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