1. On s-t paths and trails in edge-colored graphs L. Gourvès, A. Lyra*, C. Martinhon, J. Monnot, F. Protti Lagos 2009 (RS, Brazil) *alyra@ic.uff.br

2. Topics Applications Basic Definitions Paths and trails in G c without PEC closed trails Monochromatic s-t paths Conclusions and future directions

3. Topics Applications Basic Definitions Paths and trails in G c without PEC closed trails Monochromatic s-t paths Conclusions and future directions

4. Some Applications using edge colored graphs 1. Computational Biology 2. Criptography (when a color specify a type of transmission) 3. Social Sciences (where a color represents a relation between 2 individuals) etc Some Bibliography D. Dorniger, On permutations of cromossomes, In Contributions of General Algebra, 5, 95-103, 1987. D. Dorniger, Hamiltonian circuits determining the order of cromossomes, In Disc. App. Math., 5, 159-168, 1994. P. Pevzner, Computational Molecular Biology: An Algorithmic Approach, MIT Press, 2000.

5. Topics Applications Basic Definitions Paths and trails in G c without PEC closed trails Monochromatic s-t paths

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

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

8. Basic Definitions Properly edge-colored ( PEC ) closed trail. 5 start 2 3 x 4 1 (without edge repetitions!!)

9. How to find a PEC s-t Path? Basic Definitions

10. 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 q1q1 q3q3 p’p’’ pbpb pcpc p1p1 p2p2 p3p3 uaua ubub ucuc u1u1 u2u2 u3u3 st (a) 3-edge colored graph (b) non-colored (Edmond-Szeider) graph PEC s-t Paths in colored Graphs (Szeider’s Algorithm, 2003)

11. color 1 color 2 color 3 papa v’v’’u’u’’ vava vbvb v1v1 v2v2 q’q’’ qaqa qbqb q1q1 q3q3 p’p’’ pbpb pcpc p1p1 p2p2 p3p3 uaua ubub ucuc u1u1 u2u2 u3u3 st (a) 3-edge colored graph PEC s-t Paths in colored Graphs (Szeider’s Algorithm, 2003) t start u s q v p dest. (b) non-colored (Edmond-Szeider) graph

12. 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 q1q1 q3q3 p’p’’ pbpb pcpc p1p1 p2p2 p3p3 uaua ubub ucuc u1u1 u2u2 u3u3 st (a) 3-edge colored graph PEC s-t Paths in colored Graphs (Szeider’s Algorithm, 2003) (b) non-colored (Edmond-Szeider) graph

13. Equivalence between paths and trails (trail-path graph) st 1 3 2 Graph G y x X’ X’’ y’ y’’ y x X’ X’’ y’ y’’ PEC s-t trail ρ

14. Equivalence between paths and trails (trail-path graph) s t 1 3 2 s’ 1’’ 1’ t’ 2’’ 2’ 3’ Graph H We have a properly edge colored trail in G we have a properly edge colored path in H Theorem: Abouelaoualim et al., 2008 Graph G PEC s-t trail ρ PEC s-t path ρ’

15. Topics Applications Basic Definitions Paths and trails in G c without PEC closed trails Monochromatic s-t paths

16. 2 vertex/edge disjoint PEC s-t paths Theorem: Abouelaoualim et al., 2008 NP-complete, for general G c Even for Ω = (n 2 ) colors, where |V|=n s t

17. Open problem (Abouelaoualim et al., 2008) 2 vertex/edge disjoint PEC s-t paths in G c with no PEC cycles or PEC closed trails. s t

18. 18 PEC s-t paths with no PEC closed trails Theorem 1: To find 2 PEC s-t paths with length at most L> 0 is NP-complete. Even for graphs with maximum vertex degree equal to 3.

19. 19 PEC s-t paths with bounded length 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)

20. 20 NP-completeness (Proof)

21. 21 NP-completeness (Proof)

22. 22 NP-completeness (Proof) L c = 14n - 1 L v = 14mn+2m-14n+1 L = 14mn + 2m + 2 LcLc LcLc LcLc LvLv

23. NP-completeness (Proof) 23 Maximum degree 3!

24. Edge-colored graphs with no PEC closed trails Theorem 2: Find a PEC s-t trail visiting all vertices x ∈ V(G c ) exactly f(x) times, with f min (x) ≤ f(x) ≤ f max (x) s t bc de a x = a, f(a)=2 Solved in polynomial time!

25. Edge-colored graphs with no PEC closed trails Finding a PEC s-t trail passing by a vertex v is NP-complete in general edge-colored graphs. (Chou et al., 1994) Finding a PEC s-t trail visiting A ⊆ V \{s,t} is polynomial time solvable in 2-edge-colored complete graphs. (Das and Rao, 1983)

26. 26 Theorem 2 (proof) The idea is to construct the trail-path graph and the Edmonds-Szeider Graph associated to the trail-path graph. y x X’ X’’ y’ y’’ f min (x) = 1 f max (x) = 2 f min (y) = 1 f max (y) = 3 y’’’

27. 27 Theorem 2 (proof) x x’ a x’’ b x’ 1 x’ 2 x’ 3 x1x1 x2x2 x3x3 color 1 color 2 color 3 x’ a x’’ b x’ 1 x’ 2 x’ 3 x1x1 x2x2 x3x3 Subgraph H’ x associated to x ∈ S’(x) Subgraph H x associated to x ∈ S(x)

28. Corollary 3: A shortest (resp. longest) PEC s-t trail visiting vertices x of G c at least f min (x) times (resp. at most f max (x) times) Edge-colored graphs with no PEC closed trails Solved in polynomial time!

29. Corollary 3 (proof) Compute the minimum perfect matching (resp. maximum perfect matching ) M in H m. y’ a y’’ b y’ 1 y’ 2 y’ 3 y1y1 y2y2 y3y3 x’ a x’’ b x’ 1 x’ 2 x’ 3 x1x1 x2x2 x3x3 0 00 0 0 000 0 0 1 1 1 1 1 1 1

30. Edge-colored graphs with no PEC closed trails Theorem 4: The determination of a PEC s-t trail visiting all edges of E’ ⊆ E(G c ) is solved in polynomial time. s bc a e f dt E’ = {ab, bc, ca, df, fe}

31. Polynomial case (proof): Construct the trail-path graph, Construct an associated modified Edmonds- Szeider graph. xy ∈ E’ y x x1x1 x2x2 y1y1 y2y2 H x1 H x2 H y1 H y2 a xy b xy

32. PEC closed trails are allowed! 32 Edge-colored graphs with no PEC cycles 1 3 2 54 76

33. 33 Theorem 5 To find a PEC s-t trail passing by a vertex v is NP-complete. Edge-colored graphs with no PEC cycles Surprisingly, finding a PEC s-t path passing by a subset A={v 1,..., v k } is polynomial time solvable!

34. 34 Theorem 5 (proof) Use the problem Path Finding Problem in D t s x s t b x c d a

35. Theorem 5 (proof) 35 G’(v) G’’(e)

36. Theorem 5 (proof) 36 Without incoming arcs at s Without outgoing arcs at t

37. Theorem 5 (proof) 37

38. Topics Applications Basic Definitions Paths and trails in G c without PEC closed trails Monochromatic s-t paths

39. NP-completeness Theorem 6: Find 2 vertex disjoint monochromatic s-t paths with different colors in G c is NP- complete. t start s dest. The edge disjoint case is trivial

40. Theorem 6 (proof) c1c1 c3c3 c2c2 c4c4

41. s t

42. Conclusions and future directions We deal with PEC and monochromatic s-t paths and trails on c-edge colored graphs. Future directions: Given G c without PEC cycles, is there a polynomial algorithm to find two PEC s-t paths? If G c is planar, to find two monochromatic vertex disjoint s-t paths is NP-complete?

43. Thanks for your attention! You can download this presentation at http://www.ic.uff.br/~alyra My email: alyra@ic.uff.br