Minimum Dominating Set Approximation in Graphs of Bounded Arboricity

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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

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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 AµV
• DµV is dominating set (DS) iff V=N(D)
• minimum dominating set is DS of minimum size
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(log2Δ) 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

MDS on Restricted Families of Graphs

excluded

minor

Schneider & Wattenhofer, PODC \'08

bounded

independence

hard

restrictive

L. et al DISC \'08

planar

O(1) approx.

O(1) rounds

(1+²) approx.

polylog n rounds

general

bounded

degree

Θ(log n) approx.

O(log2Δ) 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

e.g. Luby SIAM J. Comp. \'86

Czygrinow & Hańćkowiak, ESA \'06

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

arbitrary graph:

n1/2 nodes

difficult to handle

star graph:

n-n1/2 nodes

center covers all

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 arboricityA of G
Where are Graphs of Bounded Arboricity?

bounded

independence

hard

restrictive

no o(A) approx. in o(log* n) rounds

• 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

bounded

arboricity

excluded

minor

planar

general

bounded

degree

unit

disc

Czygrinow & al., DISC \'08

Be Greedy!

2

4

5

8+2

Θ(log n)

5

1

2

4

7+2

3

1

4

1

7+2

3

• sequentially add nodes covering most others
• ) yields O(log Δ) approx.
• ...but in parallel?
• ) Just take all high-degree nodes!
• repeat until finished
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

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

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

How to use a Forest-Decomposition

{6}

1

{1,3,7}

{9}

{1,10}

6

2

5

{9,10}

7

10

{3,6,10}

9

8

{3,5,9}

3

4

• 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

)

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

{1,3,7}

{9}

{1,10}

6

2

5

{9,10}

7

10

{3,6,10}

9

8

{3,5,9}

3

4

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
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.
Greedy-By-Degree: Pros\'n\'Cons

general graphs:

O(log2Δ)

+ very simple

+ running timeO(log Δ)

+ message size O(log log Δ)

+ uniform & deterministic

- O(A log Δ) approx.

general graphs:

O(log Δ)

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)?