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The Internal Steiner Tree Problem: Hardness and Approximations

The Internal Steiner Tree Problem: Hardness and Approximations. Sun-Yuan Hsieh ( 謝孫源 ) Department of Computer Science and Information Engineering Institute of Medical Informatics Institute of Manufacturing Information and Systems National Cheng Kung University hsiehsy@mail.ncku.edu.tw

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The Internal Steiner Tree Problem: Hardness and Approximations

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  1. The Internal Steiner Tree Problem: Hardness and Approximations Sun-Yuan Hsieh (謝孫源) Department of Computer Science and Information Engineering Institute of Medical Informatics Institute of Manufacturing Information and Systems National Cheng Kung University hsiehsy@mail.ncku.edu.tw http://algorithm.csie.ncku.edu.tw

  2. Outline • Introduction • What is the Steiner tree? • applications of the Steiner tree • variants of the Steiner tree • MAX SNP-hardness • Approximation algorithm • ISTP(1,2) • Concluding remarks S.-Y. Hsieh @ 30th CMCT, NDHU

  3. Introduction (1/11) • What is the Steiner tree? • Given a graph G = (V, E) with a cost (or distance) function c: E R+, and a vertex subset R  V, a Steiner tree is a connected and acyclic subgraph contains all the vertices in R (terminals). 2 2 5 7 4 2 2 7 8 4 6 3 5 3 3 G  R S.-Y. Hsieh @ 30th CMCT, NDHU

  4. Introduction (2/11) • Steiner tree problem • the decision version: NP-complete [Karp, 1972 Complexity of Computer Computations] • in the Euclidean metric: NP-complete [Garey et al., 1977 SIAM J. APPL. MATH.] • in the rectilinear metric: NP-complete [Garey and Johnson, 1977 SIAM J. APPL. MATH.] • MAX SNP-hard [Bern and Plassmann, 1989 IPL] • 1.55-approximation algorithm [Robins and Zelikovsky, 2000 SODA] • 1.39-approximation algorithm [Byrka et al., 2010 STOC] • 1.25-approximation algorithm for STP(1, 2) [Berman et al., 2009 WADS] S.-Y. Hsieh @ 30th CMCT, NDHU

  5. Introduction (3/11) • Applications • VLSI design • local or global routing • placement problem • engineering change order (ECO) S.-Y. Hsieh @ 30th CMCT, NDHU

  6. Introduction (4/11) • Applications • computational biology - reconstruction of phylogenetic • minimize the tree length according to the principle of parsimony (i.e., nature always finds the paths that require a minimum evolution) extant taxa Dolphin extinct ancestral taxa Cow Pig Alpaca Cat The length of edge in T as the evolutionary time Dog Horse S.-Y. Hsieh @ 30th CMCT, NDHU

  7. Introduction (5/11) • Applications • network routing - resource allocation resource S.-Y. Hsieh @ 30th CMCT, NDHU

  8. Introduction (6/11) • Applications • telecommunications • wireless communications, transportation sender receiver transmitter S.-Y. Hsieh @ 30th CMCT, NDHU

  9. Introduction (7/11) 2 2 5 7 4 7 8 4 6 3 5 3  R G TST S.-Y. Hsieh @ 30th CMCT, NDHU

  10. Introduction (8/11) 2 2 5 7 4 7 8 2 6 3 4 5 3  R’  R G PTST S.-Y. Hsieh @ 30th CMCT, NDHU

  11. Introduction (9/11) 2 2 5 7 4 7 8 2 6 3 4 5 3 5  R G IST S.-Y. Hsieh @ 30th CMCT, NDHU

  12. Introduction (10/11) • Selected Internal Steiner tree problem (SISTP): R’  R  V • each vertex in R’ must be an internal vertex • NP-complete and MAX SNP-hard [Hsieh and Yang, 2007 TCS] • (1 + )-approximation algorithm [Li et al., 2010 Algorithmica] 2 2 5 7 4 7 8 6 3 4 5 3 5 G SIST  R’  R S.-Y. Hsieh @ 30th CMCT, NDHU

  13. Introduction (11/11) S.-Y. Hsieh @ 30th CMCT, NDHU

  14. MAX SNP-hardness (1/12) • Any MAX SNP-hard problem has a PTAS, then P = NP. • Journal of the Association for Computing Machinery 1998. • How to show a problem to be MAX SNP-hard: by providing an L-reduction from some MAX SNP-hard problem to it. • Lemma 1. STP(1, 2) is MAX SNP-hard. • Bern and Plassmann, Information Processing Letters 32(4), 1989. • We prove that ISTP is MAX SNP-hard by providing an L-reduction from STP(1, 2) to ISTP. S.-Y. Hsieh @ 30th CMCT, NDHU

  15. MAX SNP-hardness (2/12) • L-reduction: P1and P2 : two optimization problems with input instances I1 and I2 P1L-reduces to P2 if the following conditions are satisfied: • Polynomial-time Algorithm A1 produces an instance A1(I1) for P2 such that OPT(A1(I1))  c*OPT(I1) • Given any solution of A1(I1) with cost2, polynomial-time Algorithm A2 produces a solution for I1 of P1 with cost1 such that |cost1 – OPT(I1)|  d* |cost2 – OPT(A1(I1))| S.-Y. Hsieh @ 30th CMCT, NDHU

  16. MAX SNP-hardness (3/12) • Algorithm A1(I1) • Input: An instance I1 = (G1, R1, c1) of STP(1,2), where G1 = (V1, E1) and R1 = {r1, r2,…, rk} for k 2. • Output: An instance I2 = (G2, R2, c2) of ISTP. I1 = (G1, R1, c1) I2 = (G2, R2, c2) 1 1 1 1 2 1 1 2 1 1 1 1 L-reduce 1 1  dummy edge set AE  R  dummy vertex set AV S.-Y. Hsieh @ 30th CMCT, NDHU

  17. MAX SNP-hardness (4/12) S.-Y. Hsieh @ 30th CMCT, NDHU

  18. MAX SNP-hardness (5/12) • Lemma 3. If TI* is a minimum internal Steiner tree, then TI* contains at least one dummy vertex, and each dummy vertex must be a leaf in TI*. Pf: (By contradiction) TI*: minimum internal Steiner tree of I2; AV(TI*) =  According to Algorithm A1(I1), each dummy vertex must be a leaf in TI*. TI* rj v aj delete S.-Y. Hsieh @ 30th CMCT, NDHU

  19. MAX SNP-hardness (6/12) • Lemma 4. If TS* is a minimum Steiner tree of I1, then TS* contains at least one leaf-terminal. Pf: (By contradiction) delete delete TS*  R S.-Y. Hsieh @ 30th CMCT, NDHU

  20. MAX SNP-hardness (7/12) • Lemma 5. Suppose that TI* is a minimum internal Steiner tree. If we delete each dummy vertex from TI*, the resulting graph will be a minimum Steiner tree. delete dummy vertex TI* minimum internal Steiner tree minimum Steiner tree  R dummy vertex S.-Y. Hsieh @ 30th CMCT, NDHU

  21. MAX SNP-hardness (8/12) • Lemma 5. Suppose that TI* is a minimum internal Steiner tree. If we delete each dummy vertex from TI*, the resulting graph will be a minimum Steiner tree. Pf: (By contradiction) TS*: minimum Steiner tree of I1 TI* delete TS* T by c(T) – c(TS*)  1 S.-Y. Hsieh @ 30th CMCT, NDHU

  22. MAX SNP-hardness (9/12) • Algorithm A2 • Input: An internal Steiner tree TI of I2=A1(I1), where I1 = (G1, R1, c1). • Output: A Steiner tree TS of I1. TI does not contain an dummy vertex TI contains dummy vertices delete delete S.-Y. Hsieh @ 30th CMCT, NDHU

  23. MAX SNP-hardness (10/12) S.-Y. Hsieh @ 30th CMCT, NDHU

  24. MAX SNP-hardness (11/12) S.-Y. Hsieh @ 30th CMCT, NDHU

  25. MAX SNP-hardness (12/12) S.-Y. Hsieh @ 30th CMCT, NDHU

  26. Approximation Algorithm S.-Y. Hsieh @ 30th CMCT, NDHU

  27. Approximation algorithm (1/11) • Lemma 7. Suppose that T is a Steiner tree with |V(T) \ R|  2. If v is a leaf-terminal of T, then there exists an internal vertex v  V(T) such that one of the following two conditions holds: • 1) degT(v) = 2 and v  R; or • 2) degT(v)  3. R |V(T) \ R|  1 < 2  S.-Y. Hsieh @ 30th CMCT, NDHU

  28. Approximation algorithm (2/11) • Algorithm AISTP • Input: A complete graph G = (V, E) with a metric cost function c: E R+ and a proper subset R  V of terminals such that |V\R|  2. • Output: An internal Steiner tree TI. Step 1. Find a Steiner tree SA in G Use the currently best-known approximation algorithm leaf-terminal SA  R S.-Y. Hsieh @ 30th CMCT, NDHU

  29. Approximation algorithm (3/11) • Algorithm AISTP • Input: A complete graph G = (V, E) with a metric cost function c: E R+ and a proper subset R  V of terminals such that |V\R|  2. • Output: An internal Steiner tree TI. Step 2. Transform T into an internal Steiner tree leaf-terminal critical vertex v target vertex v SA  R S.-Y. Hsieh @ 30th CMCT, NDHU

  30. Approximation algorithm (4/11) • Lemma 8. v: a leaf-terminal of the current tree T selected for processing. Then, • 1) degT(v) will be reduced by 1 after the iteration. • 2) degT(v) will be unchanged after the iteration. • 3) degT(v) will be increased by 1, and fixed at 2 until the algorithm terminates. Pf: (1) degT(v) = 2 and v  R (2) degT(v)  3 critical vertex v vtarget vertex reduced by 1 unchanged increased by 1, and fixed at 2 v S.-Y. Hsieh @ 30th CMCT, NDHU

  31. Approximation algorithm (5/11) v0 vk v0 vk critical vertex v v0 vk target vertex v PT  k S.-Y. Hsieh @ 30th CMCT, NDHU

  32. Approximation algorithm (6/11) u u SA v v S.-Y. Hsieh @ 30th CMCT, NDHU

  33. Approximation algorithm (7/11) according to step 2(b), v  PT[v, v] v v v PT by Lemma 9(2) v v v S.-Y. Hsieh @ 30th CMCT, NDHU

  34. Approximation algorithm (8/11) v3 vk–1 v2 vk v1 S.-Y. Hsieh @ 30th CMCT, NDHU

  35. Approximation algorithm (9/11) • Theorem 2. Algorithm AISTP is a (2 + 1)-approximation algorithm for the internal Steiner tree problem on complete graphs, where  is the approximation ratio of the best-known algorithm for the Steiner tree problem. Pf: TI*: minimum internal Steiner tree TS*: minimum Steiner tree Use a -approximation algorithm to find a Steiner tree SA and transform it into an internal Steiner tree. c(SA)  c(TS*) Since TI* is also a Steiner tree for R. c(TS*)  c(TI*)  c(SA)  c(TI*) () S.-Y. Hsieh @ 30th CMCT, NDHU

  36. Approximation algorithm (10/11) /* by () */ S.-Y. Hsieh @ 30th CMCT, NDHU

  37. Approximation algorithm (11/11) • Theorem 3. Algorithm AISTP can be implemented to run in O(n2logn) + f(m, n) time, where f(n, m) is the time complexity of the best-known approximation algorithm for the Steiner tree problem in an input graph with n vertices and m edges. Pf: n: number of vertices; m: number of edges Step 1. f(n, m): the best-known approximation algorithm Step 2. O(mlogn): adding the needed vertices and edges by sorting the cost O(|V(T)+|E(T)|): select the nearest vertex v using a depth-first search  O(|R|(|V(T)|+|E(T)|) = O(n2) O(degT(v)): choose a vertex v  O(vRdegT(v)) = O(|E(T)|) = O(n) • O(mlogn) + O(n2) + f(n, m) = O(n2logn) + f(n, m) since an input graph is a complete graph S.-Y. Hsieh @ 30th CMCT, NDHU

  38. Approximation Algorithm for ISTP(1,2) S.-Y. Hsieh @ 30th CMCT, NDHU

  39. ISTP(1,2) (1/7) • We restrict the co-domain of the cost function to the range R+ to {1, 2}. • Lemma 14. Let T be a minimum (1,2)-internal Steiner tree and let v be an internal vertex in T. Then the following statements hold. • 1) If v is a terminal (i.e., v R), then at most one leaf will be adjacent it. • 2) If v is a Steiner vertex, then no leaf will be adjacent it. (1) (2) TI TI v v delete  R delete S.-Y. Hsieh @ 30th CMCT, NDHU

  40. ISTP(1,2) (2/7) • Lemma 15. Let T be a minimum (1,2)-internal Steiner tree and let v be an internal vertex in T. If v is a Steiner vertex, then there exists a minimum (1,2)-internal Steiner tree T’ that does not contain v. |Ti|  2 Ti contains at least one leaf delete (m – 2) + m v add 2(m – 1) • u  R •  not an internal • u is Steiner vertex • delete u, obtain another lower • cost internal Steiner tree u Tm–1 T1 Tm T2 S.-Y. Hsieh @ 30th CMCT, NDHU

  41. ISTP(1,2) (3/7) • Lemma 16. If T is a minimum (1,2)-internal Steiner tree with k 3 leaves, then there exists a minimum (1,2)-internal Steiner tree T’ with k – 1 leaves. v delete (v, t1) and (u2, l2) add (t1, u2) t1 t2 t3 u1 u2 u3 T1 T2 T3 l1 l2 l3 S.-Y. Hsieh @ 30th CMCT, NDHU

  42. ISTP(1,2) (4/7) • Lemma 17. There is an optimal solution for ISTP(1,2), which is a minimum cost path whose internal vertices are all terminals in G and whose end-vertices are two Steiner vertices in G. R S.-Y. Hsieh @ 30th CMCT, NDHU

  43. ISTP(1,2) (5/7) • ri, rj  R, u, v  Steiner vertices, let G’ = G[R  {u, v}] • Lemma 18. Let T* be a minimum (1,2)-internal Steiner tree of G. Then c(T*)  c(HCG’) – 2. Pf: c(T*) = c(HPG’[u, v])  c(HCG’) – c(u, v)  c(HCG’) – 2 S.-Y. Hsieh @ 30th CMCT, NDHU

  44. ISTP(1,2) (6/7) Lemma 19. c(PG’)  c(AG’) c(PG’) = c(AG’) – c(u, u’) – c(v, v’) + c(u’, v’)  c(AG’) v’ u AG’ v u’ PG’  R S.-Y. Hsieh @ 30th CMCT, NDHU

  45. ISTP(1,2) (7/7) S.-Y. Hsieh @ 30th CMCT, NDHU

  46. Concluding Remarks S.-Y. Hsieh @ 30th CMCT, NDHU

  47. Thank you for your attention!

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