1. Minimum Dominating Set Approximation in Graphs of Bounded Arboricity

2. 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

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(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

4. 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

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

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 arboricity A of G

7. 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

8. 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

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
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 )

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

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?