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Approximating Fault-Tolerant Domination in General Graphs

This paper discusses the intricacies of approximating fault-tolerant domination in general graphs through minimum dominating sets. The topic covers various established results, such as NP-hard lower bounds and approximation ratios for different types of dominating sets, including minimum tuple dominating sets. It analyzes the distinctions between tuple domination and standard domination, presenting key findings and methodologies, including greedy algorithms, and evaluates the effectiveness of these approaches in terms of approximation ratios. The implications for graph theory and algorithm design are also explored.

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Approximating Fault-Tolerant Domination in General Graphs

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  1. Approximating Fault-Tolerant Domination in General Graphs

  2. Minimum Dominating Set • Can be approximated with ratio • [Slavík, 1996] • [Chlebíkand Chlebíková, 2008] • NP-hard lower bound of • [Alon, Moshkovitz and Safra, 2006]

  3. 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]

  4. 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]

  5. Overview of the remaining Talk • -tuple domination vs-domination • NP-hard lower bound for -domination • Improved approximation ratio for -domination

  6. -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?

  7. -tuple Domination versus -Domination: With = 2 • At least nodes for a -tuple dom. set • But nodes suffice for a -dom. set

  8. -tuple Domination versus -Domination • : Off by a factor of nearly ! • For and Off by a factor (tight)

  9. 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

  10. 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

  11. When does the Greedy Algorithm finish? • Let a fixedoptimal solutionhavenodes • Greedydoesat least ofremainingworkper step • Ifitdoesmore, also good • Total amountofworkis • This gives an approximationratioofroughly

  12. When to stop when chopping off… • Whenischopping off oftheremainingworkineffective? • Whenremainingworkislessthan r • Thenatmost r morestepsareneeded • Stopchopping after steps • Gives an approx. ratioof

  13. Calculating the approximation ratio • does not looktoonice… • 1) • 2) • Yields: Approx. ratiooflessthan

  14. Extending the Domination Range • Insteadofdominatingthe1-neighborhood… • … dominatethe-neighborhood • Oftencalled-step domination cf. [Hage andHarary, 1996]

  15. Extending the Domination Range • The blacknodes form a 2-stepdominatingset • But not a 2-step 2-dominating set !

  16. Extending the Domination Range • Insteadofhavingdominatingnodes in the-neighborhood … • (unlessyouare in thedominatingset) • … havenode-disjointpathsoflengthatmost • Results in approximationratioof:

  17. Thank you

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