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Improved Steiner Tree Approximation in Graphs SODA 2000

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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  1. Improved Steiner Tree Approximation in GraphsSODA 2000 Gabriel Robins (University of Virginia) Alex Zelikovsky (Georgia State University)

  2. 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

  3. 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

  4. 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]

  5. 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

  6. 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)

  7. 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

  8. 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

  9. 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)

  10. 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

  11. 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)

  12. 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)

  13. 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]

  14. 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

  15. 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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