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  1. Current Trends in Theory and Practice of Comp. Science, SOFSEM09 The minimum reload s-t path/trail/walk problems L. Gourvès, A. Lyra, C. Martinhon, J. Monnot Špindlerův Mlýn / Czech Republic

  2. Topics 1.Motivation and basic definitions 2.Minimum reload s-t walk problem; 3. Paths\trails with symmetric reload costs: Polynomial and NP-hard results. 4.Paths\trails with asymmetric reload costs: Polynomial and NP-hard results. 5.Conclusions and open problems

  3. Some applications involving reload costs 1. Cargo transportation network when the colors are used to denote route subnetworks; 2. Data transmission costs in large communication networks when a color specify a type of transmission; 3. Change of technology when colors are associated to technologies; etc

  4. Basic Definitions • Paths, trails and walks with minimum reload costs c b 1 1 5 R = 1 1 1 1 1 5 a s t Reload cost matrix d

  5. Basic Definitions • Minimum reload s-t walk c b 1 1 5 R = 1 1 1 1 1 5 a s t Reload cost matrix d c(W) 3

  6. Basic Definitions • Minimum reload s-t trail c b 1 1 5 R = 1 1 1 1 1 5 a s t Reload cost matrix d c(W) ≤ c(T) 3 4

  7. Basic Definitions • Minimum reload s-t path c b 1 1 5 R = 1 1 1 1 1 5 a s t Reload cost matrix d c(W) ≤ c(T) ≤ c(P) 3 4 5

  8. Basic Definitions • Symmetric or asymmetric reload costs rij = rji rij ≠ rji or for colors “i” and “j” • Triangle inequality (between colors) 1 2 rij ≤ rjk + rik y x z for colors 1,2,3 3 w

  9. Basic Definitions NOTE: Paths (resp., trails and walks) with reload costs generalize both properly edge-colored (pec) and monochromatic paths (resp., trails and walks). rij = 0, for i j and rii = 1 ≠ s t pec s-t path cost of the minimum reload s-t path is 0

  10. Basic Definitions NOTE: Paths (resp., trails and walks) with reload costs generalize both properly edge-colored (pec) and monochromatic paths (resp., trails and walks). rij = 1, for i j and rii = 0 ≠ s t monochomatic s-t path cost of the min. reload s-t path is 0

  11. Minimum reload s-t walk s 1 v2 2 v1 3 t c Minimum reload s-t walk in G Shortest s0-t0 path in H

  12. Minimum reload s-t walk s 1 v2 2 v1 3 t All cases can be solved in polynomial time !

  13. Minimum symmetric reload s-t trail Symmetric R 1 y v 2 1 x z c a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v)

  14. Minimum symmetric reload s-t trail Symmetric R 1 y v 2 1 x z c a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v)

  15. Minimum symmetric reload s-t trail Symmetric R 1 y v 2 1 x z c a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v) Minimum symmetric reload s-t trail Minimum perfect matching

  16. Minimum symmetric reload s-t trail Symmetric R 1 y v 2 1 x z c a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v) The minimum symmetric reload s-t trail can be solved in polynomial time !

  17. NP-completeness Theorem 1 The minimum symmetric reload s–tpath problem is NP-hard if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 4.

  18. Theorem 1 (Proof) • Reduction from the (3, B2)-SAT (2-Balanced 3-SAT) • Each clause has exactly 3 literals • Each variable apears exactly 4 times (2 negated and 2 unnegated) xi is false xi is true Gadget for clause Cj Gadget for literal xi

  19. Theorem 1 (Proof) C4 C3 C6 C5 literal x7

  20. Theorem 1 (Proof) C4 C3 C6 C5 Every other entries of R are set to 1

  21. Theorem 1 (Proof) s t

  22. Theorem 1 (Proof)

  23. Theorem 1 (Proof)

  24. Non-approximation Theorem 2 In the general case, the minimum symmetric reload s–tpath problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 4. We modify the reload costs, so that: OPT(Gc)=0 I is satisfiable. OPT(Gc) >M I is not satisfiable. In this way, to distinguish between OPT(Gc)=0 or OPT(Gc) ≥M is NP-complete, otherwise P=NP!

  25. Non-approximation Theorem 2 In the general case, the minimum symmetric reload s–tpath problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 4. r1,2 = r2,1 = M r2,2 = 0 Proof: r1,3 = r3,1 = 0 r1,1 = 0 r2,3 = r3,2 = 0

  26. Non-approximation (Proof) s r1,2 = r2,1 = M t

  27. Non-approximation Theorem 3 If , for every i,j the minimum symmetric reload s–tpath problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 4. r1,2 = r2,1 = M r2,2 = 1 Proof: r1,3 = r3,1 = 1 r1,1 = 1 r2,3 = r3,2 = 1

  28. Non-approximation Theorem 3 If , for every i,j the minimum symmetric reload s–tpath problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 4. Proof: It is NP –complete to distinguish between

  29. NP-Completeness • Corollary 4: • The minimum symmetric reload s–tpath problem is NP-hard if c ≥ 4, the graph is planar, the triangle inequality holds and the maximum degree is equal to 4.

  30. Corollary4 (Proof): c c b a f a b d r1,2 = r2,1 = M d c c’ c b b’ a’ a f b a d’ d d r3,4 = r4,3 = M

  31. Some polynomial cases Theorem 5 Consider Gc with c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality. Then, the minimum symmetric reload s–tpath problem can be solved in polynomial time.

  32. Some polynomial cases Theorem 5 Consider Gc with c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality. Then, the minimum symmetric reload s–tpath problem can be solved in polynomial time. If the triangle ineq. does not hold??

  33. Some polynomial cases • The minimum toll cost s–t path problem may be solved in polynomial time. • ∀ ri,j=rj, for colors i and j andri,i=0 toll points s s 0 t auxiliar vertex and edge

  34. NP-completeness Theorem 6 The minimum asymmetric reload s–ttrail problem is NP-hard if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 4.

  35. NP-completeness (Proof) • Reduction from the (3, B2)-SAT (2-Balanced 3-SAT) • Each clause has exactly 3 literals • Each variable apears exactly 4 times (2 negated and 2 unnegated) True False Clause graph Variable graph

  36. NP-completeness (Proof) x3 Reload costs = M

  37. Non-approximation Theorem 7 (a) In the general case, the minimum asymmetric reload s–ttrail problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 3. (b) If , for every i,j the minimum asymmetric reload s–t trail problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 3.

  38. Non-approximation Theorem 7 (a) In the general case, the minimum asymmetric reload s–t trail problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 3. (b) If , for every i,j the minimum asymmetric reload s–ttrail problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 3.

  39. Non-approximation Theorem 7 (a) In the general case, the minimum asymmetric reload s–ttrail problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 3. (b) If , for every i,j the minimum asymmetric reload s–ttrail problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 3.

  40. A polynomial case Theorem 8 Consider Gc with c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality. Then, the minimum asymmetric reload s–ttrail problem can be solved in polynomial time.

  41. A polynomial case Theorem 8 Consider Gc with c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality. Then, the minimum asymmetric reload s–ttrail problem can be solved in polynomial time. If the triangle ineq. does not hold??

  42. Conclusions and Open Problems

  43. Conclusions and Open Problems Problem 1 Input:Let be 2-edge-colored graph and 2 vertices Question: Does the minimum symmetric reload s-t path problem can be solved in polynomial time? Note: Ifthetriangleineq. holds Yes!

  44. Conclusions and Open Problems Problem 2 Input:Let be 2-edge-colored graph and 2 vertices Question: Does the minimum asymmetric reload s-t trail problem can be solved in polynomial time? Note: Ifthetriangleineq. holds Yes!

  45. Thanks for your attention!!