The minimum reload s-t path/trail/walk problems Current Trends in Theory and Practice of Comp. Science, SOFSEM09 L. Gourvès, A. Lyra, C. Martinhon, J. Monnot Špindlerův Mlýn / Czech Republic

Topics 1. Motivation and basic definitions 2. Minimum reload s-t walk problem; 3. Paths\trails with symmetric reload costs: Polynomial and NP-hard results. 4. Paths\trails with asymmetric reload costs: Polynomial and NP-hard results. 5. Conclusions and open problems

1. Cargo transportation network when the colors are used to denote route subnetworks; 2. Data transmission costs in large communication networks when a color specify a type of transmission; 3. Change of technology when colors are associated to technologies; etc Some applications involving reload costs

Basic Definitions Paths, trails and walks with minimum reload costs s t Reload cost matrix R = a bc d

Basic Definitions Minimum reload s-t walk s t c(W) Reload cost matrix R = 3 a bc d

Basic Definitions Minimum reload s-t trail s t c(W) ≤ c(T) Reload cost matrix R = 34 a bc d

Basic Definitions Minimum reload s-t path s t c(W) ≤ c(T) ≤ c(P) Reload cost matrix R = 345 a bc d

Basic Definitions Symmetric or asymmetric reload costs r ij ≠ r ji Triangle inequality (between colors) z y w x 12 3 r ij ≤ r jk + r ik for colors “i” and “j” r ij = r ji or for colors 1,2,3

Basic Definitions NOTE: Paths (resp., trails and walks) with reload costs generalize both properly edge-colored (pec) and monochromatic paths (resp., trails and walks). s t r ij = 0, for i j and r ii = 1≠ pec s-t path cost of the minimum reload s-t path is 0

s t r ij = 1, for i j and r ii = 0≠ monochomatic s-t path cost of the min. reload s-t path is 0 Basic Definitions NOTE: Paths (resp., trails and walks) with reload costs generalize both properly edge-colored (pec) and monochromatic paths (resp., trails and walks).

Minimum reload s-t walk Minimum reload s-t walk in G Shortest s 0 -t 0 path in H t s v1v1 v2v2 c

Minimum reload s-t walk t s v1v1 v2v2 All cases can be solved in polynomial time !

z y v 1 2 x 1 a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v) c Symmetric R Minimum symmetric reload s-t trail

z y v 1 2 x 1 a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v) c Symmetric R Minimum symmetric reload s-t trail

z y v 1 2 x 1 a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v) c Minimum symmetric reload s-t trail Minimum perfect matching Symmetric R Minimum symmetric reload s-t trail

z y v 1 2 x 1 a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v) c Symmetric R The minimum symmetric reload s-t trail can be solved in polynomial time ! Minimum symmetric reload s-t trail

NP-completeness Theorem 1 The minimum symmetric reload s–t path problem is NP-hard if c ≥ 3, the triangle inequality holds and the maximum degree of G c is equal to 4.

x i is false Gadget for literal x i Gadget for clause C j x i is true 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) Theorem 1 (Proof)

C3C3 C6C6 C4C4 C5C5 literal x 7

Every other entries of R are set to 1 C6C6 Theorem 1 (Proof) C3C3 C4C4 C5C5

t s

We modify the reload costs, so that: OPT(G c )=0 I is satisfiable. OPT(G c ) >M I is not satisfiable. In this way, to distinguish between OPT(G c )=0 or OPT(G c ) ≥M is NP- complete, otherwise P=NP! Non-approximation Theorem 2 In the general case, the minimum symmetric reload s–t path problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of G c is equal to 4.

Non-approximation Theorem 2 In the general case, the minimum symmetric reload s–t path problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of G c is equal to 4. Proof: r 1,2 = r 2,1 = M r 1,3 = r 3,1 = 0 r 2,2 = 0 r 1,1 = 0 r 2,3 = r 3,2 = 0

t s Non-approximation (Proof) r 1,2 = r 2,1 = M

Theorem 3 If, for every i,j the minimum symmetric reload s–t path problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of G c is equal to 4. Proof: r 1,2 = r 2,1 = M r 1,3 = r 3,1 = 1 r 2,2 = 1 r 1,1 = 1 r 2,3 = r 3,2 = 1 Non-approximation

Theorem 3 If, for every i,j the minimum symmetric reload s–t path problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of G c is equal to 4. Proof: Non-approximation It is NP –complete to distinguish between

Corollary 4: The minimum symmetric reload s–t path problem is NP-hard if c ≥ 4, the graph is planar, the triangle inequality holds and the maximum degree is equal to 4. NP-Completeness

a b d c a b d c f a b d c a b d c f d’ c’ a’ b’ r 3,4 = r 4,3 = M Corollary 4 (Proof): r 1,2 = r 2,1 = M

Some polynomial cases Theorem 5 Consider G c with c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality. Then, the minimum symmetric reload s–t path problem can be solved in polynomial time.

Some polynomial cases Theorem 5 Consider G c with c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality. Then, the minimum symmetric reload s–t path problem can be solved in polynomial time. If the triangle ineq. does not hold??

Some polynomial cases The minimum toll cost s–t path problem may be solved in polynomial time. ∀ r i,j =r j, for colors i and j and r i,i =0 s t s 0 auxiliar vertex and edge toll points

NP-completeness Theorem 6 The minimum asymmetric reload s–t trail problem is NP-hard if c ≥ 3, the triangle inequality holds and the maximum degree of G c is equal to 4.

NP-completeness (Proof) Variable graph Clause graph 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) False True

x3x3 Reload costs = M NP-completeness (Proof)

(b) If, for every i,j the minimum asymmetric reload s–t trail problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of G c is equal to 3. Non-approximation Theorem 7 (a) In the general case, the minimum asymmetric reload s–t trail problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of G c is equal to 3.

(b) If, for every i,j the minimum asymmetric reload s–t trail problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of G c is equal to 3. Non-approximation Theorem 7 (a) In the general case, the minimum asymmetric reload s–t trail problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of G c is equal to 3.

(b) If, for every i,j the minimum asymmetric reload s–t trail problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of G c is equal to 3. Non-approximation Theorem 7 (a) In the general case, the minimum asymmetric reload s–t trail problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of G c is equal to 3.

A polynomial case Theorem 8 Consider G c with c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality. Then, the minimum asymmetric reload s–t trail problem can be solved in polynomial time.

A polynomial case Theorem 8 Consider G c with c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality. Then, the minimum asymmetric reload s–t trail problem can be solved in polynomial time. If the triangle ineq. does not hold??

Conclusions and Open Problems Polynomial time problemsNP-hard problems s-t walk s-t trail s-t path

Conclusions and Open Problems Input: Let be 2-edge-colored graph and 2 vertices Question: Does the minimum symmetric reload s-t path problem can be solved in polynomial time? Note: If the triangle ineq. holds Yes! Problem 1

Conclusions and Open Problems Input: Let be 2-edge-colored graph and 2 vertices Question: Does the minimum asymmetric reload s-t trail problem can be solved in polynomial time? Note: If the triangle ineq. holds Yes! Problem 2

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