On s-t paths and trails in edge-colored graphs

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

Topics • Applications • Basic Definitions • Paths and trails in Gc without PEC closed trails • Monochromatic s-t paths • Conclusions and future directions

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.

Topics • Applications • Basic Definitions • Paths and trails in Gc without PEC closed trails • Monochromatic s-t paths

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

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

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

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

PEC s-t Paths in 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 q1 q3 color 2 pa pb pc qa qb color 3 q’ q’’ p’ p’’ (a) 3-edge colored graph (b) non-colored (Edmond-Szeider) graph

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

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

2 vertex/edge disjoint PEC s-t paths • Theorem: Abouelaoualim et al., 2008 • NP-complete, for general Gc • Even for Ω = (n2) colors, where |V|=n s t

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

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.

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)

NP-completeness (Proof)

NP-completeness (Proof) Lv = 14mn+2m-14n+1 L = 14mn + 2m + 2 Lc = 14n - 1 Lv Lc Lc Lc

NP-completeness (Proof) Maximumdegree 3!

Edge-colored graphswith no PEC closed trails Theorem 2: Find a PECs-t trail visiting all vertices x∈ V(Gc) exactly f(x) times, with fmin(x) ≤ f(x) ≤ fmax(x) e d Solved in polynomial time! x = a, f(a)=2 c b a s t

Edge-colored graphswith 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)

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

Theorem 2 (proof) x’a x’’b x’a x’’b x x’1 x’1 x’3 x’2 x’2 x’3 x1 x2 x1 x2 x3 x3 color 1 Subgraph H’x associated to x ∈ S’(x) Subgraph Hx associated to x ∈ S(x) color 2 color 3

Edge-coloredgraphswith no PEC closedtrails Corollary 3: A shortest (resp. longest) PECs-t trail visiting vertices x of Gc at least fmin(x) times (resp. at most fmax(x) times) Solved in polynomial time!

Corollary 3 (proof) x’a x’’b y’a y’’b 0 0 0 0 x’1 y’1 x’3 x’2 y’2 y’3 0 0 0 0 0 0 x1 x2 y1 y2 x3 y3 1 1 1 1 1 1 1 • Compute the minimum perfect matching (resp. maximum perfect matching ) M in Hm.

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

Polynomial case (proof): • Construct the trail-path graph, • Construct an associated modified Edmonds-Szeider graph. xy ∈ E’ bxy y x Hx1 Hy1 x1 y1 axy Hx2 Hy2 x2 y2

Edge-colored graphs with no PEC cycles • PEC closed trails are allowed! 3 2 4 5 1 6 7

Theorem 5 Edge-colored graphs with no PEC cycles To find a PEC s-ttrail passing by a vertex v is NP-complete. Surprisingly, finding a PEC s-tpath passing by a subset A={v1,..., vk} is polynomial time solvable!

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

Theorem 5 (proof) G’’(e) G’(v)

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

Theorem 5 (proof)

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

Theorem 6 (proof) c1 c2 c3 c4

Theorem 6 (proof) s t

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

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

On s-t paths and trails in edge-colored graphs