Greedy Approximation with Non- submodular Potential Function

1. Greedy Approximation with Non-submodular Potential Function Ding-Zhu Du

2. Chapter 2. Greedy Strategy III. Nonsubmodular Potential Function

3. Connected Dominating Set Given a graph, find a minimum node-subset such that • each node is either in the subset or adjacent to a node in the subset and • subgraph induced by the subset is connected.

4. History • Two stage Greedy (ln Δ +3)-approximation (Guha & Khullar, 1998) where Δ is the maximum degree in input graph. • There is no polynomial time approximation with performance ratio (1-a) lnΔ 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 (Δ-1) )-approximation for any a > 0 with a nonsupmodular potential function.

5. What’s 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)

6. Example p(circles)=1 q(circles)=2

7. -p-q is not submodular x

8. Background • There 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.

9. 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.

10. How should we do with nonsubmodular functions?

11. Find a space to play your trick

12. Where is the space?

13. Why the inequality true?

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

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

17. When f is nonsubmodular

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

19. Algorithm CDS1

20. Connected Dominating Set Theorem. Algorithm CDS1 gives an (2+ ln Δ)-approximation, where Δ is the maximum node degree.

21. Proof

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

26. Algorithm

27. Analysis

28. > k <k <k <k

35. End Thanks!