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Paths and Trails in Edge Colored Graphs

Explore theoretical informatics topics on edge-colored graphs, including properly colored paths and trails, NP-completeness, and approximation algorithms. Discover applications in computational biology, cryptography, and social sciences. Learn about finding proper edge-colored paths and trails through algorithms and theorems.

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Paths and Trails in Edge Colored Graphs

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  1. Latin-American on Theoretical Informatics Symposium LATIN 2008 Paths and Trails in Edge Colored Graphs 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. Some Applications using edge colored graphs 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

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

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

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

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

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

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

  10. How to find a properly edge-colored s-t path? 2-edge-colored graph G 4 2 3 s 1 t source destination

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

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

  13. Properly s-t path in edge-colored graphs (Szeider’s Algorithm – [2003]) v’ v’’ u’ u’’ v u va ua ub uc vb t s v1 u1 u2 u3 v2 start dest. q p s t color 1 p1 p2 p3 q2 q3 color 2 pa pb pc qa qb color 3 q’ q’’ p’ p’’ (a) 3-edge colored graph (b) non-colored graph

  14. Properly s-t path in edge-colored graphs (Szeider’s Algorithm – [2003]) v’ v’’ u’ u’’ v u va ua ub uc vb t s v1 u1 u2 u3 v2 start dest. q p s t color 1 p1 p2 p3 q2 q3 color 2 pa pb pc qa qc color 3 q’ q’’ p’ p’’ (a) 3-edge colored graph (b) non-colored graph

  15. Properly s-t path in edge-colored graphs (Szeider’s Algorithm – [2003]) v’ v’’ u’ u’’ v u va ua ub uc vb t s v1 u1 u2 u3 v2 start dest. q p s t color 1 p1 p2 p3 q2 q3 color 2 pa pb pc qa qc color 3 q’ q’’ p’ p’’ (a) 3-edge colored graph (b) non-colored graph

  16. Properly s-t path in edge-colored graphs (Szeider’s Algorithm – [2003]) v’ v’’ u’ u’’ v u va ua ub uc vb t s v1 u1 u2 u3 v2 start dest. q p s t color 1 p1 p2 p3 q2 q3 color 2 pa pb pc qa qb color 3 q’ q’’ p’ p’’ (a) 3-edge colored graph (b) non-colored graph

  17. Our results: How to find a prop. edge-colored s-t trail? 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 a b s x y t Almost cycles or closed trails passing by y Cycles or closed trails passing by x

  18. Equivalence between paths and trails Graph G y x X’ y’ 3 2 X’’ y’’ y x 1 s t X’ y’ pec trail P X’’ y’’

  19. Equivalence between paths and trails Graph H Graph G 2’ 1’ 2’’ 1’ 3 2 1’ 1 s t s’ t’ 1’’ pec trail P pec 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 Graph G’ blue red s 2-edge-colored graph G 1 1 0 1’’ 1’ 4 2 3 1 1 1 0 2’’ 2’ 1 1 1 0 3’’ 3’ 1 s 1 t 0 4’’ 4’ destination source 1 t Find a minimum perferct matching (if it exists)!

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

  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. Algorithm: Delete all bridges and all nodes adjacent to edges of the same color 3 3 2 5 4 5 1 1 7 6 7 pec closed trail 1,2,3,1,5,7,6,4,1

  23. Longest prop. edge-colored path in graphs with no peccycles 2-edge-colored graph G 4 2 3 s 1 t destination source

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

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

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

  27. Longest pectrail in graphs with no pecclosed trails Construction of H y y x x X1 Y1 X1 Y1 X2 Y2 X2 Y2 Xd Yd Xd Yd Theorem: We have a pec s-t trail in G we have a pecs’-t’ path in H

  28. 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 »? k-PVDP s t Without node repetitions !!

  29. 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 »? k-PEDT s t Without edge repetitions !!

  30. 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 »? k-PEDT s t Without edge repetitions !!

  31. 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 »? k-PEDT s t Without edge repetitions !!

  32. 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 »? k-PEDT s t Without edge repetitions !!

  33. 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 »? k-PEDT s t Without edge repetitions !!

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

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

  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 Additional color t 2-edge-colored graph G Complete graph Kn with colors s x

  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. s3 s2 s3 6 7 6 s1 t1 4 s1 t1 9 1 2 3 8 s2 s3 s2 t2 4 5 t2 t3 t3 1 Variable x1 s1 t1 Variable x2 2 3 s2 11 t2 t3 7 s1 t1 8 11 s3 t3 5 9 10 10 t2 Variable x3 The k-PVDP is NP-Complete in graphs with no pec cycles Example:

  39. s2 s3 s1 t1 1 2 3 t2 t3 Variable x1 s2 7 s1 t1 8 11 s3 t3 9 10 t2 Variable x3 The k-PVDP is NP-Complete in graphs with no pec cycles Example: s s3 6 6 7 s1 t1 4 s2 t2 9 5 8 s2 s3 4 t3 Variable x2 1 s1 t1 2 3 11 t2 t3 5 10 t

  40. s2 s3 s1 t1 1 2 3 t2 t3 Variable x1 s2 7 s1 t1 8 11 s3 t3 9 10 t2 Variable x3 The k-PVDP is NP-Complete in graphs with no pec cycles Example: s s3 6 6 7 s1 t1 4 s2 t2 9 5 8 s2 s3 4 t3 Variable x2 1 s1 t1 2 3 11 t2 t3 5 10 t

  41. s2 s3 s1 t1 1 2 3 t2 t3 Variable x1 s2 7 s1 t1 8 11 s3 t3 9 10 t2 Variable x3 The k-PVDP is NP-Complete in graphs with no pec cycles Example: s s3 6 6 7 s1 t1 4 s2 t2 9 5 8 s2 s3 4 t3 Variable x2 1 s1 t1 2 3 11 t2 t3 5 10 t

  42. NP-Completeness in graphs with no pec cycles k-PEDT is also NP-complete !! Grid G(x) s s s2 s2 s1 t1 s1 t1 t2 t2 t t

  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 Additional color a t e 2-edge-colored graph G s b d c Kn 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 Additional color a t e 2-edge-colored graph G s b d c Kn 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 peccycles (closed trail) and c=Ω(n) colors Additional color a t e 2-edge-colored graph G s b d c Kn 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 Additional color a t e 2-edge-colored graph G s b d c Kn 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 Additional color a t e 2-edge-colored graph G s b d c Kn with n-1 colors

  48. 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) Approximation Algorithm for the MPEDT Theorem: The Greedy-ED has performance ratio equal to for the MPEDT problem

  49. 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) Approximation Algorithm for the MPVDP Theorem: The Greedy-VD has performance ratio equal to for the MPVDP problem

  50. Approximation ratio for MPEDT s t Greedy solution  ZH = 1

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