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Improved Steiner Tree Approximation in Graphs SODA 2000. Gabriel Robins (University of Virginia) Alex Zelikovsky (Georgia State University). Overview. Steiner Tree Problem Results: Approximation Ratios general graphs quasi-bipartite graphs graphs with edge-weights 1 & 2

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Improved Steiner Tree Approximation in Graphs SODA 2000

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Improved Steiner Tree Approximation in GraphsSODA 2000

Gabriel Robins (University of Virginia)

Alex Zelikovsky (Georgia State University)


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


Euclidean metric

Steiner Point

Cost = 2

1

1

Cost = 3

Terminals

1

1

1

Rectilinear metric

Cost = 6

Cost = 4

1

1

1

1

1

1

1

1

1

1

Steiner Tree Problem

Given: A set S of points in the plane = terminals

Find:Minimum-cost tree spanning S = minimum Steiner tree


Approximation Ratio = sup

achieved cost

optimal cost

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

GeometricSTP NP-hard [Garey & Johnson, 1977]

but has PTAS [Arora, 1996]


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]

1+ln2 = 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


Steiner points = P

Terminals = S

Steiner tree

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)


Steiner tree

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


Terminal-Spanning Trees

Terminal-spanning tree = Steiner tree without Steiner points

Minimum terminal-spanning tree = minimum spanning tree

=> efficient greedy algorithm in any metric space

Theorem: MST-heuristic is a 2-approximation

Proof: MST<Shortcut TourTour = 2 • OPTIMUM


T

K

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(T+K)


FC K

Loss(K)

C[K]

FC K

Full Steiner Trees: Loss

  • Loss of FC K = cost of connection Steiner points to terminals

  • Loss-contracted FC C[K] = K with contracted loss


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) < 1/2 cost(K)


mst

mst

gain(K1)

gain(K1)

gain(K2)

loss(K1)

gain(K3)

gain(K2)

reduction

of T cost

reduction

MST(H)

gain(K4)

loss(K2)

gain(K3)

loss(K4)

loss(K3)

loss(K3)

gain(K4)

loss(K2)

loss(K4)

loss(K1)

opt

opt

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 + C[K])

Output MST(H)


mst - optk

Approx  optk + lossk ln (1+ )

lossk

Approximation Ratio

Theorem: Loss-Contracting Algorithm output tree

Proof idea:

New Lower Bound

Let H = Steiner tree and gain C[H] (K)  0 for any k-restricted FC K

Then cost(C[H]) = 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, 1993/1+ln 2-approximation]


mst - optk

Approx  optk + lossk ln (1+ )

lossk

Quasi-bipartite and complete with edge costs 1 & 2  1.28

mst  2•(optk - lossk) - it is not true for all graphs :-(

Approx  optk + lossk ln ( - 1)

partial derivative by lossk = 0 if x = lossk / (optk - lossk ) is root of 1 + ln x + x = 0

then upper bound is equal 1 + x = 1.279

optk

lossk

Derivation of Approximation Ratios

General graphs  1+ ln3 /2

mst  2•opt

partial derivative by lossk is always positive

lossk 1/2 optk

maximum is for lossk= 1/2 optk


Open Questions

  • Better upper bound (<1.55)

    • combine Hougardy-Promel approach with LCA

    • speed of improvement 3-4% per year

  • Better lower bound (>1.004)

    • 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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