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
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