Greedy Approximation with Non- submodular Potential Function

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Greedy Approximation with Greedy Approximation with Non-submodular Potential Non-submodular Potential Function Function Ding-Zhu Du Ding-Zhu Du
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Greedy Approximation with Non- submodular Potential Function. Ding-Zhu Du. Chapter 2. Greedy Strategy. III. Nonsubmodular Potential Function. Connected Dominating Set. Given a graph, find a minimum node-subset such that - PowerPoint PPT Presentation

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  • Greedy Approximation with Non-submodular Potential Function

    Ding-Zhu Du

  • Chapter 2. Greedy StrategyIII. Nonsubmodular Potential Function

  • Connected Dominating SetGiven a graph, find a minimum node-subset such that each node is either in the subset or adjacent to a node in the subset andsubgraph induced by the subset is connected.

  • HistoryTwo stage Greedy (ln +3)-approximation (Guha & Khullar, 1998) where is the maximum degree in input graph. There is no polynomial time approximation with performance ratio (1-a) ln unless NP in DTIME(n^{log log n}) for any a > 0. Now, we show that there is a one-stage Greedy (1+a)(1+ln (-1) )-approximation for any a > 0 with a nonsupmodular potential function.

  • Whats the Potential Function? f(A)=p(A)+q(A) p(A) = # of connected components after adding edges incident to A (supmodular) q(A) = # of connected components of the subgraph induced by A (nonsupmodular)

  • Example p(circles)=1 q(circles)=2

  • x-p-q is not submodular

  • BackgroundThere exist many greedy algorithms in the literature.Some have theoretical analysis. But, most of them do not.A greedy algorithm with theoretical analysis usually has a submodular potential function.

  • Is it true?Every previously known one-stage greedy approximation with theoretical analysis has a submodular (or supermodular) potential function. Almost, only one exception which is about Steiner tree.

  • How should we do with nonsubmodular functions?

  • Find a space to play your trick

  • Where is the space?

  • Why the inequality true?

  • ExampleIn a Steiner tree, all Steiner nodes form a dominating set.In a full Steiner tree, all Steiner nodes form a connected dominating set.

  • ObservationsThe submodularity has nothing to do with sequence chosen by the greedy algorithm. It is only about X1, , XoptThe ordering of X1, , Xopt is free to choose.

  • When f is nonsubmodular

  • IdeaOrganize optimal solution in such an ordering such that every prefix subsequence induces a connected subgraph, i.e.,For prefix subsequences A and B, xq(B)-xq(A) 1 for A supset of B

  • Algorithm CDS1

  • Connected Dominating SetTheorem. Algorithm CDS1 gives an (2+ ln )-approximation, where is the maximum node degree.

  • Proof

  • Connected Dominating SetTheorem. For any a >1, there is a a(1+ ln )-approximation for the minimum connected dominating set, where is the maximum node degree.

  • Algorithm

  • Analysis

  • EndThanks!