1. Paths and Trails in Edge Colored Graphs Latin-American on Theoretical Informatics Symposium LATIN 2008 Abouelaoualim, K. Das, L. Faria, Y. Manoussakis, C. Martinhon, R. Saad Buzios-RJ - Brazil

2. Topics 1. Motivation and basic definitions 2. Properly edge-colored s-t path/trail and extensions 3. NP-completeness 4. Approximation Algorithms for associated maximization problems 5. Some instances solved in polynomial time 6. Conclusions and open problems

3. 1. Computational Biology when the colors are used to denote a sequence of chromosomes; 2. Cryptography when a color specify a type of transmission; 3. Social Sciences where a color represents a relation between 2 individuals; etc Some Applications using edge colored graphs

4. Basic Definitions Prop. edge-colored path between « s » and « t » t source destination 2 3 s 4 (without node repetitions!!) 1

5. Basic Definitions Prop. edge-colored trail between « s » and « t » t source destination 2 3 s 4 (without edge repetitions!!) 1

6. Basic Definitions Properly edge-colored cycle passing by « x » 5 start 2 3 x 4 (without node repetitions!!) 1

7. Basic Definitions Prop. edge-colored closed trail passing by « x » 5 start 2 3 x 4 (without edge repetitions!!) 1

8. Basic Definitions Almost prop. edge-colored cycle passing by « x » (without node repetitions!!) 5 start 2 3 x 4 1

9. Basic Definitions Almost properly edge-colored closed trail passing by « x » (without edge repetitions!!) 5 start 2 3 x 4 1

10. How to find a properly edge-colored s-t path? source destination 23 s 4 1 2-edge-colored graph G t

11. source destination 23 s 4 1 2-edge-colored graph G Graph G’ bluered 3’’ s 2’’ 3’ 4’’ 4’ 1’ t 1’’ 2’ t We find a perfect matching (if possible) !! How to find a properly edge-colored s-t path?

12. source destination 23 s 4 1 2-edge-colored graph G Graph G’ bluered 3’’ s 2’’ 3’ 4’’ 4’ 1’ t 1’’ 2’ t How to find a properly edge-colored s-t path? t a pec s-t path in G G’ contains a perfect matching Therem: Jensen&Gutin[1998]

13. t start u s q v p dest. color 1 color 2 color 3 papa v’v’’u’u’’ vava vbvb v1v1 v2v2 q’q’’ qaqa qbqb q2q2 q3q3 p’p’’ pbpb pcpc p1p1 p2p2 p3p3 uaua ubub ucuc u1u1 u2u2 u3u3 st (a) 3-edge colored graph (b) non-colored graph Properly s-t path in edge-colored graphs (Szeider’s Algorithm – [2003])

14. color 1 color 2 color 3 papa v’v’’u’u’’ vava vbvb v1v1 v2v2 q’q’’ qaqa qcqc q2q2 q3q3 p’p’’ pbpb pcpc p1p1 p2p2 p3p3 uaua ubub ucuc u1u1 u2u2 u3u3 st (a) 3-edge colored graph (b) non-colored graph Properly s-t path in edge-colored graphs (Szeider’s Algorithm – [2003]) t start u s q v p dest.

15. color 1 color 2 color 3 papa v’v’’u’u’’ vava vbvb v1v1 v2v2 q’q’’ qaqa qcqc q2q2 q3q3 p’p’’ pbpb pcpc p1p1 p2p2 p3p3 uaua ubub ucuc u1u1 u2u2 u3u3 st (a) 3-edge colored graph (b) non-colored graph Properly s-t path in edge-colored graphs (Szeider’s Algorithm – [2003]) t start u s q v p dest.

16. t start u s q v p dest. color 1 color 2 color 3 papa v’v’’u’u’’ vava vbvb v1v1 v2v2 q’q’’ qaqa qbqb q2q2 q3q3 p’p’’ pbpb pcpc p1p1 p2p2 p3p3 uaua ubub ucuc u1u1 u2u2 u3u3 st (a) 3-edge colored graph (b) non-colored graph Properly s-t path in edge-colored graphs (Szeider’s Algorithm – [2003])

17. Our results: Lemma: Consider a c-edge-colored graph G, and an arbitrary pec trail T between « s » and « t ». Further, suppose that at least one node in T is visited 3 times or more. Then, there exists another pec trail T’ where no nodes are visited more than 2 times s x t y Cycles or closed trails passing by x Almost cycles or closed trails passing by y ab How to find a prop. edge-colored s-t trail?

18. Equivalence between paths and trails st 1 3 2 Graph G pec trail P y x X’ X’’ y’ y’’ y x X’ X’’ y’ y’’

19. Equivalence between paths and trails st 1 3 2 s’ 1’’ 1’ t’ 2’’ 2’ 1’ Graph G Graph H pec trail Ppec path P’ Theorem: We have a pec s-t trail in G we have a pec s’-t’ path in H

20. Shortest properly edge-colored s-t Path destination 2-edge-colored graph G Graph G’ bluered s 2’’ 3’ 4’’ 4’ 1’ 1’’ 2’ source 23 s 4 1t 1 1 1 1 1 1 1 1 1 1 0 0 0 0 t 3’’ Find a minimum perferct matching (if it exists)!

21. Shortest properly edge-colored s-t trail Algorithm: Shortest prop. edge-colored s-t Trail 1.Construct H=(V’,E’) associated to G 2.Find a short. pec path P (if possible) between « s’ » and « t’ » in H 3.Return trail T in G, and size(T)=size(P)/3 Input: A 2-edge colored graph G=(V,E), and 2 nodes s,t in V Output: A shortest prop. edge-colored trail T between « s » and « t ». Construction of H y x X’ X’’ y’ y’’ y x X’ X’’ y’ y’’ H xy

22. Existence of prop. edge-colored closed trails Theorem: Let G a c-edge colored graph, such that every vertex of G is incident with at least two edges of different colors. Then either G has a bridge, or G has a prop. edge-colored closed trail. 1 3 2 Algorithm: Delete all bridges and all nodes adjacent to edges of the same color 54 76 1 3 5 7 pec closed trail 1,2,3,1,5,7,6,4,1

23. Longest prop. edge-colored path in graphs with no pec cycles destination 2-edge-colored graph G source 23 s 4 1t

24. destination 2-edge-colored graph G source 23 s 4 1t Graph G’ bluered s 2’’ 3’ 4’’ 4’ 1’ 1’’ 2’ 1 1 1 1 1 1 1 1 1 0 0 0 0 t 3’’ Find a maximum perfect matching (if it exists)! Longest prop. edge-colored path in graphs with no pec cycles

25. Longest pec trail in graphs with no pec closed trails s x t y Cycles or closed trails passing by x (not possible !!) Almost cycles or closed trails passing by y We can visit node « y » several times !! FACT: Node « y » can be visited at most times!

26. s x t y Cycles or closed trails passing by x (not possible !!) Almost cycles or closed trails passing by y We can visit node « y » several times !! FACT: Node « y » can be visited at most times! Longest pec trail in graphs with no pec closed trails

27. y x X1X1 X2X2 XdXd Y1Y1 Y2Y2 YdYd y x X1X1 X2X2 XdXd Y1Y1 Y2Y2 YdYd Construction of H Theorem: We have a pec s-t trail in G we have a pec s’-t’ path in H

28. s k-Properly Vertex Disjoint Path problem Input: Given a 2-edge colored graph G, a const. k and nodes s,t V. Question: Does G contains k pec vertex disjoint paths between « s » and « t »? t k-PVDP Without node repetitions !!

29. s k-Properly Edge Disjoint Trails problem Question: Does G contains k pec edge disjoint trails between « s » and « t »? t k-PEDT Without edge repetitions !! Input: Given a 2-edge colored graph G, a const. k and nodes s,t V.

30. s k-Properly Edge Disjoint Trails problem Question: Does G contains k pec edge disjoint trails between « s » and « t »? t k-PEDT Without edge repetitions !! Input: Given a 2-edge-colored graph G, a const. k and nodes s,t V.

31. s k-Properly Edge Disjoint Trails problem Question: Does G contains k pec edge disjoint trails between « s » and « t »? t k-PEDT Without edge repetitions !! Input: Given a 2-edge-colored graph G, a const. k and nodes s,t V.

32. s k-Properly Edge Disjoint Trails problem Question: Does G contains k pec edge disjoint trails between « s » and « t »? t k-PEDT Without edge repetitions !! Input: Given a 2-edge colored graph G, a const. k and nodes s,t V.

33. s k-Properly Edge Disjoint Trails problem Input: Given a 2-edge colored graph G, a const. k and nodes s,t V. Question: Does G contains k pec edge disjoint trails between « s » and « t »? t k-PEDT Without edge repetitions !!

34. NP-Completeness uv Fortune, Hopcroft, Wylie [1980] Directed cycle problem - DC Input: A digraph D=(V,A) and a pair of nodes u,v V Output: Does exist a vertex disjoint circuit passing by « u » and « v » ? Output: Does exist an arc disjoint Circuit passing by « u » and « v » ? Theorem: DC problem is NP-Complete uv Directed Closed-Trail problem - DCT

35. NP-Completeness Theorem: Both 2-PVDP and 2-PEDT problems are NP Complete on arbitrary 2-edge-colored graphs. Reduction: DC problem 2-PVDP Reduction: DCT problem 2-PEDT Lemma: DCT problem is NP-Complete. Proof : (sketch) 1. 2. 3. 0. Both 2-PVDP and 2-PEDT are in NP

36. Both 2-PVDP and 2-PEDT in c-edge colored graphs Theorem: Both 2-PVDP and 2-PEDT problems are NP-Complete even for graphs with colors s t 2-edge-colored graph G Complete graph K n with colors x Additional color

37. The k-PVDP is NP-Complete in graphs with no pec cycles SAT k-AVDP 2-edge-colored graph G=(V,E) (with no pec cycles) and 2 nodes s,t є V True assignments for B k-Vertex Disjoint s-t Paths in G

38. The k-PVDP is NP-Complete in graphs with no pec cycles Example: Variable x 1 t2t2 s2s2 s1s1 t3t3 s3s3 t1t1 1 23 t2t2 s2s2 t3t3 s1s1 t1t1 s3s3 Variable x 2 4 6 5 t3t3 s1s1 t1t1 s2s2 s3s3 t2t2 Variable x 3 11 7 8 9 10 t2t2 s2s2 s1s1 t3t3 s3s3 t1t1 1 23 6 4 5 7 8 9 11

39. The k-PVDP is NP-Complete in graphs with no pec cycles Example: Variable x 1 t2t2 s2s2 s1s1 t3t3 s3s3 t1t1 1 23 t2t2 s2s2 t3t3 s1s1 t1t1 s3s3 Variable x 2 4 6 5 t3t3 s1s1 t1t1 s2s2 s3s3 t2t2 Variable x 3 11 7 8 9 10 t2t2 s2s2 s1s1 t3t3 s3s3 t1t1 1 23 6 4 5 7 8 9 11 s t

40. The k-PVDP is NP-Complete in graphs with no pec cycles Example: Variable x 1 t2t2 s2s2 s1s1 t3t3 s3s3 t1t1 1 23 t2t2 s2s2 t3t3 s1s1 t1t1 s3s3 Variable x 2 4 6 5 t3t3 s1s1 t1t1 s2s2 s3s3 t2t2 Variable x 3 11 7 8 9 10 t2t2 s2s2 s1s1 t3t3 s3s3 t1t1 1 23 6 4 5 7 8 9 11 s t

41. The k-PVDP is NP-Complete in graphs with no pec cycles Example: Variable x 1 t2t2 s2s2 s1s1 t3t3 s3s3 t1t1 1 23 t2t2 s2s2 t3t3 s1s1 t1t1 s3s3 Variable x 2 4 6 5 t3t3 s1s1 t1t1 s2s2 s3s3 t2t2 Variable x 3 11 7 8 9 10 t2t2 s2s2 s1s1 t3t3 s3s3 t1t1 1 23 6 4 5 7 8 9 11 s t

42. t2t2 s2s2 s1s1 t1t1 t2t2 s2s2 s1s1 t1t1 NP-Completeness in graphs with no pec cycles Grid G(x) t s s t k-PEDT is also NP-complete !!

43. Both 2-PVDP and 2-PEDT in c-edge with no properly edge-colored cycles (closed trails) Theorem: The 2-PVDP (2-PEDT) problem is NP-Complete for graphs with no pec cycles (closed trail) and c=Ω(n) colors s t 2-edge-colored graph G b Additional color a c d e K n with n-1 colors

44. Both 2-PVDP and 2-PEDT in c-edge with no properly edge-colored cycles (closed trails) Theorem: The 2-PVDP (2-PEDT) problem is NP-Complete for graphs with no pec cycles (closed trail) and c=Ω(n) colors s t 2-edge-colored graph G b Additional color a c d e K n with n-1 colors

45. Both 2-PVDP and 2-PEDT in c-edge with no properly edge-colored cycles (closed trails) Theorem: The 2-PVDP (2-PEDT) problem is NP-Complete for graphs with no pec cycles (closed trail) and c=Ω(n) colors s t 2-edge-colored graph G b Additional color a c d e K n with n-1 colors

46. Both 2-PVDP and 2-PEDT in c-edge with no properly edge-colored cycles (closed trails) Theorem: The 2-PVDP (2-PEDT) problem is NP-Complete for graphs with no pec cycles (closed trail) and c=Ω(n) colors s t 2-edge-colored graph G b Additional color a c d e K n with n-1 colors

47. Both 2-PVDP and 2-PEDT in c-edge with no properly edge-colored cycles (closed trails) Theorem: The 2-PVDP (2-PEDT) problem is NP-Complete for graphs with no pec cycles (closed trail) and c=Ω(n) colors s t 2-edge-colored graph G b Additional color a c d e K n with n-1 colors

48. Approximation Algorithm for the MPEDT Greedy-ED Procedure 1. S Ø 2. Repeat Find an pec shortest trail T between « s » and « t »; S S  E(T); E E - E(T); Until (no pec s-t trails are found) Theorem: The Greedy-ED has performance ratio equal to for the MPEDT problem

49. Approximation Algorithm for the MPVDP Greedy-VD Procedure 1. S Ø 2. Repeat Find a pec shortest path P between « s » and « t »; S S  E(P); V V - V(P); Until (no pec s-t paths are found) Theorem: The Greedy-VD has performance ratio equal to for the MPVDP problem

50. st Greedy solution  Z H = 1 Approximation ratio for MPEDT

51. st Greedy solution  Z H = 1 Approximation ratio for MPEDT Optimum solution  Opt = k/2

52. st Approximation ratio for MPEDT Approximation ratio =

53. t start u s q v p dest. color 1 color 2 color 3 papa v’v’’u’u’’ vava vbvb v1v1 v2v2 q’q’’ qaqa qbqb q2q2 q3q3 p’p’’ pbpb pcpc p1p1 p2p2 p3p3 uaua ubub ucuc u1u1 u2u2 u3u3 s1s1 (a) 3-edge colored graph (b) non-colored graph s2s2 t1t1 t2t2 Some Polynomial Cases: we have no (almost) pec cycles passing by « s » or « t ».

54. t start u s q v p dest. color 1 color 2 color 3 papa v’v’’u’u’’ vava vbvb v1v1 v2v2 q’q’’ qaqa qbqb q2q2 q3q3 p’p’’ pbpb pcpc p1p1 p2p2 p3p3 uaua ubub ucuc u1u1 u2u2 u3u3 s1s1 (a) 3-edge colored graph (b) non-colored graph s2s2 t1t1 t2t2 Some Polynomial Cases: we have no (almost) pec cycles passing by « s » or « t ».

55. Open Problems and Future Diretions Input: Given a c-edge-colored complete graph, and vertices s,t of Open question: Maximize the number of edge-disjoint pec s-t paths in is in P? Future work: What about the performance ratio of both MPVDP and MPEDT problems in graphs with no pec cycles (closed trails)? Input: Given a 2-edge-colored graph with no pec cycles, vertices s,t  V(G) and a fixed k 2. Question: Does G contains k pec vertex disjoint paths between « s » and « t »?

56. Thanks for your attention!!