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

Transcription

1 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

2 Introduction Local algorithms: output at each node depends only on the constant-radius neighbourhood of the node Assumptions: Unit-disk graph Each node knows its coordinates (Linial 1992, Naor and Stockmeyer 1995) Problems: Dominating set Vertex cover 2 / 37

3 Prior work 3 / 37 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)

4 Our contributions 4 / 37 Simple local algorithm 3-approximation Small local horizon (locality distance): Present algorithm: r = 83 Wiese and Kranakis (2007): r = for 3-approximation Quasi unit-disk graphs Weighted versions

5 Dominating set 5 / 37 Input assumed to be a unit-disk graph

6 Dominating set 6 / 37 An optimal solution

7 Dominating set: local algorithm 7 / 37

8 Dominating set: local algorithm 8 / 37 Tile the plane with 2 4 rectangles

9 Dominating set: local algorithm 9 / 37 3-colour the rectangles

10 Dominating set: local algorithm 10 / 37 For each rectangle...

11 Dominating set: local algorithm 11 / 37 For each rectangle construct an extended rectangle

12 Dominating set: local algorithm 12 / 37 Extended rectangles are non-intersecting for each colour

13 Dominating set: local algorithm 13 / 37 Extended rectangles are non-intersecting for each colour

14 Dominating set: local algorithm 14 / 37 Extended rectangles are non-intersecting for each colour

15 Dominating set: local algorithm 15 / 37 For each extended rectangle...

16 Dominating set: local algorithm 16 / 37 For each extended rectangle, form a subproblem...

17 Dominating set: local algorithm 17 / and solve the subproblem optimally

18 Dominating set: local algorithm 18 / 37 Only inside needs to be dominated

19 Dominating set: local algorithm 19 / 37 Repeat for each rectangle

20 Dominating set: local algorithm 20 / 37 Repeat for each rectangle

21 Dominating set: local algorithm 21 / 37 Repeat for each rectangle

22 22 / 37

23 Dominating set: local algorithm 23 / 37 Union of local solutions

24 Dominating set: feasibility 24 / 37 Each node is dominated in at least one subproblem

25 Dominating set: approximation ratio 25 / 37 OPT is a feasible solution to each subproblem

26 Dominating set: approximation ratio 26 / 37 OPT is a feasible solution to each subproblem

27 Dominating set: approximation ratio 27 / 37 OPT is a feasible solution to each subproblem

28 Dominating set: approximation ratio 28 / 37 OPT is a feasible solution to each subproblem

29 Dominating set: approximation ratio 29 / 37 Factor 3 approximation from 3-colouring

30 Vertex cover 30 / 37 The same basic approach applies here as well

31 Local horizon: worst case 31 / 37 Consider a shortest path within an extended rectangle

32 Local horizon: worst case 32 / 37 Pick even nodes distance between any pair > 1

33 Local horizon: worst case 33 / 37 Place disks of radius 1/2 on even nodes non-intersecting

34 Local horizon: worst case 34 / 37 Area bound: at most 42 such disks = at most 83 edges

35 Local horizon: average case 20 diameter nodes 35 / 37

36 Local horizon: average case 1.0 diameter fraction of connected graphs nodes 36 / 37

37 Conclusions 37 / 37 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! jukka.suomela@cs.helsinki.fi

38 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 , Berlin, Germany, 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), To appear. N. Linial. Locality in distributed graph algorithms. SIAM Journal on Computing, 21(1): , [DOI]

39 References (2) M. Naor and L. Stockmeyer. What can be computed locally? SIAM Journal on Computing, 24(6): , [DOI] J. Urrutia. Local solutions for global problems in wireless networks. Journal of Discrete Algorithms, 5(3): , [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 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, Springer-Verlag. To appear.