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## Approximating Fault-Tolerant Domination in General Graphs

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**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: