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 Let H Steiner
tree and gain CH (K) ? 0 for any k-restricted
FC K Then cost(CH) cost(H) - loss(H) ? optk
With every iteration, cost(T) decreases by
gain(K)loss(K) Until cost(T) finally drops below
optk The total loss does not grow too fast
Similar techniques used in Ravi Klein,
1993/1ln 2-approximation
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