The Steiner problem with edge length 1 and 2

1. The Steiner problem with edge length 1 and 2 Source: Information Process Letter 32 (1989)171-176 Author: Marshall Bern and PaulPlassmann Reporter: Chih-Ying Lin (林知瑩)

2. Outline Introduction Problem Definition Preview Algorithm Example Ratio

3. Introduction G=(V,E) with edge length≧0，and a set N V of distinguished vertices. The Steiner problem asks for a minimum length tree within G that spans all members of N. NP-complete problem G=(9,11)

4. Rayward-Smith’s average distance heuristic (ADH) is a 4/3-approximation algorithm for this problem. • It is the first proof that a polynomial-time heuristic for an NP-complete Steiner problem achieves an approximation bound better than that given by a minimum spanning tree.

5. Problem Definition Steiner(1, 2)  In complete graph All length 1 or 2 The Steiner(1, 2) asks for a minimum length tree within G that spans all members of N.

6. Preview

7. Algorithm 1.Find a vertex v (optional or terminal) and a set S of terminals (possibly containing v) that minimize the average distance over all choices of v and S v=C S={C, F} minimize the average distance =[d(C,C)+d(C,F)]/(2-1)=1

8. 2.Replace S∪{v} by a single terminal vs and for each vertex u, let d(vs, u) be the minimum distance from u to a vertex of S∪{v}.

10. Example G=(9,36) and terminal node {C, D, H, I} Bold edge length =1 and unseen edge length=2

11. V=H, S={D,H, I} minimize the average distance ={d(H, D)+d(I, H))}/(3-1) =2/2=1

12. V=C, S={C, VS1} minimize the average distance ={d(C, VS1)}/(2-1) =2

14. Instance For an instance I of STEINER(1, 2) |I|:=the number of vertices (terminal and optional) |N|:=n ratio(I)= HEU(I)／OPT(I) HEU(I) and OPT(I) mean tree length.

15. 4/3-approximation algorithm for STEINER (1,2) G(V,H) complete graph Square:=terminal node Circle:=optional Bold edge length is 1,other unseen is 2.

16. ADH=(1+1)/1=2

17. ADH=2/1=2

18. OPT(I)

21. OPT(I)=12+5=17

22. HEU(I)

25. HEU(I)=22 Ratio(I)=22/17=1.29…

26. Ratio= HEU(I)/OPT(I)=[1/2(3n-2)]/(2n-2)=4/3

27. 4/3-approximation algorithm for STEINER (1,2) For an instance I of STEINER(1, 2) |I|:=the number of vertices(terminal and optional) n:=|N| HEU(I):=the length of the tree found by ADH assuming a worst possible order of breaking ties OPT(I):= the length of an optimal Steiner tree ratio(I)=HEU(I)/OPT(I)

28. Lemma 1 • A worst-case instance I contains no pair of terminals 1 apart.

29. Lemma 2 • For instance I, the average distance in each reduction is greater than 1.

30. Lemma 3 • If I contains a K-star for K≧3, then ratio (I) ≦4/3.

31. Lemma 4 • OPT(I)≧n +|P| +1 • P:=A minimum-cardinality set of vertices (optional or terminal ) that dominates all terminals in instance I.

32. Lemma 5 • OPT(I) ≧3/2n- 1/2q-1 • q:= The number of equivalence classes in this partition that contain three terminals.

33. Theorem ADH is a 4/3-approximation algorithm for STEINER(1, 2). HEU(I) ≦ 2n-s-2 OPT(I) ≧ 3/2n- 1/2q-1

34. Thank you