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 1 Cost = 2 Steiner Point Cost =  3 1 Terminals 1 1 Euclidean metric Cost = 6Cost = 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 cost achieved cost Approximation Ratio = sup

Approximation Ratios in Graphs 2-approximation [3 independent papers, ] 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:  [Rajagopalan & Vazirani, 1999] This paper: 1.5 for the Batched 1-Steiner Heuristic [Kahng & Robins, 1992] 1.28 for Loss-Contracting Heuristic, runtime O(S 2 P) Terminals = S Steiner points = P Steiner tree

Approximation in Complete Graphs with Edge Costs 1 & 2 Approximation ratios: Rayward-Smith Heuristic [Bern & Plassmann, 1989] using Lovasz’ algorithm for parity matroid problem [Furer, Berman & Zelikovsky, TR 1997] This paper:  PTAS of k-restricted Loss-Contracting Heuristics Steiner tree

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

KT 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, gain T (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 opt k = Cost(optimal k-restricted Steiner tree) opt = Cost(optimal Steiner tree) Fact: opt k  (1+ 1/log 2 k) opt [Du et al, 1992] loss k = 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 = gain T (K) / loss(K) If r  0 then exit repeat H = H + K T = MST(T + C[K]) Output MST(H) mst opt gain(K 1 ) gain(K 2 ) loss(K 4 ) loss(K 3 ) gain(K 3 ) gain(K 4 ) loss(K 2 ) loss(K 1 ) reduction of T cost mst opt gain(K 1 ) gain(K 2 ) loss(K 1 ) loss(K 2 ) gain(K 3 ) gain(K 4 ) loss(K 3 ) loss(K 4 ) reduction MST(H)

Approximation Ratio Theorem: Loss-Contracting Algorithm output tree Approx  opt k + loss k ln (1+ ) loss k mst - opt k Proof idea –New Lower Bound: mst - gain(H) - loss(H)  opt k for any non-improvable steiner tree H –The total loss does not grow too fast: loss(H)  loss k ln (1+(mst-opt k )/loss k ) since greedy choice - similar to greedy for setcovering

Derivation of Approximation Ratios Approx  opt k + loss k ln (1+ ) loss k mst - opt k General graphs  1+ ln3 /2 mst  2 opt partial derivative by loss k is always positive loss k  1/2 opt k maximum is for loss k = 1/2 opt k Quasi-bipartite and complete with edge costs 1 & 2  1.28 mst  2 (opt k - loss k ) - it is not true for all graphs :-( Approx  opt k + loss k ln ( - 1) partial derivative by loss k = 0 if x = loss k / (opt k - loss k ) is root of 1 + ln x + x = 0 then upper bound is equal 1 + x = loss k opt k

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