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Graph-Theoretical Approach to Shunting Problems at Train Depots

This paper discusses the application of graph theory in solving train depot algorithms, focusing on train assignment and marshalling problems. It covers constraints, methods, and various topologies, and explores the equivalence with ordering problems. The complexity of coloring permutation and complement graphs is also examined. Conclusions are drawn regarding the online and offline aspects of train assignment.

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Graph-Theoretical Approach to Shunting Problems at Train Depots

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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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