Minimum Dominating Set • Can be approximated with ratio • [Slavík, 1996] • [Chlebíkand 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 orhave 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 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
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
When does the Greedy Algorithm finish? • Let a fixedoptimal solutionhavenodes • Greedydoesat least ofremainingworkper step • Ifitdoesmore, also good • Total amountofworkis • This gives an approximationratioofroughly
When to stop when chopping off… • Whenischopping off oftheremainingworkineffective? • Whenremainingworkislessthan r • Thenatmost r morestepsareneeded • Stopchopping after steps • Gives an approx. ratioof
Calculating the approximation ratio • does not looktoonice… • 1) • 2) • Yields: Approx. ratiooflessthan
Extending the Domination Range • Insteadofdominatingthe1-neighborhood… • … dominatethe-neighborhood • Oftencalled-step domination cf. [Hage andHarary, 1996]
Extending the Domination Range • The blacknodes form a 2-stepdominatingset • But not a 2-step 2-dominating set !
Extending the Domination Range • Insteadofhavingdominatingnodes in the-neighborhood … • (unlessyouare in thedominatingset) • … havenode-disjointpathsoflengthatmost • Results in approximationratioof: