Minimum Dominating Set Approximation in Graphs of Bounded Arboricity

Minimum Dominating Sets (MDS) important in theory and practice dominating set in a social network minimum dominating set 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

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(log2 Δ) 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

MDS on Restricted Families of Graphs excluded minor planar L. et al DISC '08 Schneider & Wattenhofer, PODC '08 O(1) approx. O(1) rounds (1+²) approx. polylog n rounds bounded degree general bounded independence Θ(log n) approx. O(log2 Δ) 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 restrictive hard e.g. Luby SIAM J. Comp. '86 Czygrinow & Hańćkowiak, ESA '06

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 star graph: n-n1/2 nodes center covers all arbitrary graph: n1/2 nodes difficult to handle

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 arboricity A of G

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

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

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

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

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

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

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 {9} {1,3,7} 6 {1,10} 5 2 7 {9,10} 10 9 8 {3,6,10} 3 {3,5,9} 4

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 )

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. )

Greedy-By-Degree: Pros'n'Cons general graphs: O(log2 Δ) + very simple + running time O(log Δ) + message size O(log log Δ) + uniform & deterministic - O(A log Δ) approx. general graphs: O(log Δ)

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)?

Thank You! Questions & Comments?