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AMORE meeting, 1-4 October, Leiden, Holland

A graph theoretical approach to shunting problems. L. Koci, G. Di Stefano Dipartimento di Ingegneria Elettrica, Università dell’Aquila, Italy. AMORE meeting, 1-4 October, Leiden, Holland. Train depot algorithms. Instance. Outcoming trains. Depot. Incoming trains.

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AMORE meeting, 1-4 October, Leiden, Holland

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  1. A graph theoretical approach to shunting problems L. Koci, G. Di Stefano Dipartimento di Ingegneria Elettrica, Università dell’Aquila, Italy AMORE meeting, 1-4 October, Leiden, Holland 3rd AMORE meeting, Leiden

  2. Train depot algorithms • Instance • Outcoming trains Depot Incoming trains 3rd AMORE meeting, Leiden

  3. Train depot algorithms • Goal • Arrange the trains on a minimum number of depot tracks • Put the trains in a “correct” order to minimize shunting operations 3rd AMORE meeting, Leiden

  4. Train depot algorithms • Constraints • Trains • sequences, types, lengths • Tracks • topologies, lengths 3rd AMORE meeting, Leiden

  5. Train depot algorithms • Methods • Combinatorics • Graph theory (reduction to graph problems) • Heuristics 3rd AMORE meeting, Leiden

  6. Train scheduling • Train arrival and departure time A: first incoming train B: second C: third B: first outcoming train A: second C: third C A B time Evening Midnight Morning 3rd AMORE meeting, Leiden

  7. Train numbering • Train arrival and departure time 2: first incoming train 1: second 3: third 1: first outcoming train 2: second 3: third 3 3 2 2 1 1 time Evening Midnight Morning 3rd AMORE meeting, Leiden

  8. Train depot algorithms Train assignment • Trains are numbered from 1 to N • Incoming train permutation  = [1 , 2 … N] • Each train i is represented by an integer • Outgoing train sequence S = [1, 2, ... N] • [1, 2, ... N] Depot [1 , 2 … N] ex. [3, 7, …, 5] 3rd AMORE meeting, Leiden

  9. Depot topologies • Shunting area 3rd AMORE meeting, Leiden

  10. Depot topologies • Marshalling area 3rd AMORE meeting, Leiden

  11. Train depot algorithms Train depot algorithms • Train assignment • Marshalling problem Evening 4 1 8 5 7 2 6 3 3rd AMORE meeting, Leiden

  12. Train depot algorithms Train depot algorithms • Train assignment • Marshalling problem 4 1 8 5 7 2 6 3 3rd AMORE meeting, Leiden

  13. Train depot algorithms Train depot algorithms • Train assignment • Marshalling problem 4 1 8 5 7 2 6 3 3rd AMORE meeting, Leiden

  14. Train depot algorithms Train depot algorithms • Train assignment • Marshalling problem 4 8 1 5 7 2 6 3 3rd AMORE meeting, Leiden

  15. Train depot algorithms Train depot algorithms • Train assignment • Marshalling problem Night 4 8 1 5 7 2 6 3 3rd AMORE meeting, Leiden

  16. Train depot algorithms Train depot algorithms • Train assignment • Marshalling problem Morning 4 8 5 7 1 2 6 3 3rd AMORE meeting, Leiden

  17. Train depot algorithms Train depot algorithms • Train assignment • Marshalling problem 4 8 5 7 1 2 6 3 3rd AMORE meeting, Leiden

  18. Train depot algorithms Train depot algorithms • Train assignment • Marshalling problem 1 2 3 4 5 6 7 8 3rd AMORE meeting, Leiden

  19. Ordering problem (1) The storage of N trains in a marshalling depot using the minimum number of tracks is equivalent to The ordering of a sequence of N numbers using the minimun number of queues 3rd AMORE meeting, Leiden

  20. Ordering problem (1) 4 4 1 8 5 7 2 6 3 1 S  • Train assignment  = [ 4 1 8 5 7 2 6 3 ] S = [ 1 2 3 4 5 6 7 8 ] 3rd AMORE meeting, Leiden

  21. Graph equivalence • Train assignment 4 3 1 6  = [ 4 1 8 5 7 2 6 3 ] 8 2 7 5 S = [ 1 2 3 4 5 6 7 8 ] Permutation graph Ordering problem Permutation graph coloring 3rd AMORE meeting, Leiden

  22. Colouring solution • Minimum colouring (colour = track) of a general graph is NP-complete  = [ 4, 1, 8, 5, 7, 2, 6, 3 ] 4 3 1 • Minimum colouring of a permutation graph is solved in O (n lg n) time 6 8 2 7 5 3rd AMORE meeting, Leiden

  23. Colouring solution • Train assignment  = [ 4, 1, 8, 5, 7, 2, 6, 3 ] 4 3 4 8 1 1 5 7 6 6 2 8 2 3 7 5 Depot Permutation graph 3rd AMORE meeting, Leiden

  24. Ordering problem (2) The storage of N trains in a shunting depot using the minimum number of tracks is equivalent to The ordering of a sequence of N numbers using the minimun number of stacks 3rd AMORE meeting, Leiden

  25. Ordering problem (2) 4 1 4 1 8 5 7 2 6 3  • Train assignment 4 3 1  = [ 4 1 8 5 7 2 6 3 ] 6 8 2 7 5 S = [ 1 2 3 4 5 6 7 8 ] 3rd AMORE meeting, Leiden

  26. Complement graph • Colouring 4 4 3 3 1 1 6 6 8 8 2 2 7 5 7 5 Complement permutation graph Permutation graph 3rd AMORE meeting, Leiden

  27. Coloring complexity • What is the complement graph of a permutation graph? Permutation graph (Marshalling area) Complement permutation graph (Shunting area) Colouring in O (n lg n) time 3rd AMORE meeting, Leiden

  28. 1 2 3 4 5 6 7 8 4 1 8 5 7 2 6 3 Train depot algorithms Train depot algorithms 4 3 • Train assignment • Shunting area 1 6 8 4 1 2 8 5 2 7 5 7 6 3 3rd AMORE meeting, Leiden

  29. Online Problem (3) • Train assignment • Offline • The algorithm is given the entire sequence of trains to store in the depot • Coloring of permutation graph • Online • When assigning a train to a depot track, there is no knowledge of the remaining incoming trains • Greedy assignment to tracks 3rd AMORE meeting, Leiden

  30. Train depot algorithms • Train assignment • conclusions • These train storage problems are ordering problems • The problems are equiv. to coloring of permutation graphs • The coloring is solvable in O (n lg n) [Pnueli et al., ‘71] • Offline solution  Online solution 3rd AMORE meeting, Leiden

  31. Circle Graphs • Definition: Intersection graphs of chords in a circle. a b b c d a c d 3rd AMORE meeting, Leiden

  32. with an equator equator Circle Graphs • Permutation Graphs are Circle Graphs  = [ 4 1 8 5 7 2 6 3 ] S = [ 1 2 3 4 5 6 7 8 ] 3rd AMORE meeting, Leiden

  33. Removing the “night” in a shunting area C A B time Evening Midnight Morning 3rd AMORE meeting, Leiden

  34. D Removing the “night” in a shunting area C A B time 3rd AMORE meeting, Leiden

  35. D Removing the “night” in a shunting area C A B time Let X and Y be two trains, and let IX and IY be the relative intervals If IX and IYoverlap (i.e. IX IY  but neitherIXIY nor IYIX) Then two different tracks for X and Y A B C 3rd AMORE meeting, Leiden

  36. D Transf. into a circle graph C A B time C B A D No equator that is No night 3rd AMORE meeting, Leiden

  37. D Transf. into a circle graph C A B time C B B A D A B A C C D Shunting tracks Circle graph 3rd AMORE meeting, Leiden

  38. D Transf. into a circle graph C A B time C B B A D D A C C D Shunting tracks Circle graph 3rd AMORE meeting, Leiden

  39. Train depot algorithms • Assignament on a shunting area without “night” • conclusions • This train storage problem is equiv. coloring of circle graphs • Coloring of circle graphs is NP-complete [GT4] • Is 2-Approx. but not (3/2)-Approx. • 3-coloring is in P; 4-coloring is NP-complete [Unger, 88] • The same problem for a marshalling area is solvable in O(n lg n) 3rd AMORE meeting, Leiden

  40. Generalized problems • Track access constraints • Single Input Single Output (SISO) • Double Input Single Output (DISO) • Single Input Double Output (SIDO) • Double Input Double Output (DIDO) station 3rd AMORE meeting, Leiden

  41. H Hypergraphs H = ( V , E ) V is a set of vertices E is a set of subsets (hyperedges) of V If all hyperedges have size k, H is called k-uniform 2-uniform hypergraphs are normal graphs 3rd AMORE meeting, Leiden

  42. Single Input Double Output • Train assignment • Two tracks are enough 4 8 7 6 3 4 1 8 5 7 2 6 3 8 7 6 5 4 3 2 1 1 5 2 3rd AMORE meeting, Leiden

  43. SIDO constraint • Train assignment • SIDO triple constraint 3 3 Can we use a single track? 2 2 2 1 3 1 1 No Evening Midnight Morning 3rd AMORE meeting, Leiden

  44. Single Input Double Output • Why? • triple constraint: three trains in the input sequence form a valley. Forbidden sequences: [2 1 3] and [3 1 2] admissible sequences : [1 2 3], [1 3 2], [2 3 1], and [3 2 1] 3rd AMORE meeting, Leiden

  45. Single Input Double Output • Input sequence representation:  =[ 4,1,8,5,7,2,6,3 ] 3rd AMORE meeting, Leiden

  46. Single Input Double Output • Modelling as a 3-uniform hypergraph: Valley hypergraph H() 3rd AMORE meeting, Leiden

  47. Single Input Double Output • Track assignament = coloring of H() At least two nodes in a hyperedge have a different color. 3rd AMORE meeting, Leiden

  48. 4 8 7 6 3 1 5 2 Single Input Double Output • SIDO track assignament  coloring of H() 3rd AMORE meeting, Leiden

  49. SIDO vs. DISO • These two problems are equivalent • Relation SIDO/DISO • Given an arbitrary train permutation  = [1 , 2 ... N] , the permutation index -1 = [1-1 , 2-1 , ... , N-1], and the time reversing operator R, s.t. ( -1) R = [N-1 , ... , 2-1, 1-1], then SIDO ()  DISO (-1)R 3rd AMORE meeting, Leiden

  50. SIDO/DISO conclusions • Coloring 3-uniform hypergraphs: • NP-hard • k-coloring is approx. within O(n/( lgk-1 n)2) [Hofmaister, Lefmann, ’98; ] • 2-coloring is NP-complete for 3-uniform hypergraphs [Approx. results: Krivelevich et al., 2001] • 2-coloring is in P for valley hypergraphs • k-coloring of valley hypergraphs ? Open! 3rd AMORE meeting, Leiden

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