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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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slide1

Minimum Dominating Set

Approximation

in Graphs of Bounded Arboricity

minimum dominating sets mds
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)?