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Finding Light Spanners in Bounded Pathwidth Graphs [article]

Michelangelo Grigni, Hao-Hsiang Hung
2012 arXiv   pre-print
In this paper we make some progress towards the conjecture that light spanners exist for every minor-closed graph family. In particular, we show that they exist for graphs with bounded pathwidth.  ...  If G is from certain minor-closed graph families (at least bounded genus graphs and apex graphs), then we know that light spanners exist.  ...  Bounded pathwidth graphs have light spanners, computable by a greedy algorithm. We prove this in Section 3.  ... 
arXiv:1104.4669v2 fatcat:ovxe3r27ojg5jl7w3mvzjo3oii

Light Spanners in Bounded Pathwidth Graphs [chapter]

Michelangelo Grigni, Hao-Hsiang Hung
2012 Lecture Notes in Computer Science  
We have developed an efficient algorithm for light spanners in bounded pathwidth graphs, based on an intermediate data structure called monotone tree.  ...  In approximation algorithm design, light spanners has applications in graph-metric problems such as metric TSP (the traveling salesman problem).  ...  We have discussed the major difficulty in handling bounded treewidth graphs in [9] : we have no control over the MST topology inside of the bounded treewidth graphs, and a trimming might introduce too  ... 
doi:10.1007/978-3-642-32589-2_42 fatcat:md5obquxgvfdjjmxps7ht4srti

Light spanners for bounded treewidth graphs imply light spanners for H-minor-free graphs [article]

Glencora Borradaile, Hung Le
2017 arXiv   pre-print
We also prove that the greedy spanner of a bounded pathwidth graph is light and discuss the possibility of extending our proof to bounded treewidth graphs.  ...  We take a further step toward proving this conjecture by showing that if the bounded treewidth graphs have light greedy spanners, then the conjecture is true.  ...  The greedy spanner for bounded pathwidth graphs is light Grigni and Hung proved that graphs of pathwidth pw have a (1 + )-spanner of lightness O(pw 3 / ) [10] .  ... 
arXiv:1703.10633v1 fatcat:53nk65ug2nbi7glnr2nkxhuwju

Light Spanners [article]

Michael Elkin, Ofer Neiman, Shay Solomon
2014 arXiv   pre-print
A t-spanner of a weighted undirected graph G=(V,E), is a subgraph H such that d_H(u,v)< t· d_G(u,v) for all u,v∈ V.  ...  Specifically, it is shown that for any parameters k> 1 and ϵ>0, any weighted graph G on n vertices admits a (2k-1)·(1+ϵ)-stretch spanner of weight at most w(MST(G))· O_ϵ(kn^1/k/ k), where w(MST(G)) is  ...  Currently, some of the known results are for planar graph [ADM + 95], bounded-genus graphs [Gri00] , unit disk graphs [KPX08] , and bounded pathwidth graphs [GH12] .  ... 
arXiv:1404.7703v1 fatcat:cwt35vchlzc23a4p2xqxsibrhe

Light Spanners [chapter]

Michael Elkin, Ofer Neiman, Shay Solomon
2014 Lecture Notes in Computer Science  
The bound on the number of edges is known to be asymptotically tight for certain small values of k, and for all k assuming Erdős' girth conjecture. In this paper we focus on the weight of a spanner.  ...  Light weight spanners were also found useful for various data gathering and dissemination tasks in overlay networks [BKR + 02, KV02], in wireless and sensor networks [SS10], for network design [MP98, SCRS01  ...  Currently, some of the known results are for planar graph [ADM + 95], bounded-genus graphs [Gri00] , unit disk graphs [KPX08] , and bounded pathwidth graphs [GH12] .  ... 
doi:10.1007/978-3-662-43948-7_37 fatcat:hhdxvydgujdubhhr537gxwyt6q

Light Spanners

Michael Elkin, Ofer Neiman, Shay Solomon
2015 SIAM Journal on Discrete Mathematics  
A t-spanner of a weighted undirected graph G = (V, E), is a subgraph H such that dH (u, v) ≤ t · dG(u, v) for all u, v ∈ V .  ...  The sparseness of the spanner can be measured by its size (the number of edges) and weight (the sum of all edge weights), both being important measures of the spanner's quality -in this work we focus on  ...  Currently, some of the known results are for planar graph [ADM + 95], bounded-genus graphs [Gri00] , unit disk graphs [KPX08] , and bounded pathwidth graphs [GH12] .  ... 
doi:10.1137/140979538 fatcat:hxqgt45xxbg6linwkejrh4tdi4

The Greedy Spanner is Existentially Optimal [article]

Arnold Filtser, Shay Solomon
2020 arXiv   pre-print
as good) as the worst performance of an optimal spanner over all graphs in G.  ...  Roughly speaking, the greedy spanner is said to be existentially optimal (or near-optimal) for a graph family G if the worst performance of the greedy spanner over all graphs in G is just as good (or nearly  ...  We remark that light spanners were extensively studied in various graph families such as planar graphs [ADD + 93, Kle05] , apex graphs [GS02] , bounded pathwidth graphs [GH12] , bounded genus graphs  ... 
arXiv:1605.06852v2 fatcat:hnfwncky75f4jor5vxmfiowg7y

Minor-free graphs have light spanners [article]

Glencora Borradaile and Hung Le and Christian Wulff-Nilsen
2017 arXiv   pre-print
Our lightness bound is O(σ_H/ϵ^31/ϵ) where σ_H = |V(H)|√( |V(H)|) is the sparsity coefficient of H-minor-free graphs. That is, it has a practical dependency on the size of the minor H.  ...  We show that every H-minor-free graph has a light (1+ϵ)-spanner, resolving an open problem of Grigni and Sissokho and proving a conjecture of Grigni and Hung.  ...  Later Grigni and Hung [12] , in showing that graphs of bounded pathwidth have light spanners, conjectured that H-minor-free graphs also have light spanners; that is, that the dependence on n can be  ... 
arXiv:1711.00821v1 fatcat:23fsvgew3ncv7gywloqpd3yyqq

A light metric spanner [article]

Lee-Ad Gottlieb
2015 arXiv   pre-print
In this paper, we resolve the question in the affirmative, and show that doubling spaces admit (1+ϵ)-stretch spanners with lightness W_D = (ddim/ϵ)^O(ddim).  ...  Important in its own right, our result also implies a much faster polynomial-time approximation scheme for the traveling salesman problemin doubling metric spaces, improving upon the bound presented in  ...  These include planar graphs [ADD + 93, ACC + 96], unit disk graphs [KPX08] , graphs which are snowflakes of metrics [GS14] , and graphs of bounded pathwidth [GH12] and bounded genus [DHM10] .  ... 
arXiv:1505.03681v1 fatcat:ehlgvtsgrvg6lem2m535klqtky

Algorithms for Optimization Problems in Planar Graphs (Dagstuhl Seminar 16221)

Jeff Erickson, Philip N. Klein, Dániel Marx, Claire Mathieu, Marc Herbstritt
2016 Dagstuhl Reports  
This report contains abstracts for the recent developments in planar graph algorithms discussed during the seminar as well as summaries of open problems in this area of research.  ...  This report documents the program and the outcomes of Dagstuhl Seminar 16221 "Algorithms for Optimization Problems in Planar Graphs". The seminar was held from May 29 to June 3, 2016.  ...  This is in contrast with the popular existing approach for planar PTASs of constructing light-weight spanners in one iteration, which notably requires a constant-factor approximate solution to start from  ... 
doi:10.4230/dagrep.6.5.94 dblp:journals/dagstuhl-reports/EricksonKMM16 fatcat:wasdfgivt5fqdppfxo3iqqs2ta

On Light Spanners, Low-treewidth Embeddings and Efficient Traversing in Minor-free Graphs [article]

Vincent Cohen-Addad and Arnold Filtser and Philip N. Klein and Hung Le
2020 arXiv   pre-print
Construction of a light subset spanner.  ...  capacity in minor-free metrics; (3) the first efficient approximation scheme for vehicle routing with bounded capacity on bounded genus metrics.  ...  Over the years, light spanners with constant lightness have been shown to exist in Euclidean metrics [RS98, LS19], doubling metrics [Got15, BLW19], planar graphs [ADD + 93], bounded genus graphs TSP in  ... 
arXiv:2009.05039v1 fatcat:ilbpen2dt5ao3bmwqps6ehiov4

On Strong Diameter Padded Decompositions

Arnold Filtser, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization  
Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles.  ...  Given a weighted graph G = (V, E, w), a partition of V is ∆-bounded if the diameter of each cluster is bounded by ∆.  ...  The tradeoff between stretch and sparsity/lightness of spanners had been the focus of an intensive research effort, and low stretch graph spanners were used in a plethora of applications.  ... 
doi:10.4230/lipics.approx-random.2019.6 dblp:conf/approx/Filtser19 fatcat:swgqqzo7sjd4tpj2jb7hfcnfvu

Page 2800 of Mathematical Reviews Vol. , Issue 91E [page]

1991 Mathematical Reviews  
Halld6orsson, Approx- imating maximum independent sets by excluding subgraphs (pp. 13-25); Ingo Althéfer, Gautam Das, David Dobkin and Debo- rah Joseph, Generating sparse spanners for weighted graphs (  ...  Zaroliagis, Efficient parallel algorithms for short- est paths in planar graphs (pp. 288-300); Hans L. Bodlaender and Rolf H.  ... 

Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane [article]

Hadrien Cambazard, Nicolas Catusse
2017 arXiv   pre-print
In this paper, a fixed-parameter algorithm for the Rectilinear TSP is presented and relies on techniques for solving TSP on bounded-treewidth graphs.  ...  Both bounds improve over the best time bounds known for these problems.  ...  It is indeed intended for graph problems with a bounded treewidth/pathwidth and a global connectivity property such as the Hamiltonian Cycle, Steiner tree or TSP.  ... 
arXiv:1512.06649v3 fatcat:a3hk3u3rmbhnjm4usmrevsonhy

Low Treewidth Embeddings of Planar and Minor-Free Metrics [article]

Arnold Filtser, Hung Le
2022 arXiv   pre-print
Important corollaries of our result include a bicriteria PTAS for metric Baker's problems and a PTAS for the vehicle routing problem with bounded capacity in planar graphs, both run in almost-linear time  ...  In this work, we devise a new embedding technique to improve the treewidth bound of Cohen-Addad et al. exponentially to O_ϵ(loglog n)^2.  ...  They also admit spanners with multiplicative distortion 1 + and Õr ( −3 ) lightness [BLW17] . On the other hand, there are other graph families that embed well into bounded treewidth graphs.  ... 
arXiv:2203.15627v1 fatcat:3fkdetjlivctvcfrmsnfexgyka
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