# Greedy Approximation with Non- submodular Potential Function

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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!