Minimum weight connectivity augmentation for planar straight-line graphs

Akitaya, Hugo A.; Inkulu, R.; Nichols, Torrie L.; Souvaine, Diane L.; Tóth, Csaba D.; Winston, Charles R.

doi:10.1016/j.tcs.2018.05.031

Cited by 4 publications

(15 citation statements)

References 19 publications

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“…Similarities can be drawn to 2‐edge augmentation for planar graphs with a straight‐line embedding. This problem was recently shown to have a polynomial time algorithm which uses the key property that augmentation can be performed on each face of the graph independently of the other faces [2]. It is possible that both the LSDAP and planar augmentation for more complex disasters have the same property, which could lead to polynomial time exact algorithms.…”

Section: Discussionmentioning

confidence: 99%

“…This plane graph has two types of faces. Every face f is either a subset of   0 for some l-cut  0 (for instance, the face with boundary (12, 7, 4) is a subset of  E 1 in Figure 12), or f is not a subset of any disaster region   0 (for instance, the face with boundary (2,16,15,4,3) in Figure 12). Since p is required to not intersect the interior of the destruction zones of the l-cuts in , only the second kind of face is considered as a potential face within which to construct p. Also, since G ∪ {p} must be non-intersecting, both end-nodes of p must occur on the boundary of a single face of A.…”

Section: Algorithm 4 Findshortestchainmentioning

confidence: 99%

“…A recent breakthrough in this area has resulted in an approximation algorithm with ratio below 2 (see [6]), which was a long sought‐after result. Another key recent result is the development of a polynomial time algorithm for augmenting from 1‐connectivity to 2‐connectivity in the case where the input graph is connected and planar with a straight‐line embedding [2].…”

Section: Introductionmentioning

confidence: 99%

“…FIGURE2 The figure shows a geometric network G on the left as well as a disaster D of length l. The graph on the right shows the augmented graph G * , an l-resilient network. This network is constructed by adding the edge set E * = {(1, 5), (2, 3)}, where (1, 5) is a polygonal chain with 2 segments, and (2, 3) is a polygonal chain with one segment.…”

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Network augmentation for disaster‐resilience against geographically correlated failure

et al. 2022

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We introduce a formal framework for the study of augmenting networks in the plane for disaster‐resilience, where a disaster is modeled by a straight‐line segment. We generalize various graph structures from classical 2‐edge‐connectivity, including minimal cuts and blocks. The key concept that we introduce is that of an l$$ l $$‐leaf, which builds on the fundamental “leaf‐block” concept from classical augmentation. We present a number of algorithms for constructing the above‐mentioned graph structures, including a sweep‐line algorithm that finds all edge‐cuts that can be destroyed by a single disaster. We also present an algorithm which optimally adds a single edge between a pair of l$$ l $$‐leaves or blocks while avoiding certain disaster regions. Finally, we present a number of heuristic schemes for solving the disaster‐resilient network augmentation problem and perform extensive experiments to demonstrate the power of the l$$ l $$‐leaf concept within heuristic design.

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Section: Discussionmentioning

confidence: 99%

Section: Algorithm 4 Findshortestchainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Network augmentation for disaster‐resilience against geographically correlated failure

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“…For plane geometric graphs, these problems remain hard even if restricted to trees [28]. The reader is referred to [1,6,7,20,25,28,29] for different results about connectivity augmentation problems in plane geometric graphs, and the survey [21] for more details and related topics. Recent research on compatible plane graphs can be found in [2-4, 19, 22].…”

Section: Related Previous Workmentioning

confidence: 99%

Plane augmentation of plane graphs to meet parity constraints

Catana

García

Tejel

et al. 2020

Applied Mathematics and Computation

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A plane topological graph G = (V, E) is a graph drawn in the plane whose vertices are points in the plane and whose edges are simple curves that do not intersect, except at their endpoints. Given a plane topological graph G = (V, E) and a set CG of parity constraints, in which every vertex has assigned a parity constraint on its degree, either even or odd, we say that G is topologically augmentable to meet CG if there exits a plane topological graph H on the same set of vertices, such that G and H are edge-disjoint and their union is a plane topological graph that meets all parity constraints. In this paper, we prove that the problem of deciding if a plane topological graph is topologically augmentable to meet parity constraints is N P-complete, even if the set of vertices that must change their parities is V or the set of vertices with odd degree. In particular, deciding if a plane topological graph can be augmented to a Eulerian plane topological graph is N P-complete. Analogous complexity results are obtained, when the augmentation must be done by a plane topological perfect matching between the vertices not meeting their parities. We extend these hardness results to planar graphs, when the augmented graph must be planar, and to plane geometric graphs (plane topological graphs whose edges are straight-line segments). In addition, when it is required that the augmentation is made by a plane geometric perfect matching between the vertices not meeting their parities, we also prove that this augmentation problem is N P-complete for plane geometric trees and paths. For the particular family of maximal outerplane graphs, we characterize maximal outerplane graphs that are topological augmentable to satisfy a set of parity constraints. We also provide a polynomial time algorithm that decides if a maximal outerplane graph is topologically augmentable to meet parity constraints, and if so, produces a set of edges with minimum cardinality.

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The complexity landscape of disaster‐aware network extension problems

Vass,

Nagy,

Brányi

et al. 2023

Networks

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This article deals with the complexity of problems related to finding cost‐efficient, disaster‐aware cable routes. We overview various mathematical problems studied to augment a backbone network topology to make it more robust against regional failures. These problems either consider adding a single cable, multiple cables, or even nodes too. They adapt simplistic or more sophisticated regional failure models. Their objective is to identify the network's weak points or minimize the investment cost concerning the risk of a network outage. We investigate the tradeoffs in mathematical modeling for the same real‐world scenario, where more sophisticated models face more computationally challenging problems. We have seen how efficiently computational geometry algorithms can be used to solve simplified problems even for sufficiently large networks. In this article, we aim to understand why different mathematical models formulated for the same real‐world scenario can or cannot be solved efficiently. In particular, we show simplistic mathematical models that formulate NP‐hard problems.

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Minimum weight connectivity augmentation for planar straight-line graphs

Cited by 4 publications

References 19 publications

Network augmentation for disaster‐resilience against geographically correlated failure

Network augmentation for disaster‐resilience against geographically correlated failure

Plane augmentation of plane graphs to meet parity constraints

The complexity landscape of disaster‐aware network extension problems

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