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

Christoph 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

Christoph 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

Christoph 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

Christoph 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

Christoph 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.

Christoph 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

Christoph 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! Θ(log n) 1 2

Christoph 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

Christoph 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

Christoph 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

Christoph 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 {1,10} {1,3,7} {3,5,9} {9,10} {3,6,10} {9} {6} )

Christoph 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} {9} {6}

Christoph 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... )

Christoph 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 )

Christoph 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)

Christoph 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) )

ETH Zurich – Distributed Computing Group Roger Wattenhofer 18ETH Zurich – Distributed Computing – Christoph Lenzen Roger Wattenhofer Thank You! Questions & Comments?