ETH Zurich – Distributed Computing – www.disco.ethz.ch

Klaus-Tycho Förster

Approximating Fault-Tolerant Domination in General Graphs

Minimum Dominating Set

• Can be approximated with ratio– [Slavík, 1996]– [Chlebík and Chlebíková, 2008]

• NP-hard lower bound of – [Alon, Moshkovitz and Safra, 2006]

Minimum -tuple Dominating Set

minimum 2-tuple dominating set

• -tuple dominating set: – Every node should have dominating nodes in its neighborhood

[Harary and Haynes, 2000 and Haynes, Hedetniemi and Slater, 1998]

• Can be approximated with ratio– [Klasing and Laforest, 2004]

Minimum -Dominating Set

minimum 2-dominating set• -dominating set:

– Every node should be in the dominating set or have dominating nodes in its neighborhood[Fink and Jacobson, 1985]

• Best known approx.-ratio[Kuhn, Moscibroda and Wattenhofer, 2006]

Overview of the remaining Talk

• -tuple domination vs -domination

• NP-hard lower bound for -domination

• Improved approximation ratio for -domination

-tuple Domination versus -Domination

• -tuple dominating set only exists if min. degree

• Every -tuple dominating set is a -dominating set

• But how “bad” can a -tuple dom. set be in comparison?

-tuple Domination versus -Domination: With = 2

• At least nodes for a -tuple dom. set• But nodes suffice for a -dom. set

-tuple Domination versus -Domination

• : Off by a factor of nearly !

• For and Off by a factor (tight)

NP−hard   lower   bound   for  𝑘−domination• NP-hard lower bound for 1-domination

– [Alon, Moshkovitz and Safra, 2006]

• If we could approx. -dom. set with ratio of – Then build a -multiplication graph:

Example for

• NP-hard lower bound for -domination

Improved   approximation   ratio   for  𝑘−domination

• Utilizes a greedy-algorithm

• Use “degree” of per node– , if in the -dominating set– else #neighbors in the -dominating set, but at most

• Pick a node that improves total sum of degree the most

• Let a fixed optimal solution have nodes

• Greedy does at least of remaining work per step

• If it does more, also good

• Total amount of work is

• This gives an approximation ratio of roughly

When does the Greedy Algorithm finish?

• When is chopping off of the remaining work ineffective?

• When remaining work is less than r

• Then at most r more steps are needed

• Stop chopping after steps

• Gives an approx. ratio of

When to stop when chopping off…

• does not look too nice…

• 1)

• 2)

• Yields: Approx. ratio of less than

•

Calculating the approximation ratio

Extending the Domination Range

• Instead of dominating the 1-neighborhood…

• … dominate the -neighborhood

• Often called -step domination cf. [Hage and Harary, 1996]

Extending the Domination Range

• The black nodes form a 2-step dominating set

• But not a 2-step 2-dominating set !

Extending the Domination Range

• Instead of having dominating nodes in the -neighborhood …– (unless you are in the dominating set)

• … have node-disjoint paths of length at most

• Results in approximation ratio of:

ETH Zurich – Distributed Computing – www.disco.ethz.ch

Klaus-Tycho Förster

Thank you