Distributed Construction of Minimum Connected Dominating ...
Minimum Dominating Set Approximation in Graphs of Bounded Arboricity
18
ETH Zurich – Distributed Computing Group Roger Wattenhofer 1 ETH Zurich – Distributed Computing – www.disco.ethz.ch Christoph Lenzen Roger Wattenhofer Minimum Dominating Set Approximation in Graphs of Bounded Arboricity
description
Minimum Dominating Set Approximation in Graphs of Bounded Arboricity. Minimum Dominating Sets (MDS). important in theory and practice. minimum dominating set. dominating set in a social network. graph G=(V,E) N(A) denotes inclusive neighborhood of A µ V - PowerPoint PPT Presentation
Transcript of Minimum Dominating Set Approximation in Graphs of Bounded Arboricity
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?
