Degree Bounded Network Design with Metric Costs

Finding low-cost spanning subgraphs with given degree and connectivity requirements is a fundamental problem in the area of network design. We consider the problem of finding d-regular spanning subgraphs (or d-factors) of minimum weight with connectivity requirements. For the case of k-edge-connectedness, we present approximation algorithms that achieve constant approximation ratios for all d ≥ 2 • k/2 . For the case of k-vertex-connectedness, we achieve constant approximation ratios for d ≥ 2k − 1. Our algorithms also work for arbitrary degree sequences if the minimum degree is at least 2 • k/2 (for k-edgeconnectivity) or 2k − 1 (for k-vertex-connectivity). Problem Definitions and PreliminariesGraphs and Connectivity. All graphs in this paper are undirected and simple. Let G = (V, E) a graph. In the following, n = |V | is the number of vertices.

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“…Also Min-kReg-kVertex and Min-kReg-kEdge admit constant factor approximations for all k ≥ 1 [1]. We refer to Tables 1 and 2 for an overview.…”

Section: Previous and Related Resultsmentioning

confidence: 99%

“…Most variants of such problems are NP-hard. Because of this, finding good approximation algorithms for such network design problems has been the topic of a significant amount of research [1,[4][5][6][7][8][9][10]14,[16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Approximation Algorithms for k-Connected Graph Factors

Manthey

Waanders

2015

Approximation and Online Algorithms

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“…Finally, select the E spl i−1 valid candidate set for the splitting off whose weighted sum of the above metric is maximal. The weights depend on heuristic design principles such as avoiding parallel edges [18] or introducing loop edges, discussed in Section V.…”

Section: Heuristic Approach To Select Edge Pairs For Splitting-offmentioning

confidence: 99%

“…our framework is integrating Integer Linear Program (ILP) formulations with a graph theoretical approach namely edge splitting-off -which can be used to prove various properties of graphs with given global [15] and local [16] connectivity, often called k-connected graph characterization [17] -for achieving the desired efficiency and flexibility. Note that edge splittingoff was already successfully applied for different network design problems [18], [19], including Edmonds' arborescence construction [20], [21].…”

Section: Introductionmentioning

confidence: 99%

Resilient Routing Table Computation Based on Connectivity Preserving Graph Sequences

Tapolcai,

Babarczi,

et al. 2023

IEEE INFOCOM 2023 - IEEE Conference on Computer Communications

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Fast reroute (FRR) mechanisms that can instantly handle network failures in the data plane are gaining attention in packet-switched networks. In FRR no notification messages are required as the nodes adjacent to the failure are prepared with a routing table such that the packets are re-routed only based on local information. However, designing the routing algorithm for FRR is challenging because the number of possible sets of failed network links and nodes can be extremely high, while the algorithm should keep track of which nodes are aware of the failure. In this paper, we propose a generic algorithmic framework that combines the benefits of Integer Linear Programming (ILP) and an effective approach from graph theory related to constructive graph characterization of k-connected graphs, i.e., edge splittingoff. We illustrate these benefits through arborescence design for FRR and show that (i) due to the ILP we have great flexibility in defining the routing problem, while (ii) the problem can still be solved very fast. We demonstrate through simulations that our framework outperforms state-of-the-art FRR mechanisms and provides better resilience with shorter paths in the arborescences.

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“…It was the converse that attracted us. Chan et al [2] give a very closely related argument, presented very efficiently. Our theorem is used significantly in the next section, so we include our slightly modified version of their proof.…”

Section: Introductionmentioning

confidence: 99%

Liftings in Finite Graphs and Linkages in Infinite Graphs with Prescribed Edge-Connectivity

Richter

Thomassen

2016

Graphs and Combinatorics

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Let G be a graph and let s be a vertex of G. We consider the structure of the set of all lifts of two edges incident with s that preserve edge-connectivity. Mader proved that two mild hypotheses imply there is at least one pair that lifts, while Frank showed (with the same hypotheses) that there are at least (deg(s) − 1)/2 disjoint pairs that lift. We consider the lifting graph: its vertices are the edges incident with s, two being adjacent if they form a liftable pair. We have three main results, the first two with the same hypotheses as for Mader's Theorem. (i) Let F be a subset of the edges incident with s. We show that F is independent in the lifting graph of G if and only if there is a single edge-cut C in G of size at most r + 1 containing all the edges in F , where r is the maximum number of edge-disjoint paths from a vertex (not s) in one component of G − C to a vertex (not s) in another component of G − C.(ii) In the k-lifting graph, two edges incident with s are adjacent if their lifting leaves the resulting graph with the property that any two vertices different from s are joined by k pairwise edge-disjoint paths. If both deg(s) and k are even, then the k-lifting graph is a connected complete multipartite graph. In all other cases, there are at most two components. If there are exactly two components, then each component is a complete multipartite graph. If deg(s) is odd and there are two components, then one component is a single vertex. (iii) Huck proved that if k is odd and G is (k + 1)-edge-connected, then G is weakly k-linked (that is, for any k pairs {x i , y i }, there are k edge-disjoint paths P i , with P i joining x i and y i ). We use our results to extend a slight weakening of Huck's theorem to some infinite graphs: if k is odd, every (k + 2)-edge-connected, locally finite, 1-ended, infinite graph is weakly k-linked.

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Degree Bounded Network Design with Metric Costs

Cited by 8 publications

References 34 publications

Approximation Algorithms for k-Connected Graph Factors

Approximation Algorithms for k-Connected Graph Factors

Resilient Routing Table Computation Based on Connectivity Preserving Graph Sequences

Liftings in Finite Graphs and Linkages in Infinite Graphs with Prescribed Edge-Connectivity

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