ETH Zurich – Distributed Computing Group Roger Wattenhofer 1ETH Zurich – Distributed Computing – www.disco.ethz.ch
Christoph LenzenRoger Wattenhofer
Minimum Dominating SetApproximation
in Graphs of Bounded Arboricity
Christoph Lenzen@DISC 2010
Minimum Dominating Sets (MDS)
• 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
• important in theory and practice
minimum dominating setdominating set in a social network
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(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?
Garey & Johnson, '79
Feige, JACM '98 Raz & Safra, STOC '97
Kuhn & al., PODC '04
Kuhn & al., SODA '06
Christoph Lenzen@DISC 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
generalboundeddegree
O(1) approx.O(1) rounds
planar
O(1) approx.O(1) rounds
unitdisc
O(1) approx.Θ(log* n) rounds
boundedindependence
O(1) approx.O(log n) rounds
Θ(log n) approx.O(log2 Δ) rounds(log Δ) rounds
excludedminor
(1+²) approx.polylog n rounds
e.g. Luby SIAM J. Comp. '86
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:
+star graph:
n-n1/2 nodes
center covers all
arbitrary graph:
n1/2 nodes
difficult to handle
O(n) edges
=same lower
bounds as in
general case
Christoph Lenzen@DISC 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 = E1 [ E2 [ ... [ Ef 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 Lenzen@DISC 2010
Where are Graphs of Bounded Arboricity?
restrictive hard
generalboundeddegree
planar
unitdisc
boundedindependence
excludedminor
boundedarboricity
• 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
boundedarboricity
Czygrinow & al., DISC '08
no o(A) approx. in o(log* n) rounds
Christoph Lenzen@DISC 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!
8+2
7+2 7+2
5
5
4 4
4
3 3
2
11
Θ(log n)
1
2
Christoph Lenzen@DISC 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 Lenzen@DISC 2010
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
Greedy-By-Degree: Details
Christoph Lenzen@DISC 2010
• ...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 Lenzen@DISC 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
5
1
2
34
6
7
89
10
{1,10} {1,3,7}
{3,5,9}
{9,10}
{3,6,10}
{9}
{6}
)
Christoph Lenzen@DISC 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}
1
2
34
{9}6
7
89
10
{6}
Christoph Lenzen@DISC 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 Lenzen@DISC 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(A2) approx. in O(log n) rounds w.h.p.
Algorithm: Parent Dominating Set
)
Christoph Lenzen@DISC 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(log2 Δ)
general graphs:O(log Δ)
Christoph Lenzen@DISC 2010
Parent Dominating Set: Pros'n'Cons
+ simple
+ O(A2) approx. (deterministic)
+/- running time O(log n) (randomized)
• open question:
Are there faster O(1) approx. for A2O(1)?
general graphs:O(log Δ)
)
ETH Zurich – Distributed Computing Group Roger Wattenhofer 18ETH Zurich – Distributed Computing – www.disco.ethz.ch
Christoph LenzenRoger Wattenhofer
Thank You!Questions & Comments?
