Improved Approximation Algorithms for (Budgeted) Node-Weighted Steiner Problems

Bateni, MohammadHossein; Hajiaghayi, MohammadTaghi; Liaghat, Vahid

doi:10.1007/978-3-642-39206-1_8

Cited by 23 publications

(22 citation statements)

References 16 publications

(44 reference statements)

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“…In addition, each node provides opportunities for encounters with other, competing species traveling in the same vegetation [68, 69, 70, 58]. Our results here suggest that, as in engineered networks [71, 72], the cost of including a node in the turtle ant network may vary among nodes, because whether other ants follow an exploring ant that leaves the trail at that node depends on the node’s physical configuration. We assigned transition indices here by visual inspection of the vegetation, and we hope in future work to investigate further the effects of node configuration on path formation using biophysical models based on 3D scanning data.…”

Section: Discussionmentioning

confidence: 80%

“…Our results here suggest that, as in engineered networks [64,65], the cost of including a node in the turtle ant network may vary among nodes, because whether other ants follow an exploring ant that leaves the trail at that node depends on the node's physical configuration. The costs of additional nodes include the loss of ants from the trail, and the pursuit of fruitless paths, which may detract from the colony's ability to distribute resources among its many nests, and make fewer ants available to recruit effectively when a new food source is discovered.…”

Section: Turtle Ants Form Loops To Promote Coherencementioning

confidence: 77%

“…We evaluated networks generated by heuristic algorithms designed to optimize each objective. Finding the network that optimizes total nodes is a variant of the classic Steiner tree problem (an NP-complete problem) [71, 72], and so we applied a standard approximation algorithm [80]. To optimize average length, we used the following algorithm: Input: Graph G = ( V G , E G ) and terminals X ⊆ V G Create an empty graph T . Add edges T in order of increasing length until T contains all terminals X , and all terminals are in the same connected component. Remove all nodes and edges not part of the same connected component as the terminals X Remove edges from T in decreasing order of length; only remove edges that decrease the average length without disconnecting T …”

Section: Methodsmentioning

confidence: 99%

Section: Approximating Optimal Networkmentioning

confidence: 99%

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Better tired than lost: turtle ant trail networks favor coherence over short edges

Chandrasekhar

Marshall

Austin

et al. 2019

Preprint

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1To create and maintain a backbone routing network is a basic challenge in many engineered and 2 biological systems [1][2][3], from wireless sensor networks and robot swarms to neural circuits 3 and blood circulation. Optimal routing in such networks often seeks to minimize transport 4 delay [4][5][6][7][8], but routing decisions may be influenced by variability in the terrain [9][10][11][12]. Here 5 we find that turtle ants build trail networks that emphasize coherence, keeping the ants together 6 on the trail in a heterogeneous environment, rather than minimizing the distance travelled. The 7 routing backbone of turtle ants (Cephalotes goniodontus), an arboreal species that forages in the 8 1 tree canopy of tropical forests, connects many nests [13][14][15], using trail pheromone that the ants 9 put down continuously, not just on the way back from a food source. Unlike species that forage 10 on the ground, arboreal ants are constrained to travel within the vegetation network of branches 11 and vines. We compared observed turtle ant trails with random, hypothetical trails in the same 12 surrounding vegetation. Strikingly, the trails do not minimize distance travelled, but instead 13 minimize the total number of nodes in the backbone, and favor nodes with 3d configurations 14 that are easily reinforced with pheromone. Thus, rather than forming the shortest paths, turtle 15 ants take advantage of natural variation in the environment to build coherent trails. This ensures 16 that the nests and food sources stay connected, at the expense of longer travel time. This design 17 principle may be beneficial in applications where distributed agents, such as swarms of robots, 18 must coordinate using a communication backbone in complex environments to collectively 19 solve a task, such as building a structure, searching for resources, or surveying terrain [16]. 20 Introduction 21The goal of a backbone routing network is to ensure that there is a path for any two 22 devices on the network to communicate. In real-world systems, however, physical variation in the 23 environment can affect the accuracy and rate of communication [9][10][11][12][17][18][19]. Understanding the 24 physical structure of the environment may improve the design of routing algorithms [16,[20][21][22][23][24], for 25 example, by reducing the search space of possible routing paths and steering network construction 26 away from parts of the terrain that are difficult to reach. 27The 14,000 species of ants have evolved diverse distributed routing and search algorithms to 28 search for, obtain, and distribute resources [25] in the diverse environments that they inhabit [26-29 29]. For example, species such as Formica and Argentine ants, which forage and build trails in 30 a 2d-plane, have minimal constraints on trail geometry [30][31][32], and can minimize the distance 31 travelled by forming trails with branch points that approximate Steiner trees [27, 33, 34], an NP-32 complete generalization of the minimal spanning tree concept. By contrast, many specie...

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Section: Discussionmentioning

confidence: 80%

Section: Turtle Ants Form Loops To Promote Coherencementioning

confidence: 77%

Section: Methodsmentioning

confidence: 99%

Section: Approximating Optimal Networkmentioning

confidence: 99%

See 2 more Smart Citations

Better tired than lost: turtle ant trail networks favor coherence over short edges

Chandrasekhar

Marshall

Austin

et al. 2019

Preprint

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“…For other problems we use the results for the plain connectivity quota problem, that achieves ratio O(log min{n, Q}) [4]. We note that often the graph H arising from the reduction is the incidence graph of the paths and edges/nodes of a tree, and thus such instances may admit better (constant) ratios than the ones known for general graphs.…”

Section: Subset Steiner Connected Dominating Setmentioning

confidence: 99%

2-Node-Connectivity Network Design

Nutov

2021

Approximation and Online Algorithms

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We consider network design problems in which we are given a graph G = (V, E) with edge costs, and seek a min-cost/size 2-nodeconnected subgraph G = (V , E ) that satisfies a prescribed property.-In the Block-Tree Augmentation problem the goal is to augment a tree T by a min-size edge setWe break the natural ratio of 2 for this problem and show that it admits approximation ratio 1.91. This result extends to the related Crossing Family Augmentation problem. -In the 2-Connected Dominating Set problem G should dominate V . We give the first non-trivial approximation algorithm for this problem, with expected ratio Õ(log 4 |V |). -In the 2-Connected Quota Subgraph problem we are given node profits p(v) and G should have profit at least a given quota Q. We show expected ratio Õ(log 2 |V |), almost matching the best known ratio O(log 2 |V |). Our algorithms are very simple, and they combine three main ingredients: 1. A probabilistic spanning tree embedding with distortion Õ(log |V |) results in a variant of the Block-Tree Augmentation problem. 2. An approximation ratio preserving reduction of Block-Tree Augmentation variants to Node Weighted Steiner Tree problems. 3. Using existing approximation algorithms for variants of the Node Weighted Steiner Tree problem.

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