Improved Steiner Tree Approximation in Graphs

8/3/2019 Improved Steiner Tree Approximation in Graphs

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Improved Steiner TreeApproximation in Graphs

Prof. Rushen Chahal

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Overview

Steiner Tree Problem

Results: Approximation Ratios general graphs

quasi-bipartite graphs

graphs with edge

-we

ights 1 & 2

Terminal-Spanning trees = 2-approximation

Full Steiner Components: Gain & Loss

k-restricted Steiner Trees

Loss-Contra

cting Algor

ithm Der ivation of Approximation Ratios

Open Questions

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Steiner Tree Problem

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

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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 cost

Approximation Ratio = sup

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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 / 2Cannot be approximated better than 1.004

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Approximation in Quasi-Bipartite

GraphsQuasi-bipartite graphs = all Steiner points are pairwisedisjoint

Approximation ratios:

1.5 +I [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

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Approximation in Complete Graphs

with Edge Costs 1 & 2

Approximation ratios:

1.333Rayward-Smith Heuristic [Bern & Plassmann, 1989]

1.295 using Lovasz algorithm for parity matroid problem[Furer, Berman & Zelikovsky, TR 1997]

This paper:

1.279 + I PTAS of k-restricted Loss-Contracting Heuristics

Steiner tree

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Terminal-Spanning TreesTerminal-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