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Improved Steiner Tree Approximation in Graphs SODA 2000

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Improved Steiner Tree Approximation in Graphs. SODA 2000. Gabriel Robins (University of Virginia) ... 1.5 for the Batched 1-Steiner Heuristic [Kahng & Robins, 1992] ... – PowerPoint PPT presentation

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Title: Improved Steiner Tree Approximation in Graphs SODA 2000


1
Improved Steiner Tree Approximation in
GraphsSODA 2000
  • Gabriel Robins (University of Virginia)
  • Alex Zelikovsky (Georgia State University)

2
Overview
  • Steiner Tree Problem
  • Results Approximation Ratios
  • general graphs
  • quasi-bipartite graphs
  • graphs with edge-weights 1 2
  • Terminal-Spanning trees 2-approximation
  • Full Steiner Components Gain Loss
  • k-restricted Steiner Trees
  • Loss-Contracting Algorithm
  • Derivation of Approximation Ratios
  • Open Questions

3
Steiner Tree Problem
  • Given A set S of points in the plane
    terminals
  • Find Minimum-cost tree spanning S minimum
    Steiner tree

4
Steiner Tree Problem in Graphs
  • Given Weighted graph G(V,E,cost) and
    terminals S ? V
  • Find Minimum-cost tree T within G spanning S
  • Complexity Max SNP-hard Bern Plassmann, 1989
  • even in complete graphs with
    edge costs 1 2
  • Geometric STP NP-hard Garey
    Johnson, 1977
  • but has PTAS Arora, 1996

5
Approximation Ratios in Graphs
  • 2-approximation 3
    independent papers, 1979-81
  • Last decade of the second millennium
  • 11/6 1.84 Zelikovsky
  • 16/9 1.78 Berman Ramayer
  • PTAS with the limit ratios
  • 1.73 Borchers Du
  • 1ln2 1.69 Zelikovsky
  • 5/3 1.67 Promel Steger
  • 1.64 Karpinski Zelikovsky
  • 1.59 Hougardy Promel
  • This paper
  • 1.55 1 ln3 / 2
  • Cannot be approximated better than 1.004

6
Approximation in Quasi-Bipartite Graphs
  • Quasi-bipartite graphs all Steiner points are
    pairwise disjoint

Approximation ratios 1.5 ? Rajagopalan
Vazirani, 1999 This paper 1.5 for the Batched
1-Steiner Heuristic Kahng Robins, 1992 1.28
for Loss-Contracting Heuristic, runtime O(S2P)
7
Approximation in Complete Graphs with Edge Costs
1 2
  • Approximation ratios
  • 1.333 Rayward-Smith Heuristic Bern Plassmann,
    1989
  • 1.295 using Lovasz algorithm for parity
    matroid problem
  • Furer, Berman Zelikovsky, TR 1997
  • This paper
  • 1.279 ? PTAS of k-restricted Loss-Contracting
    Heuristics

8
Terminal-Spanning Trees
  • Terminal-spanning tree Steiner tree without
    Steiner points
  • Minimum terminal-spanning tree minimum spanning
    tree
  • gt efficient greedy algorithm in any metric space

Theorem MST-heuristic is a 2-approximation Proof
MST lt Shortcut Tour ? Tour 2 OPTIMUM
9
Full Steiner Trees Gain
  • Full Steiner Tree all terminals are leaves
  • Any Steiner tree union of full components (FC)
  • Gain of a full component K, gainT(K) cost(T) -
    mst(TK)

10
Full Steiner Trees Loss
  • Loss of FC K cost of connection Steiner points
    to terminals
  • Loss-contracted FC CK K with contracted loss

11
k-Restricted Steiner Trees
  • k-restricted Steiner tree any FC has ? k
    terminals
  • optk Cost(optimal k-restricted Steiner tree)
  • opt Cost(optimal Steiner tree)
  • Fact optk ? (1 1/log2k) opt Du et al,
    1992
  • lossk Loss(optimal k-restricted Steiner
    tree)
  • Fact loss (K) lt 1/2 cost(K)

12
Loss-Contracting Algorithm
  • Input weighted complete graph G
  • terminal node set S
  • integer k
  • Output k-restricted Steiner tree spanning S
  • Algorithm
  • T MST(S)
  • H MST(S)
  • Repeat forever
  • Find k-restricted FC K maximizing
  • r gainT(K) / loss(K)
  • If r ? 0 then exit repeat
  • H H K
  • T MST(T CK)
  • Output MST(H)

13
Approximation Ratio
  • Theorem Loss-Contracting Algorithm output tree
  • Proof idea
  • New Lower Bound mst - gain(H) - loss(H) ? optk
    for any non-improvable steiner tree H
  • The total loss does not grow too fast
    loss(H) ? losskln
    (1(mst-optk)/lossk )
    since greedy choice - similar
    to greedy for setcovering

14
Derivation of Approximation Ratios
General graphs ? 1 ln3 /2 mst ? 2opt partial
derivative by lossk is always positive lossk ?
1/2 optk maximum is for lossk 1/2 optk
15
Open Questions
  • Better upper bound (lt1.55)
  • combine Hougardy-Promel approach with LCA
  • speed of improvement 3-4 per year
  • Better lower bound (gt1.004)
  • really difficult
  • thinnest gap 1.279,1.004
  • More time-efficient heuristics
  • Tradeoffs between runtime solution quality
  • Special cases of Steiner problem
  • so far LCA is the first working better for all
    cases
  • Empirical benchmarking / comparisons
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