### A sharp threshold for minimum bound

A sharp threshold for
minimum bound-depth/diameter
spanning and Steiner trees
in random networks
arXiv:0810.4908
Omer Angel Abraham Flaxman David B. Wilson
U British Columbia
U Washington
Microsoft Research
Minimum spanning tree (MST)
• Graph with nonnegative edge weights
• Connected acyclic subgraph,
minimizes sum of edge weights (costs)
• Classical optimization problem
electric network, communication network, etc.
• Efficiently computable:
Prim’s algorithm (explore tree from start vertex)
Kruskal’s algorithm (add edges in order by weight)
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MST on graph with random weights
Weight distribution irrelevant to MST
Clique K4
12 trees like
4 trees like
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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)
• 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]
Minimum bounded-depth/diameter
spanning 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]
Greedy Tree
Depth 2 Greedy Tree
Depth 3 Greedy Tree
Sharp threshold for depth bound
Sliced and spliced tree
Lower bound ingredients
Concentration of level weights
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.
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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
Minimum bounded-depth/diameter
Steiner tree
• Generalizes two different NP-hard problems, is
NP-hard
• Solvable by integer programming [AchuthanCaccetta, Gruber-Raidl]
• Fast approximation algorithms [Bar-IlanKortsarz-Peleg, Althus-Funke-Har-PeledKönemann-Ramos-Skutella]
• Heuristics [Abdalla-Deo-Franceschini, DahlGouveia-Requejo, Voß, Gouveia, CostaCordeauc-Laporte, Raidl-Julstrom, Gruber-Raidl,
Gruber-Van-Hemert-Raidl, Kopinitsch, Putz,
Zaubzer, Bayati-Borgs-Braunstein-ChayesRamezanpour-Zecchina, …]
Same threshold for Steiner trees
(with linear number of terminals)
Everything works for Steiner trees
(with linear number of terminals)
Steiner trees with
sub-linear number of terminals
Don’t know asymptotic weight when depth bound is
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
Optimization problems
with side-constraints
Side-constraint (depth or diameter bound)
has almost no effect on optimization
(up to a point)
http://arXiv.org/abs/0810.4908