Preserving and Increasing Local Edge-Connectivity in Mixed Graphs

Bang‐Jensen, Jørgen; Frank, András; Jackson, Bill

doi:10.1137/s0036142993226983

Cited by 59 publications

(74 citation statements)

References 20 publications

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“…We can build a tree T of cost O(L) such that the expected profit of the contracted nodes connected by T is in fact at least c∈C θ c · z * c which is in Ω(opt/ log n). This can be done by using results of Bang-Jensen et al [BJFJ95], which gives another way of obtaining a 2-approximation algorithm for Prize Collecting Steiner Tree (PCST). The result of Bang-Jensen et al [BJFJ95] implies that from x * we can build a convex combination of trees such that each client c appears in z * c fractional number of the trees.…”

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confidence: 99%

Approximation Algorithms for Movement Repairmen

Hajiaghayi

Khandekar

Khani

et al. 2013

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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Abstract. In the Movement Repairmen (MR) problem we are given a metric space (V, d) along with a set R of k repairmen r1, r2, . . . , r k with their start depots s1, s2, . . . , s k ∈ V and speeds v1, v2, . . . , v k ≥ 0 respectively and a set C of m clients c1, c2, . . . , cm having start locations s 1 , s 2 , . . . , s m ∈ V and speeds v 1 , v 2 , . . . , v m ≥ 0 respectively. If t is the earliest time a client cj is collocated with any repairman (say, ri) at a node u, we say that the client is served by ri at u and that its latency is t. The objective in the (Sum-MR) problem is to plan the movements for all repairmen and clients to minimize the sum (average) of the clients latencies. The motivation for this problem comes, for example, from Amazon Locker Delivery [Ama10] and USPS gopost [Ser10]. We give the first O(log n)-approximation algorithm for the Sum-MR problem. In order to solve Sum-MR we formulate an LP for the problem and bound its integrality gap. Our LP has exponentially many variables, therefore we need a separation oracle for the dual LP. This separation oracle is an instance of Neighborhood Prize Collecting Steiner Tree (NPCST) problem in which we want to find a tree with weight at most L collecting the maximum profit from the clients by visiting at least one node from their neighborhoods. The NPCST problem, even with the possibility to violate both the tree weight and neighborhood radii, is still very hard to approximate. We deal with this difficulty by using LP with geometrically increasing segments of the time line, and by giving a tricriteria approximation for the problem. The rounding needs a relatively involved analysis. We give a constant approximation algorithm for Sum-MR in Euclidean Space where the speed of the clients differ by a constant factor. We also give a constant approximation for the makespan variant.

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confidence: 99%

Approximation Algorithms for Movement Repairmen

Hajiaghayi

Khandekar

Khani

et al. 2013

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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show abstract

“…It is known that for a skew-supermodular function p this is an (integer) contrapolymatroid (for details see [1]). In order to turn our proof into polynomial algorithms we need to assume that we can test membership in polynomial time in C(p − d G ) for any graph G: though this is not necessarily true in general (as an example, let p(X 0 ) = 0 for a fixed X 0 and −inf ty otherwise), but it will always hold in the applications given below.…”

Section: Preliminariesmentioning

confidence: 99%

“…We note that, by the properties of a contrapolymatroid, a polynomial algorithm to the degree specified covering problem will give rise to a solution to the minimum version of the problem, and to more general versions such as the minimum node-cost problem. For more details we refer to [1]. Define the greedy bound by gb(p) = max{ t X∈X p(X) : X is a subpartition of V } = min{1 · x : x ∈ C(p)}: this is obviously a lower bound for the minimum total size of any hypergraph covering p.…”

Section: Preliminariesmentioning

confidence: 99%

“…First we consider a special case of the theorem of Benczúr and Frank [2], that already includes the classical splitting lemma of Lovász. Then we give a simple proof for the classical splitting theorem of Mader [11] (used by Frank in [7]) and the undirected splitting theorem in mixed graphs used by Bang-Jensen, Frank and Jackson in [1].…”

Section: Introductionmentioning

confidence: 99%

“…This problem is a natural generalization of many connectivity augmentation problems. Examples include the local edge-connectivity augmentation problem in graphs solved by Frank [7], the global arc-connectivity augmentation problem in mixed graphs solved by Bang-Jensen, Frank and Jackson [1], and the node-to-area connectivity augmentation problem solved by Ishii and Hagiwara [10]. The general problem of covering a symmetric skew-supermodular set function p : 2 V → Z ∪ {−∞} with a minimum number of graph edges is known to be NPcomplete (see e.g.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A New Approach to Splitting-Off

Bernáth

Király

Integer Programming and Combinatorial Optimization

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A new approach to undirected splitting-off is presented in this paper. We study the behaviour of splitting-off algorithms when applied to the problem of covering a symmetric skew-supermodular set function by a graph. This hard problem is a natural generalization of many solved connectivity augmentation problems, such as local edge-connectivity augmentation of graphs, global arcconnectivity augmentation of mixed graphs with undirected edges, or the nodeto-area connectivity augmentation problem in graphs. Using a simple lemma we characterize the situation when a splitting-off algorithm can be stuck. This characterization enables us to give very simple proofs for the classical results mentioned above. Finally we apply our observations in generalizations of the above problems: we consider three connectivity augmentation problems in hypergraphs with hyperedges of minimum total size without increasing the rank. The first is local edge-connectivity augmentation of undirected hypergraphs. The second is global arc-connectivity augmentation of mixed hypergraphs with undirected hyperedges. The third is a hypergraphic generalization of the nodeto-area connectivity augmentation problem. We show that a greedy approach (almost) works for these cases. IntroductionLet us be given a finite ground set V . A set function p : 2 V → Z ∪ {−∞} is called skew-supermodular if at least one of the following two inequalities holds for every X, Y ⊆ V :An extended abstract version of this paper has appeared in the proceedings of IPCO 2008, pages 401-415

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A note on mixed graphs and directed splitting off

Enni¹

1998

J. Graph Theory

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We give counterexamples to two conjectures of Bill Jackson in Some remarks on arc‐connectivity, vertex splitting, and orientation in graphs and digraphs (Journal of Graph Theory 12(3):429–436, 1988) concerning orientations of mixed graphs and splitting off in digraphs, and prove the first conjecture in the (di‐) Eulerian case(s). Beside that we solve a degree constrained non‐uniform directed augmentation problem for di‐Eulerian mixed graphs. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 213–221, 1998

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Preserving and Increasing Local Edge-Connectivity in Mixed Graphs

Cited by 59 publications

References 20 publications

Approximation Algorithms for Movement Repairmen

Approximation Algorithms for Movement Repairmen

A New Approach to Splitting-Off

A note on mixed graphs and directed splitting off

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