A sharp threshold for minimum bound-depth/diameter spanning and Steiner trees in random networks

1. A sharp threshold forminimum bound-depth/diameterspanning and Steiner treesin random networks arXiv:0810.4908 Omer Angel Abraham Flaxman David B. Wilson U British Columbia U Washington Microsoft Research

2. Minimum spanning tree (MST) • Graph with nonnegative edge weights • Connected acyclic subgraph, minimizes sum of edge weights (costs) • Classical optimization problemelectric network, communication network, etc. • Efficiently computable: Prim’s algorithm (explore tree from start vertex) Kruskal’s algorithm (add edges in order by weight) 1 3 5 7 4 2 6

3. 2 1 3 3 2 1 4 MST on graph with random weights Clique K4 Weight distribution irrelevant to MST 12 trees like 4 trees like

4. MST on graph with random weights • Weight distribution irrelevant to MST • Not same as uniform spanning tree (UST) (e.g. non-uniform on K4) • Diameter of MST on Kn is (n1/3)[Addario-Berry, Broutin, Reed] • Diameter of UST on Kn is (n1/2) [Rényi, Szekeres] • Weight of MST with Exp(1) weights on Kn tends to (3) a.a.s. [Frieze] • PDF of edge weights 1 at 0  weight (3) [Steele]

5. Minimum bounded-depth/diameterspanning tree • Data in communication network, delay for each link, put a limit on number of links. • Also known as “MST with hop constraints” • Tree with depth  k from specified root has diameter  2k. Tree with diameter  2k has “center” from which depth is  k • NP-hard for any diameter bound between 4 and n-2, poly-time solvable for 2,3, & n-1 [Garey & Johnson] • Inapproximable within factor of O(log n) unless P=NP [Bar-Ilan, Kortsarz, Peleg]

6. Greedy Tree

9. Depth 2 Greedy Tree

10. Depth 3 Greedy Tree

11. Weight of tree vs. depth bound

13. Weight of tree vs. depth bound

14. Sharp threshold for depth bound

15. Sliced and spliced tree

16. Lower bound ingredients

17. Concentration of level weights

18. Minimum Steiner tree • In addition to graph, set of terminals is specified. Tree must connect terminals, may or may not connect other vertices. • Another classical optimization problem. • NP-hard to solve. 1 3 5 7 4 2 6

19. Steiner trees on Kn When there are m terminals and Exp(1) weights, the Steiner tree weight tends to when 2  m  o(n) [Bollobás, Gamarnik, Riordin, Sudakov] When m=(n), weight is unknown constant

20. Minimum bounded-depth/diameterSteiner tree • Generalizes two different NP-hard problems, is NP-hard • Solvable by integer programming [Achuthan-Caccetta, Gruber-Raidl] • Fast approximation algorithms [Bar-Ilan- Kortsarz-Peleg, Althus-Funke-Har-Peled- Könemann-Ramos-Skutella] • Heuristics [Abdalla-Deo-Franceschini, Dahl-Gouveia-Requejo, Voß, Gouveia, Costa-Cordeauc-Laporte, Raidl-Julstrom, Gruber-Raidl, Gruber-Van-Hemert-Raidl, Kopinitsch, Putz, Zaubzer, Bayati-Borgs-Braunstein-Chayes-Ramezanpour-Zecchina, …]

21. Same threshold for Steiner trees(with linear number of terminals)

22. Everything works for Steiner trees(with linear number of terminals)

23. Steiner trees withsub-linear number of terminals Don’t know asymptotic weight when depth bound is

24. Minimum bounded depth/diameter spanning subgraph • If depth-constrained, best subgraph is a tree, we give minimum weight • If diameter-constrained, best subgraph is not a tree, possible to get smaller weight

25. Optimization problemswith side-constraints Side-constraint (depth or diameter bound) has almost no effect on optimization (up to a point) http://arXiv.org/abs/0810.4908