Nearly ETH-tight Algorithms for Planar Steiner Tree with Terminals on Few Faces

Kisfaludi-Bak, Sándor; Nederlof, Jesper; Leeuwen, Erik Jan van

doi:10.1145/3371389

Cited by 6 publications

(5 citation statements)

References 39 publications

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“…4.2. If the conditions (24) are satisfied for a vertex v ∈ V \T , one can pseudo-eliminate [12] or replace [31] vertex v, i.e., delete v and connect any two vertices u, w ∈ N (v) by a new edge {u, w} of weight c({v, u}) + c({v, w}). The SPG depicted in Fig.…”

Section: C(e(s )) ≤ C(e(s))mentioning

confidence: 99%

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Implications, conflicts, and reductions for Steiner trees

Rehfeldt

Koch

2021

Math. Program.

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The Steiner tree problem in graphs (SPG) is one of the most studied problems in combinatorial optimization. In the past 10 years, there have been significant advances concerning approximation and complexity of the SPG. However, the state of the art in (practical) exact solution of the SPG has remained largely unchallenged for almost 20 years. While the DIMACS Challenge 2014 and the PACE Challenge 2018 brought renewed interest into Steiner tree problems, even the best new SPG solvers cannot match the state of the art on the vast majority of benchmark instances. The following article seeks to advance exact SPG solution once again. The article is based on a combination of three concepts: Implications, conflicts, and reductions. As a result, various new SPG techniques are conceived. Notably, several of the resulting techniques are (provably) stronger than well-known methods from the literature that are used in exact SPG algorithms. Finally, by integrating the new methods into a branch-and-cut framework, we obtain an exact SPG solver that is not only competitive with, but even outperforms the current state of the art on an extensive collection of benchmark sets. Furthermore, we can solve several instances for the first time to optimality.

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Section: C(e(s )) ≤ C(e(s))mentioning

confidence: 99%

“…See e.g. [5,15] for approximation, and [24,29,47] for complexity results. However, when it comes to (practical) exact algorithms, the picture is significantly more bleak.…”

Section: Introductionmentioning

confidence: 99%

Implications, conflicts, and reductions for Steiner trees

Rehfeldt

Koch

2021

Math. Program.

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“…Grid Tiling has also been extended to higher dimensions [15], and successfully used to establish lower bounds for coloring unit disk and unit ball graphs [16]. Most recently, it has been used to study the parameterized complexity of Steiner Tree in planar graphs [17,18]. The problem is also starting to be applied outside the strictly geometric setting: in computational topology [19] and for studying H -free graphs [20].…”

Section: Related Workmentioning

confidence: 99%

The Homogeneous Broadcast Problem in Narrow and Wide Strips II: Lower Bounds

2019

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Let P be a set of nodes in a wireless network, where each node is modeled as a point in the plane, and let s ∈ P be a given source node. Each node p can transmit information to all other nodes within unit distance, provided p is activated. The (homogeneous) broadcast problem is to activate a minimum number of nodes such that in the resulting directed communication graph, the source s can reach any other node. We study the complexity of the regular and the hop-bounded version of the problem-in the latter s must be able to reach every node within a specified number of hops-where we also consider how the complexity depends on the width w of the strip. We prove the following two lower bounds. First, we show that the regular version of the problem is W[1]-complete when parameterized by the solution size k. More precisely, we show that the problem does not admit an algorithm with running time f (k)n o(√ k) , unless ETH fails. The construction can also be used to show an f (w)n Ω(w) lower bound when we parameterize by the strip width w. Second, we prove that the hop-bounded version of the problem is NP-hard in strips of width 40. These results complement the algorithmic results in a companion paper (de Berg et al. in Algorithmica, submitted).

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“…We also note that while the running time of our algorithm is reminiscent of the algorithm by Kisfaludi-Bak et al [39] for Steiner Tree parameterized by the terminal face cover number, our algorithm is and needs to be substantially more involved. The intuition of their algorithm is to show that a minimum Steiner tree for a set of terminals that can be covered by k faces has bounded treewidth.…”

Section: Introductionmentioning

confidence: 99%

“…The minimum number of faces of the input planar graph that cover all the terminals was termed the terminal face cover number γ(G) by Krauthgamer et al [41]. The terminal face cover number is of broad interest as a parameter and has been studied with respect to several cut and flow problems [41,42,13,47], shortest path problems [32,14], finding non-crossing walks [28], and the minimum Steiner tree problem [39]. In particular, Planar Steiner Tree has an algorithm running in time 2 O(γ(G) log γ(G)) n O( √ γ(G)) .…”

Section: Introductionmentioning

confidence: 99%

Planar Multiway Cut with Terminals on Few Faces

Pandey¹,

Leeuwen²

2022

Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

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We consider the Edge Multiway Cut problem on planar graphs. It is known that this can be solved in n O( √ t) time [Klein, Marx, ICALP 2012] and not in n o( √ t) time under the Exponential Time Hypothesis [Marx, ICALP 2012], where t is the number of terminals. A generalization of this parameter is the number k of faces of the planar graph that jointly cover all terminals. For the related Steiner Tree problem, an n O( √ k) time algorithm was recently shown [Kisfaludi-Bak et al., SODA 2019]. By a completely different approach, we prove in this paper that Edge Multiway Cut can be solved in n O( √ k) time as well. Our approach employs several major concepts on planar graphs, including homotopy and sphere-cut decomposition. We also mix a global treewidth dynamic program with a Dreyfus-Wagner style dynamic program to locally deal with large numbers of terminals.

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Nearly ETH-tight Algorithms for Planar Steiner Tree with Terminals on Few Faces

Cited by 6 publications

References 39 publications

Implications, conflicts, and reductions for Steiner trees

Implications, conflicts, and reductions for Steiner trees

The Homogeneous Broadcast Problem in Narrow and Wide Strips II: Lower Bounds

Planar Multiway Cut with Terminals on Few Faces

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