$ terminals $K\subseteq V$, we need to output the minimum-cost arborescence in $G$ that contains an $r$\textrightarrow $t$ path for every $t \in K$. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasi-polynomial time $O(\log^2k/\log \log k)$-approximation algorithms for the problem, which are tight under popular complexity assumptions. In this paper, we consider the more general Degree-Bounded Directed Steiner Tree (DB-DST) problem, where we are additionally given a degree bound $d_v$ on each vertex $v \in V$, and we require that every vertex $v$ in the output tree has at most $d_v$ children. We give a quasi-polynomial time $(O(\log n \log k), O(\log^2 n))$-bicriteria approximation: The algorithm produces a solution with cost at mos...
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In the Steiner k -Forest problem, we are given an edge weighted graph, a collection D of node pairs, and an integer k ⩽ | D |. The goal is to find a min-weight subgraph that connects at least k pairs. The best known ratio for this problem... more
In the Steiner k -Forest problem, we are given an edge weighted graph, a collection D of node pairs, and an integer k ⩽ | D |. The goal is to find a min-weight subgraph that connects at least k pairs. The best known ratio for this problem is min { O (√ n ), O (√ k )} [Gupta et al. 2010]. In Gupta et al. [2010], it is also shown that ratio ρ for Steiner k -Forest implies ratio O (ρ · log 2 n ) for the related Dial-a-Ride problem. The only other algorithm known for Dial-a-Ride, besides the one resulting from Gupta et al. [2010], has ratio O (√ n ) [Charikar and Raghavachari 1998]. We obtain approximation ratio n 0.448 for Steiner k -Forest and Dial-a-Ride with unit weights, breaking the O (√ n ) approximation barrier for this natural case. We also show that if the maximum edge-weight is O ( n ϵ ), then one can achieve ratio O ( n (1 + ϵ) · 0.448 ), which is less than √ n if ϵ is small enough. The improvement for Dial-a-Ride is the first progress for this problem in 15 years. To prove ...
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ABSTRACT Given a graph with edge costs, the power of a node is the maximum cost of an edge leaving it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider several... more
ABSTRACT Given a graph with edge costs, the power of a node is the maximum cost of an edge leaving it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider several fundamental undirected network design problems under the power minimization criteria. The Minimum-Power Edge-Cover (MPEC) problem is: given a graph G=(V,E) with edge costs {c(e):e∈E} and a subset S⊆V of nodes, find a minimum-power subgraph H of G containing an edge incident to every node in S. We give a 3/2-approximation algorithm for MPEC, improving over the 2-approximation by [M.T. Hajiaghayi, G. Kortsarz, V.S. Mirrokni, Z. Nutov, Power optimization for connectivity problems, Mathematical Programming 110 (1) (2007) 195–208]. For the Min-Powerk-Connected Subgraph () problem we obtain the following results. For k=2 and k=3, we improve the previously best known ratios of 4 [G. Calinescu, P.J. Wan, Range assignment for biconnectivity and k-edge connectivity in wireless ad hoc networks, Mobile Networks and Applications 11 (2) (2006) 121–128] and 7 [M.T. Hajiaghayi, G. Kortsarz, V.S. Mirrokni, Z. Nutov, Power optimization for connectivity problems, Mathematical Programming 110 (1) (2007) 195–208] to and , respectively. Finally, we give a 4rmax-approximation algorithm for the Minimum-Power Steiner Network (MPSN) problem: find a minimum-power subgraph that contains r(u,v) pairwise edge-disjoint paths for every pair u,v of nodes.
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We present two new approximation algorithms for the problem of finding a k-node connected spanning subgraph (directed or undirected) of minimum cost. The best known approximation guarantees for this problem were O(min (k,n/√n-k)) for both... more
We present two new approximation algorithms for the problem of finding a k-node connected spanning subgraph (directed or undirected) of minimum cost. The best known approximation guarantees for this problem were O(min (k,n/√n-k)) for both directed and undirected graphs, and O(ln k) for undirected graphs with n ≥ 6k2, where n is the number of nodes in the input graph. Our first algorithm has approximation ratio O(k/n-kln 2 k, which is O(ln2 k) except for very large values of k, namely, k=n-o(n). This algorithm is based on a new result on l-connected p-critical graphs, which is of independent interest in the context of graph theory. Our second algorithm uses the primal-dual method and has approximation ratio O(√n ln k) for all values of n,k. Combining these two gives an algorithm with approximation ratio O(ln k • min (√k, k/n-k ln k)), which asymptotically improves the best known approximation guarantee for directed graphs for all values of n,k, and for undirected graphs for k&amp;gt; √n⁄6. Moreover, this is the first algorithm that has an approximation guarantee better than Θ(k) for all values of n,k. Our approximation ratio also provides an upper bound on the integrality gap of the standard LP-relaxation to the problem.As a byproduct, we also get the following result which is of independent interest. To get a faster implementation of our algorithms, we consider the problem of adding a minimum-cost edge set to increase the outconnectivity of a directed graph by Δ a graph is said to be l-outconnected from its node r if it contains l internally disjoint paths from r to any other node. The best known time complexity for the later problem is O(m3). For the particular case of Δ=1, we give a primal-dual algorithm with running time O(m2).
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In [1], we gave an algorithm for the data migration and non-deterministic open shop scheduling problems in the minimum sum version, that was claimed to achieve a 5.06-approximation. Unfortunately, it was pointed to us by Maxim Sviridenko... more
In [1], we gave an algorithm for the data migration and non-deterministic open shop scheduling problems in the minimum sum version, that was claimed to achieve a 5.06-approximation. Unfortunately, it was pointed to us by Maxim Sviridenko that the argument contained an unfounded assumption that has eluded all of its readers until now. We detail in this document how this error can be amended by modifying the analysis. A side effect of this is that the approximation ratio is improved to 4.96.
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Abstract: In the group Steiner problem we are given a graph with edge weights w(e) and m subsets ofvertices fg i gi=1 . Each subset g i is called a group and the vertices inSg i are called terminals. Itis required to find a minimum weight... more
Abstract: In the group Steiner problem we are given a graph with edge weights w(e) and m subsets ofvertices fg i gi=1 . Each subset g i is called a group and the vertices inSg i are called terminals. Itis required to find a minimum weight tree that contains at least one terminal from every group.
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... 4. KP Eswaran and RE Tarjan, Augmentation Problems, SIAM J. Computing, 5 (1976), 653665. ... 9. Kamal Jain, Factor 2 Approximation Algorithm for the Generalized Steiner Net-work Problem, FOCS 1998, 448-457. Page 12. A... more
... 4. KP Eswaran and RE Tarjan, Augmentation Problems, SIAM J. Computing, 5 (1976), 653665. ... 9. Kamal Jain, Factor 2 Approximation Algorithm for the Generalized Steiner Net-work Problem, FOCS 1998, 448-457. Page 12. A 3/2-Approximation Algorithm 101 10. ...
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ABSTRACT In the Steiner Network problem we are given a graph G with edge-costs and connectivity requirements r uv between node pairs u,v. The goal is to find a minimum-cost subgraph H of G that contains r uv edge-disjoint paths for all... more
ABSTRACT In the Steiner Network problem we are given a graph G with edge-costs and connectivity requirements r uv between node pairs u,v. The goal is to find a minimum-cost subgraph H of G that contains r uv edge-disjoint paths for all u,v ∈ V. In Prize-Collecting Steiner Network problems we do not need to satisfy all requirements, but are given a penalty function for violating the connectivity requirements, and the goal is to find a subgraph H that minimizes the cost plus the penalty. The case when r uv ∈ {0,1} is the classic Prize-Collecting Steiner Forest problem. In this paper we present a novel linear programming relaxation for the Prize-Collecting Steiner Network problem, and by rounding it, obtain the first constant-factor approximation algorithm for submodular and monotone non-decreasing penalty functions. In particular, our setting includes all-or-nothing penalty functions, which charge the penalty even if the connectivity requirement is slightly violated; this resolves an open question posed in [SSW07]. We further generalize our results for element-connectivity and node-connectivity.
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In this paper, we consider the following red-blue median problem which is a generalization of the well-studied k-median problem. The input consists of a set of red facilities, a set of blue facilities, and a set of clients in a metric... more
In this paper, we consider the following red-blue median problem which is a generalization of the well-studied k-median problem. The input consists of a set of red facilities, a set of blue facilities, and a set of clients in a metric space and two integers k r ,k b ≥0. The problem is to open at most k r red facilities and at most k b blue facilities and minimize the sum of distances of clients to their respective closest open facilities. We show, somewhat surprisingly, that the following simple local search algorithm yields a constant factor approximation for this problem. Start by opening any k r red and k b blue facilities. While possible, decrease the cost of the solution by closing a pair of red and blue facilities and opening a pair of red and blue facilities. We also improve the approximation factor for the prize-collecting k-median problem from 4 (Charikar et al. in Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pp. 642–641, 2001) to 3+ϵ, which matches the current best approximation factor for the k-median problem.
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We present algorithms with poly-logarithmic approximation ratios for the buy-at-bulk network design problem in the node-weighted setting. We obtain the following results where h is the number of pairs in the input. • An O(logh)... more
We present algorithms with poly-logarithmic approximation ratios for the buy-at-bulk network design problem in the node-weighted setting. We obtain the following results where h is the number of pairs in the input. • An O(logh) approximation for the single-sink non- uniform buy-at-bulk network design. Unless P = NP this ratio is tight up to constant factors. • An O(log4 h)
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In the k-center problem, the input is a bound k and n points with the distance between every two of them, such that the distances obey the triangle inequality. The goal is to choose a set of k points to serve as centers, so that the... more
In the k-center problem, the input is a bound k and n points with the distance between every two of them, such that the distances obey the triangle inequality. The goal is to choose a set of k points to serve as centers, so that the maximum distance from the centers C to any point is as small as possible. This fundamental facility location problem is NP-hard. The symmetric case is well-understood from the viewpoint of approximation; it admits a 2{approximation, but not better. We address the approximability of the asymmetric k-center problem. Our rst result shows that the linear program used by Archer (Arc01) to devise O(log k){approximation has integrality ratio that is at least (1 o(1)) log n; this improves on the previous bound 3 of (Arc01). Using a similar construction, we then prove that the problem cannot be approximated within a ratio of 1 4 log n, unless NP DTIME(nlog log log n). These are the rst lower bounds for this problem that are tight, up to constant factors, with the O(log n){approximation due to (PV98, Arc01).