Improved Steiner Tree Approximation in Graphs SODA 2000 Gabriel Robins (University of Virginia) Alex...

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

• Terminal-Spanning trees = 2-approximation• Full Steiner Components: Gain & Loss• k-restricted Steiner Trees• Loss-Contracting Algorithm • Derivation of Approximation Ratios

• Open Questions

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

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

1

1Cost = 2 Steiner PointCost = 31Terminals

1

1

Euclidean metric

11

1

1

11

1

11

1

Cost = 6 Cost = 4

Rectilinear metric

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]

optimal costachieved costApproximation Ratio = sup

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

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)

Terminals = S

Steiner points = P

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

Steiner tree

Terminal-Spanning Trees Terminal-spanning tree = Steiner tree without Steiner pointsMinimum terminal-spanning tree = minimum spanning tree

=> efficient greedy algorithm in any metric space

Theorem: MST-heuristic is a 2-approximationProof: MST < Shortcut Tour Tour = 2 • OPTIMUM

K T

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

C[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

FC K

Loss(K)

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)

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

opt

gain(K1)

gain(K2)

loss(K4)

loss(K3)

gain(K3)

gain(K4)

loss(K2)

loss(K1)

reductionof T cost

mst

opt

gain(K1)

gain(K2)

loss(K1)

loss(K2)

gain(K3)

gain(K4)

loss(K3)

loss(K4)

reductionMST(H)

Approximation RatioTheorem: Loss-Contracting Algorithm output tree

Approx optk + lossk ln (1+ ) lossk

mst - optk

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]

Derivation of Approximation Ratios

Approx optk + lossk ln (1+ ) lossk

mst - optk

General graphs 1+ ln3 /2mst 2•opt

partial derivative by lossk is always positive

lossk 1/2 optk

maximum is for lossk = 1/2 optk

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

lossk

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