Minimum Dominating Set Approximation in Graphs of Bounded

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

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 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 Kuhn & al. , SODA '06 Raz & Safra, STOC '97 Kuhn & al. , PODC '04 Christoph Lenzen@DISC 2010

MDS on Restricted Families of Graphs excluded planar Schneider & Wattenhofer, minor PODC '08 O(1) approx. O(1) rounds bounded degree O(1) approx. O(1) rounds unit disc O(1) approx. Θ(log* n) rounds (1+²) approx. polylog n rounds bounded independence O(1) approx. O(log n) rounds restrictive general Θ(log n) approx. O(log 2 Δ) rounds (log Δ) rounds L. '08 L. et et al al SPAA DISC '08 hard e. g. Luby SIAM J. Comp. '86 Czygrinow & Hańćkowiak, ESA '06 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: O(n) edges = + star graph: n-n 1/2 nodes center covers all same lower bounds as in general case arbitrary graph: n 1/2 nodes difficult to handle Christoph Lenzen@DISC 2010

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 = E 1 [ E 2 [. . . [ Ef into f forests minimum number of forests is arboricity A of G Christoph Lenzen@DISC 2010

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

Be Greedy! • sequentially add nodes covering most others ) yields O(log Δ) approx. 45 8+2 2 Θ(log n) 51 2 7+2 3 4 1 7+2 41 3 • . . . but in parallel? ) Just take all high-degree nodes! • repeat until finished Christoph Lenzen@DISC 2010

Why does Greedy-By-Degree work? V D C M 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 ¸ 4 A and A 2 O(1) ) |D| 2 O(|M|) Christoph Lenzen@DISC 2010

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

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 Barenboim & Elkin, PODC '08 • note: we cannot handle each forest individually Christoph Lenzen@DISC 2010

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 {6} 1 6 5 9 {1, 3, 7} {9} ) {9, 10} 7 10 4 2 {1, 10} {3, 6, 10} 8 3 {3, 5, 9} Christoph Lenzen@DISC 2010

Acting Greedily again • sequentially, an A approx. is trivial: Ø Ø 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 6 5 9 {1, 3, 7} {9, 10} 7 10 4 2 {1, 10} {3, 6, 10} 8 3 {3, 5, 9} Christoph Lenzen@DISC 2010

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 ) Christoph Lenzen@DISC 2010

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(A 2) approx. in O(log n) rounds w. h. p. ) Christoph Lenzen@DISC 2010

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

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

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