Local 3-approximation algorithms for weighted dominating set and - - PowerPoint PPT Presentation

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Local 3-approximation algorithms for weighted dominating set and - - PowerPoint PPT Presentation

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Local 3-approximation algorithms for weighted dominating set and vertex cover in quasi unit-disk graphs Marja Hassinen, Valentin Polishchuk, Jukka Suomela HIIT, University of Helsinki, Finland LOCALGOS 14 June 2008 Introduction Local

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Local 3-approximation algorithms for weighted dominating set and vertex cover in quasi unit-disk graphs

Marja Hassinen, Valentin Polishchuk, Jukka Suomela

HIIT, University of Helsinki, Finland

LOCALGOS 14 June 2008

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Introduction

Local algorithms: output at each node depends only

  • n the constant-radius neighbourhood of the node

(Linial 1992, Naor and Stockmeyer 1995)

Assumptions:

◮ Unit-disk graphs ◮ Each node knows its coordinates

Problems:

◮ Dominating set ◮ Vertex cover

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Prior work

Dominating set:

◮ 15-approximation

(Urrutia 2007)

◮ 5-approximation

(Czyzowicz et al. 2008)

◮ (1 + ǫ)-approximation

(Wiese and Kranakis 2007)

Vertex cover:

◮ 12-approximation trivial ◮ (1 + ǫ)-approximation

(Wiese and Kranakis 2008)

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Our contributions

Simple local algorithm 3-approximation Small local horizon (locality distance):

◮ Present algorithm: r = 83 ◮ Wiese and Kranakis (2007):

r = 46814 for 3-approximation Quasi unit-disk graphs Weighted versions

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Dominating set

Input — assumed to be a unit disk graph

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Dominating set

An optimal solution

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Dominating set: local algorithm

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Dominating set: local algorithm

Divide the plane into 2 × 4 rectangles

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Dominating set: local algorithm

3-colour the rectangles

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Dominating set: local algorithm

For each rectangle. . .

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Dominating set: local algorithm

For each rectangle construct an extended rectangle

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Dominating set: local algorithm

Extended rectangles are non-intersecting for each colour

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Dominating set: local algorithm

Extended rectangles are non-intersecting for each colour

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Dominating set: local algorithm

Extended rectangles are non-intersecting for each colour

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Dominating set: local algorithm

For each extended rectangle. . .

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Dominating set: local algorithm

For each extended rectangle, form a subproblem. . .

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Dominating set: local algorithm

. . . and solve the subproblem optimally

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Dominating set: local algorithm

Only inside needs to be dominated

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Dominating set: local algorithm

Repeat for each rectangle

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Dominating set: local algorithm

Repeat for each rectangle

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Dominating set: local algorithm

Repeat for each rectangle

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Dominating set: local algorithm

Union of local solutions

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Dominating set: feasibility

Each node is dominated in at least one subproblem

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Dominating set: approximation ratio

OPT is a feasible solution to each subproblem

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Dominating set: approximation ratio

OPT is a feasible solution to each subproblem

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Dominating set: approximation ratio

OPT is a feasible solution to each subproblem

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Dominating set: approximation ratio

OPT is a feasible solution to each subproblem

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Dominating set: approximation ratio

Factor 3 approximation from 3-colouring

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Vertex cover

The same basic approach applies here as well

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Local horizon: worst case

Consider a shortest path within an extended rectangle

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Local horizon: worst case

Pick even nodes — distance between any pair > 1

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Local horizon: worst case

Place disks of radius 1/2 on even nodes — non-intersecting

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Local horizon: worst case

Area bound: at most 42 such disks =

⇒ at most 83 nodes

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Local horizon: average case

50 100 150 200 5 10 15 20 nodes diameter

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Local horizon: average case

50 100 150 200 5 10 15 20 nodes diameter 0.0 0.2 0.4 0.6 0.8 1.0 fraction of connected graphs

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Conclusions

Local 3-approximation algorithm for dominating set and vertex cover Assumptions: (quasi) unit-disk graphs, coordinates known Unweighted case: local and poly-time Weighted case: local — but not necessarily poly-time!

◮ Other complexity measures for local algorithms

besides the local horizon? Challenge: apply the same idea to other problems!

http://www.hiit.fi/ada/geru — jukka.suomela@cs.helsinki.fi

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References (1)

  • J. Czyzowicz, S. Dobrev, T. Fevens, H. González-Aguilar, E. Kranakis,
  • J. Opatrny, and J. Urrutia. Local algorithms for dominating and

connected dominating sets of unit disk graphs with location aware

  • nodes. In Proc. 8th Latin American Theoretical Informatics

Symposium (LATIN, Búzios, Brazil, April 2008), volume 4957 of Lecture Notes in Computer Science, pages 158–169, Berlin, Germany, 2008. Springer-Verlag. [DOI]

  • M. Hassinen, V. Polishchuk, and J. Suomela. Local 3-approximation

algorithms for weighted dominating set and vertex cover in quasi unit-disk graphs. In Proc. 2nd International Workshop on Localized Algorithms and Protocols for Wireless Sensor Networks (LOCALGOS, Santorini Island, Greece, June 2008), 2008. To appear.

  • N. Linial. Locality in distributed graph algorithms. SIAM Journal on

Computing, 21(1):193–201, 1992. [DOI]

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References (2)

  • M. Naor and L. Stockmeyer. What can be computed locally? SIAM

Journal on Computing, 24(6):1259–1277, 1995. [DOI]

  • J. Urrutia. Local solutions for global problems in wireless networks.

Journal of Discrete Algorithms, 5(3):395–407, 2007. [DOI]

  • A. Wiese and E. Kranakis. Local PTAS for dominating and connected

dominating set in location aware unit disk graph. Technical Report TR-07-17, Carleton University, School of Computer Science, Ottawa, Canada, Oct. 2007.

  • A. Wiese and E. Kranakis. Local PTAS for independent set and vertex

cover in location aware unit disk graphs. In Proc. 4th IEEE/ACM International Conference on Distributed Computing in Sensor Systems (DCOSS, Santorini Island, Greece, June 2008), Berlin, Germany,

  • 2008. Springer-Verlag. To appear.