Minimum Weight 2-Edge-Connected Spanning Subgraphs in Planar Graphs

Berger, André; Grigni, Michelangelo

doi:10.1007/978-3-540-73420-8_10

Cited by 14 publications

(24 citation statements)

References 7 publications

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“…The general approach used for TSP has proved useful in obtaining approximation schemes for other problems in planar graphs, including minimum-weight two-edge-connected spanning multi-subgraph, 4 TSP on a subset of the nodes [31], minimum-weight two-edge-connected spanning subgraph [8], and Steiner tree [10]. As mentioned above, the basic technique has been generalized [19] to apply to bounded-genus graphs, giving rise to new approximation schemes for such graphs.…”

Section: Combining Stepmentioning

confidence: 99%

“…Thus the dual of a planar embedded graph is a planar embedded graph. 8 Let T be a spanning tree of G. For an edge e ∈ T , there is a unique simple cycle consisting of e and the unique path in T between the endpoints of e. This cycle is called the elementary cycle of e with respect to T in G. 6 For the purpose of the current result, all we need is that every graph embeddable on an orientable surface of genus zero has a combinatorial embedding that satisfies Euler's formula. However, it is known (see, .e.g, [35]) more generally that for any graph embedded on a closed, orientable surface, the corresponding combinatorial embedding determines the geometric embedding up to homeomorphism.…”

Section: Dualitymentioning

confidence: 99%

“…See [12] for a discussion of later work. 8 For disconnected graphs, this definition of dual diverges from the geometric definition in that it assigns multiple dual nodes to a single region of the sphere/plane. According to the geometric definition, the dual of a graph is always connected.…”

Section: Dualitymentioning

confidence: 99%

See 2 more Smart Citations

A Linear-Time Approximation Scheme for TSP in Undirected Planar Graphs with Edge-Weights

Klein¹

2008

SIAM J. Comput.

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Section: Combining Stepmentioning

confidence: 99%

Section: Dualitymentioning

confidence: 99%

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A Linear-Time Approximation Scheme for TSP in Undirected Planar Graphs with Edge-Weights

Klein¹

2008

SIAM J. Comput.

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“…This version of the problem, like the other variants, is SNP-hard in general graphs [6]. In [3], Berger and Grigni gave a polynomial-time approximation scheme (PTAS) for {1, 2}-edge connectivity (ie. the spanning case) in planar multigraphs.…”

Section: Introductionmentioning

confidence: 99%

“…the subset case) for planar multigraphs. The running time is significantly lower than that of [3]. In the following, OPT denotes the weight of the optimal solution to the problem at hand.…”

Section: Introductionmentioning

confidence: 99%

The Two-Edge Connectivity Survivable Network Problem in Planar Graphs

Borradaile

Klein

2008

Automata, Languages and Programming

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Abstract. Consider the following problem: given a graph with edgeweights and a subset Q of vertices, find a minimum-weight subgraph in which there are two edge-disjoint paths connecting every pair of vertices in Q. The problem is a failure-resilient analog of the Steiner tree problem, and arises in telecommunications applications. A more general formulation, also employed in telecommunications optimization, assigns a number (or requirement) rv ∈ {0, 1, 2} to each vertex v in the graph; for each pair u, v of vertices, the solution network is required to contain min{ru, rv} edge-disjoint u-to-v paths.We address the problem in planar graphs, considering a popular relaxation in which the solution is allowed to use multiple copies of the input-graph edges (paying separately for each copy). The problem is SNP-hard in general graphs and NP-hard in planar graphs. We give the first polynomial-time approximation scheme in planar graphs. The running time is O(n log n).Under the additional restriction that the requirements are in {0, 2} for vertices on the boundary of a single face of a planar graph, we give a linear-time algorithm to find the optimal solution.

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Light Spanners in Bounded Pathwidth Graphs

Grigni¹,

Hung²

2012

Mathematical Foundations of Computer Science 2012

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Abstract. Given an edge-weighted graph G and > 0, a (1+ )-spanner is a spanning subgraph G whose shortest path distances approximate those of G within a factor of 1+ . For G from certain graph families (such as bounded genus graphs and apex graphs), we know that light spanners exist. That is, we can compute a (1+ )-spanner G with total edge weight at most a constant times the weight of a minimum spanning tree. This constant may depend on and the graph family, but not on the particular graph G nor on the edge weighting. The existence of light spanners is essential in the design of approximation schemes for the metric TSP (the traveling salesman problem) and similar graph-metric problems.In this paper we make some progress towards the conjecture that light spanners exist for every minor-closed graph family: we show that light spanners exist for graphs with bounded pathwidth, and they are computed by a greedy algorithm. We do this via the intermediate construction of light monotone spanning trees in such graphs.

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Minimum Weight 2-Edge-Connected Spanning Subgraphs in Planar Graphs

Cited by 14 publications

References 7 publications

A Linear-Time Approximation Scheme for TSP in Undirected Planar Graphs with Edge-Weights

A Linear-Time Approximation Scheme for TSP in Undirected Planar Graphs with Edge-Weights

The Two-Edge Connectivity Survivable Network Problem in Planar Graphs

Light Spanners in Bounded Pathwidth Graphs

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