1. ETH Zurich – Distributed Computing Group Roger Wattenhofer 1ETH Zurich – Distributed Computing – www.disco.ethz.ch Christoph Lenzen Roger Wattenhofer Minimum Dominating Set Approximation in Graphs of Bounded Arboricity

2. Christoph Lenzen@DISC 2010 Minimum Dominating Sets (MDS) graph G=(V,E) N(A) denotes inclusive neighborhood of A µ V D µ V is dominating set (DS) iff V=N(D) minimum dominating set is DS of minimum size important in theory and practice minimum dominating set dominating set in a social network

3. Christoph Lenzen@DISC 2010 MDS on General Graphs finding an MDS is NP-hard ) we're looking for approximations O(log Δ) approx. in O(log n) rounds...but for reasonable message size O(log 2 Δ) rounds o(log Δ) approx. is NP-hard polylog. approx. needs  (log Δ) and  (log 1/2 n) rounds ) maybe "simpler" graphs are easier? Garey & Johnson, '79 Feige, JACM '98 Raz & Safra, STOC '97 Kuhn & al., PODC '04 Kuhn & al., SODA '06

4. Christoph Lenzen@DISC 2010 MDS on Restricted Families of Graphs L. et al DISC '08 Schneider & Wattenhofer, PODC '08 L. et al SPAA '08 Czygrinow & Hańćkowiak, ESA '06 restrictive hard general bounded degree O(1) approx. O(1) rounds planar O(1) approx. O(1) rounds unit disc O(1) approx. Θ(log * n) rounds bounded independence O(1) approx. O(log n) rounds Θ(log n) approx. O(log 2 Δ) rounds  (log Δ) rounds excluded minor (1+ ² ) approx. polylog n rounds e.g. Luby SIAM J. Comp. '86

5. Christoph Lenzen@DISC 2010 What's a Good Compromise?...or: what have many "easy" graphs in common? ) They are sparse! This is not good enough: + star graph: n-n 1/2 nodes center covers all arbitrary graph: n 1/2 nodes difficult to handle O(n) edges = same lower bounds as in general case

6. Christoph Lenzen@DISC 2010 Arboricity A "good" property is preserved under taking subgraphs. ) Demand sparsity in every subgraph! This property is called bounded arboricity. graph G=(V,E) partition E = E 1 [ E 2 [... [ E f into f forests minimum number of forests is arboricity A of G 3-forest decomp. of the Peterson graph......whose arboricity is however only 2.

7. Christoph Lenzen@DISC 2010 Where are Graphs of Bounded Arboricity? restrictive hard general bounded degree planar unit disc bounded independence excluded minor bounded arboricity arboricity 2 permits K √n minor no strong lower bounds  o(log A) approx. is NP-hard  no (5- ² ) approximation in o(log * n) time bounded arboricity Czygrinow & al., DISC '08 no o(A) approx. in o(log * n) rounds

8. Christoph Lenzen@DISC 2010 sequentially add nodes covering most others ) yields O(log Δ) approx....but in parallel? ) Just take all high-degree nodes! repeat until finished Be Greedy! 8+2 7+2 5 5 4 4 4 3 3 2 1 1 Θ(log n) 1 2

9. Christoph Lenzen@DISC 2010 D = nodes of (current) max. deg. Δ C = nodes (freshly) covered by D M = optimum solution |D|Δ/2 · |E(C [ D)| < A(|C [ D|) · A(|C|+|D|) ) (Δ/2-A)|D| < A|C| · A(Δ+1)|M| if Δ ¸ 4A and A 2 O(1) ) |D| 2 O(|M|) Why does Greedy-By-Degree work? D C M V

10. Christoph Lenzen@DISC 2010 Q:What about Δ < 4A ? A:Each c 2 C elects one deg. Δ neighbor into D! Q:How avoid time complexity  (Δ)? A:Take all nodes of degree Δ/2 at once! Q:How deal with unknown Δ? A:It's enough to check up to distance 2! ) uniform O(log Δ) approx. in O(log Δ) rounds Greedy-By-Degree: Details

11. Christoph Lenzen@DISC 2010...we would like to have an O(1) approx. for A 2 O(1) What about using a (rooted) forest decomposition? decomposition into f 2 O(A) forests takes Θ(log n) time note: we cannot handle each forest individually Neat, but... Barenboim & Elkin, PODC '08

12. Christoph Lenzen@DISC 2010 For an MDS M, · (A+1)|M| nodes are not covered by parents. ) These have · A(A+1)|M| parents. ) Let's try to cover all nodes (that have one) by parents! ) set cover instance with each element in · A sets How to use a Forest-Decomposition 5 1 2 3 4 6 7 8 9 10 {1,10} {1,3,7} {3,5,9} {9,10} {3,6,10} {9} {6} )

13. Christoph Lenzen@DISC 2010 sequentially, an A approx. is trivial:  pick any uncovered node  choose all of its parents  repeat until finished  for every node, one of its parents is in an optimum solution Acting Greedily again {1,10} 5 {1,3,7} {3,5,9} {9,10} {3,6,10} 1 2 3 4 {9} 6 7 8 9 10 {6}

14. Christoph Lenzen@DISC 2010 any sequence of nodes that share no parents is feasible the order is irrelevant for the outcome define H:=(V,E') by {v,w} 2 E', v and w share a parent ) we need a maximal independent in H And now more quickly... )

15. Christoph Lenzen@DISC 2010 compute O(A) forest decomp. (O(log n) rounds) simulate MIS algorithm on H (O(log n) rounds w.h.p. output parents of MIS nodes and nodes w/o parents ) O(A 2 ) approx. in O(log n) rounds w.h.p. Algorithm: Parent Dominating Set )

16. Christoph Lenzen@DISC 2010 Greedy-By-Degree: Pros'n'Cons +very simple +running time O(log Δ) +message size O(log log Δ) +uniform & deterministic -O(A log Δ) approx. general graphs: O(log 2 Δ) general graphs: O(log Δ)

17. Christoph Lenzen@DISC 2010 Parent Dominating Set: Pros'n'Cons +simple +O(A 2 ) approx. (deterministic) +/-running time O(log n) (randomized) open question: Are there faster O(1) approx. for A 2 O(1)? general graphs: O(log Δ) )

18. ETH Zurich – Distributed Computing Group Roger Wattenhofer 18ETH Zurich – Distributed Computing – www.disco.ethz.ch Christoph Lenzen Roger Wattenhofer Thank You! Questions & Comments?