On the restricted k-Steiner tree problem

Bose, Prosenjit; D’Angelo, Anthony; Durocher, Stéphane

doi:10.1007/s10878-021-00808-z

Cited by 5 publications

(5 citation statements)

References 37 publications

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“…Our result shows that Connectivity for line segments is fixed-parameter tractable when parameterized by the number k of segments. The running time of our algorithms are a large improvement over the best previous time bound of by Chambers et al [ 9 ] and the bound of that could be obtained adapting the approach of Bose et al [ 6 ].…”

Section: Introductionmentioning

confidence: 66%

“…Closer to our setting is the work of Bose et al [ 6 ], who considered the version of the k -Steiner tree problem where the points have to lie on given lines, and showed how to solve it in time. Like the work of Brazil et al [ 7 ], their technique can be adapted to the problem Connectivity with uncertain segment regions.…”

Section: Introductionmentioning

confidence: 99%

“…Then they show that there are critical paths. Compared to the work of Bose et al [ 6 ] and Brazil et al [ 7 ], we first note that they are minimizing the sum of the length of the edges. Optimizing the sum is usually harder than optimizing the bottleneck value.…”

Section: Introductionmentioning

confidence: 99%

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Connectivity with Uncertainty Regions Given as Line Segments

Cabello,

Gajser

2024

Algorithmica

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For a set $${\mathcal {Q}}$$ Q of points in the plane and a real number $$\delta \ge 0$$ δ ≥ 0 , let $${\mathbb {G}}_\delta ({\mathcal {Q}})$$ G δ ( Q ) be the graph defined on $${\mathcal {Q}}$$ Q by connecting each pair of points at distance at most $$\delta $$ δ .We consider the connectivity of $${\mathbb {G}}_\delta ({\mathcal {Q}})$$ G δ ( Q ) in the best scenario when the location of a few of the points is uncertain, but we know for each uncertain point a line segment that contains it. More precisely, we consider the following optimization problem: given a set $${\mathcal {P}}$$ P of $$n-k$$ n - k points in the plane and a set $${\mathcal {S}}$$ S of k line segments in the plane, find the minimum $$\delta \ge 0$$ δ ≥ 0 with the property that we can select one point $$p_s\in s$$ p s ∈ s for each segment $$s\in {\mathcal {S}}$$ s ∈ S and the corresponding graph $${\mathbb {G}}_\delta ( {\mathcal {P}}\cup \{ p_s\mid s\in {\mathcal {S}}\})$$ G δ ( P ∪ { p s ∣ s ∈ S } ) is connected. It is known that the problem is NP-hard. We provide an algorithm to exactly compute an optimal solution in $${{\,\mathrm{{\mathcal {O}}}\,}}(f(k) n \log n)$$ O ( f ( k ) n log n ) time, for a computable function $$f(\cdot )$$ f ( · ) . This implies that the problem is FPT when parameterized by k. The best previous algorithm uses $${{\,\mathrm{{\mathcal {O}}}\,}}((k!)^k k^{k+1}\cdot n^{2k})$$ O ( ( k ! ) k k k + 1 · n 2 k ) time and computes the solution up to fixed precision.

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Section: Introductionmentioning

confidence: 66%

Section: Introductionmentioning

confidence: 99%

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Connectivity with Uncertainty Regions Given as Line Segments

Cabello,

Gajser

2024

Algorithmica

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“…A similar result was shown in Monma and Suri [34, Lemma 4.1, pg. 277] 6. A simple cycle is a cycle where no vertex is repeated except the first vertex.…”

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confidence: 99%

On the Restricted 1-Steiner Tree Problem

Bose¹,

D’Angelo²,

Durocher³

2020

Lecture Notes in Computer Science

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Given a set P of n points in R 2 and an input line γ in R 2 , we present an algorithm that runs in optimal Θ(n log n) time and Θ(n) space to solve a restricted version of the 1-Steiner tree problem. Our algorithm returns a minimum-weight tree interconnecting P using at most one Steiner point s ∈ γ, where edges are weighted by the Euclidean distance between their endpoints. We then extend the result to j input lines. Following this, we show how the algorithm of Brazil et al.[10] that solves the k-Steiner tree problem in R 2 in O(n 2k ) time can be adapted to our setting. For k > 1, restricting the (at most) k Steiner points to lie on an input line, the runtime becomes O(n k ). Next we show how the results of Brazil et al. [10] allow us to maintain the same time and space bounds while extending to some non-Euclidean norms and different tree cost functions. Lastly, we extend the result to j input curves.

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“…Details appear in Chapter 2. A preliminary version of this result appeared in COCOON 2020 under the title "On the restricted 1-Steiner tree problem" [21], and the full version of this result appeared in the Journal of Combinatorial Optimization in 2021 under the title "On the restricted k-Steiner tree problem" [22].…”

Section: -Steiner Tree Problem Restricted To a Linementioning

confidence: 99%

Constrained Geometric Optimization Problems

D'Angelo¹

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On the restricted k-Steiner tree problem

Cited by 5 publications

References 37 publications

Connectivity with Uncertainty Regions Given as Line Segments

Connectivity with Uncertainty Regions Given as Line Segments

On the Restricted 1-Steiner Tree Problem

Constrained Geometric Optimization Problems

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