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Minimum Dominating Set Approximation in Graphs of Bounded Arboricity

Minimum Dominating Set Approximation in Graphs of Bounded Arboricity. Minimum Dominating Sets (MDS). important in theory and practice. minimum dominating set. dominating set in a social network. graph G=(V,E) N(A) denotes inclusive neighborhood of A µ V

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Minimum Dominating Set Approximation in Graphs of Bounded Arboricity

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  1. Minimum Dominating Set Approximation in Graphs of Bounded Arboricity

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

  3. 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(log2Δ) rounds • o(log Δ) approx. is NP-hard • polylog. approx. needs (log Δ)and (log1/2 n) rounds • ) maybe "simpler" graphs are easier? Kuhn & al., SODA '06 Garey & Johnson, '79 Raz & Safra, STOC '97 Feige, JACM '98 Kuhn & al., PODC '04

  4. MDS on Restricted Families of Graphs excluded minor Schneider & Wattenhofer, PODC '08 bounded independence hard restrictive L. et al DISC '08 planar O(1) approx. O(1) rounds (1+²) approx. polylog n rounds general bounded degree Θ(log n) approx. O(log2Δ) rounds (log Δ) rounds O(1) approx. O(1) rounds unit disc O(1) approx. O(log n) rounds O(1) approx. Θ(log*n) rounds L. et al SPAA '08 e.g. Luby SIAM J. Comp. '86 Czygrinow & Hańćkowiak, ESA '06

  5. What's a Good Compromise? • ...or: what have many "easy" graphs in common? • ) They are sparse! • This is not good enough: O(n) edges = + same lower bounds as in general case arbitrary graph: n1/2 nodes difficult to handle star graph: n-n1/2 nodes center covers all

  6. Arboricity • A "good" property is preserved under taking subgraphs. • ) Demand sparsity in every subgraph! • This property is called bounded arboricity. 3-forest decomp. of the Peterson graph... ...whose arboricity is however only 2. • graph G=(V,E) • partition E=E1 [E2 [...[Ef into f forests • minimum number of forests is arboricityA of G

  7. Where are Graphs of Bounded Arboricity? bounded independence hard restrictive no o(A) approx. in o(log* n) rounds • 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 bounded arboricity excluded minor planar general bounded degree unit disc Czygrinow & al., DISC '08

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

  9. Why does Greedy-By-Degree work? V • 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|) D C M

  10. Greedy-By-Degree: Details Q: What about Δ < 4A ? A: Each c2C 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

  11. Neat, but... • ...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 Barenboim & Elkin, PODC '08

  12. How to use a Forest-Decomposition {6} 1 {1,3,7} {9} {1,10} 6 2 5 {9,10} 7 10 {3,6,10} 9 8 {3,5,9} 3 4 • 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 )

  13. Acting Greedily again • 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 {6} 1 {1,3,7} {9} {1,10} 6 2 5 {9,10} 7 10 {3,6,10} 9 8 {3,5,9} 3 4

  14. And now more quickly... ) • 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

  15. Algorithm: Parent Dominating Set ) • 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(A2) approx. in O(log n) rounds w.h.p.

  16. Greedy-By-Degree: Pros'n'Cons general graphs: O(log2Δ) + very simple + running timeO(log Δ) + message size O(log log Δ) + uniform & deterministic - O(A log Δ) approx. general graphs: O(log Δ)

  17. Parent Dominating Set: Pros'n'Cons ) general graphs: O(log Δ) • + simple • + O(A2) approx. (deterministic) • +/- running time O(log n) (randomized) • open question: • Are there faster O(1) approx. for A2O(1)?

  18. Thank You!Questions & Comments?

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