Report

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) 1 3 5 4 2 6 7 MST on graph with random weights Weight distribution irrelevant to MST Clique K4 12 trees like 4 trees like 1 2 3 3 1 2 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] 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. 1 3 4 2 5 6 7 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