The Rectilinear Steiner Tree Problem is $NP$-Complete

Garey, M. R.; Johnson, David S.

doi:10.1137/0132071

Cited by 962 publications

(409 citation statements)

References 12 publications

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“…Given a graph G = (V, E) and a non-negative integer k, the Connected Vertex Cover problem asks for a set C of at most k vertices such that G[C] is connected and C is a vertex cover of G. This problem is NPcomplete on planar graphs [13]. Until now only an exponential-size kernel in general graphs is known [16].…”

Section: Case Studiesmentioning

confidence: 99%

Linear Problem Kernels for NP-Hard Problems on Planar Graphs

Guo

Niedermeier

Automata, Languages and Programming

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Abstract. We develop a generic framework for deriving linear-size problem kernels for NP-hard problems on planar graphs. We demonstrate the usefulness of our framework in several concrete case studies, giving new kernelization results for Connected Vertex Cover, Minimum Edge Dominating Set, Maximum Triangle Packing, and Efficient Dominating Set on planar graphs. On the route to these results, we present effective, problem-specific data reduction rules that are useful in any approach attacking the computational intractability of these problems.

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Section: Case Studiesmentioning

confidence: 99%

Linear Problem Kernels for NP-Hard Problems on Planar Graphs

Guo

Niedermeier

Automata, Languages and Programming

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“…We apply Reduction 25 to Vertex Cover for planar cubic graphs, which is known to be NPcomplete [1,32]. 4 Given a planar cubic graph G, it is easy to check that G is planar, has maximum degree 3, and is bipartite, since the edge gadget for edge e = {x, y} is bipartite, with x, y in the same part.…”

Section: Reductions From Min Vertex Covermentioning

confidence: 99%

Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

Foucaud

2015

Journal of Discrete Algorithms

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An identifying code is a subset of vertices of a graph with the property that each vertex is uniquely determined (identified) by its nonempty neighbourhood within the identifying code. When only vertices out of the code are asked to be identified, we get the related concept of a locating-dominating set. These notions are closely related to a number of similar and well-studied concepts such as the one of a test cover. In this paper, we study the decision problems Identifying Code and Locating-Dominating Set (which consist in deciding whether a given graph admits an identifying code or a locating-dominating set, respectively, with a given size) and their minimization variants Minimum Identifying Code and Minimum LocatingDominating Set. These problems are known to be NP-hard, even when the input graph belongs to a number of specific graph classes such as planar bipartite graphs. Moreover, it is known that they are approximable within a logarithmic factor, but hard to approximate within any sub-logarithmic factor. We extend the latter result to the case where the input graph is bipartite, split or co-bipartite: both problems remain hard in these cases. Among other results, we also show that for bipartite graphs of bounded maximum degree (at least 3), the two problems are hard to approximate within some constant factor, a question which was open. We summarize all known results in the area, and we compare them to the ones for the related problem Dominating Set. In particular, our work exhibits important graph classes for which Dominating Set is efficiently solvable, but Identifying Code and Locating-Dominating Set are hard (whereas in all previous works, their complexity was the same). We also introduce graph classes for which the converse holds, and for which the complexities of Identifying Code and Locating-Dominating Set differ.

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“…The reduction is from minimum vertex cover for planar graphs with maximum degree three [8]. Let G = (V, E) be an instance of such graphs.…”

Section: Lemmamentioning

confidence: 99%

Closest Pair and the Post Office Problem for Stochastic Points

Kamousi

Chan

Suri

2011

Lecture Notes in Computer Science

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Abstract. Given a (master) set M of n points in d-dimensional Euclidean space, consider drawing a random subset that includes each point mi ∈ M with an independent probability pi. How difficult is it to compute elementary statistics about the closest pair of points in such a subset? For instance, what is the probability that the distance between the closest pair of points in the random subset is no more than , for a given value ? Or, can we preprocess the master set M such that given a query point q, we can efficiently estimate the expected distance from q to its nearest neighbor in the random subset? We obtain hardness results and approximation algorithms for stochastic problems of this kind.

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The Rectilinear Steiner Tree Problem is $NP$-Complete

Cited by 962 publications

References 12 publications

Linear Problem Kernels for NP-Hard Problems on Planar Graphs

Linear Problem Kernels for NP-Hard Problems on Planar Graphs

Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

Closest Pair and the Post Office Problem for Stochastic Points

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