The minimum reload s-t path/trail/walk problems

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## The minimum reload s-t path/trail/walk problems

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**Current Trends in Theory and Practice of Comp. Science,**SOFSEM09 The minimum reload s-t path/trail/walk problems 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**Some applications involving reload costs**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**Basic Definitions**• Paths, trails and walks with minimum reload costs c b 1 1 5 R = 1 1 1 1 1 5 a s t Reload cost matrix d**Basic Definitions**• Minimum reload s-t walk c b 1 1 5 R = 1 1 1 1 1 5 a s t Reload cost matrix d c(W) 3**Basic Definitions**• Minimum reload s-t trail c b 1 1 5 R = 1 1 1 1 1 5 a s t Reload cost matrix d c(W) ≤ c(T) 3 4**Basic Definitions**• Minimum reload s-t path c b 1 1 5 R = 1 1 1 1 1 5 a s t Reload cost matrix d c(W) ≤ c(T) ≤ c(P) 3 4 5**Basic Definitions**• Symmetric or asymmetric reload costs rij = rji rij ≠ rji or for colors “i” and “j” • Triangle inequality (between colors) 1 2 rij ≤ rjk + rik y x z for colors 1,2,3 3 w**Basic Definitions**NOTE: Paths (resp., trails and walks) with reload costs generalize both properly edge-colored (pec) and monochromatic paths (resp., trails and walks). rij = 0, for i j and rii = 1 ≠ s t pec s-t path cost of the minimum 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). rij = 1, for i j and rii = 0 ≠ s t monochomatic s-t path cost of the min. reload s-t path is 0**Minimum reload s-t walk**s 1 v2 2 v1 3 t c Minimum reload s-t walk in G Shortest s0-t0 path in H**Minimum reload s-t walk**s 1 v2 2 v1 3 t All cases can be solved in polynomial time !**Minimum symmetric reload s-t trail**Symmetric R 1 y v 2 1 x z c a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v)**Minimum symmetric reload s-t trail**Symmetric R 1 y v 2 1 x z c a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v)**Minimum symmetric reload s-t trail**Symmetric R 1 y v 2 1 x z c a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v) Minimum symmetric reload s-t trail Minimum perfect matching**Minimum symmetric reload s-t trail**Symmetric R 1 y v 2 1 x z c a) Neighbourhood of “v” in G b) Weighted non-colored subgraph G(v) The minimum symmetric reload s-t trail can be solved in polynomial time !**NP-completeness**Theorem 1 The minimum symmetric reload s–tpath problem is NP-hard if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 4.**Theorem 1 (Proof)**• 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) xi is false xi is true Gadget for clause Cj Gadget for literal xi**Theorem 1 (Proof)**C4 C3 C6 C5 literal x7**Theorem 1 (Proof)**C4 C3 C6 C5 Every other entries of R are set to 1**Non-approximation**Theorem 2 In the general case, the minimum symmetric reload s–tpath problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 4. We modify the reload costs, so that: OPT(Gc)=0 I is satisfiable. OPT(Gc) >M I is not satisfiable. In this way, to distinguish between OPT(Gc)=0 or OPT(Gc) ≥M is NP-complete, otherwise P=NP!**Non-approximation**Theorem 2 In the general case, the minimum symmetric reload s–tpath problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 4. r1,2 = r2,1 = M r2,2 = 0 Proof: r1,3 = r3,1 = 0 r1,1 = 0 r2,3 = r3,2 = 0**Non-approximation (Proof)**s r1,2 = r2,1 = M t**Non-approximation**Theorem 3 If , for every i,j the minimum symmetric reload s–tpath problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 4. r1,2 = r2,1 = M r2,2 = 1 Proof: r1,3 = r3,1 = 1 r1,1 = 1 r2,3 = r3,2 = 1**Non-approximation**Theorem 3 If , for every i,j the minimum symmetric reload s–tpath problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 4. Proof: It is NP –complete to distinguish between**NP-Completeness**• Corollary 4: • The minimum symmetric reload s–tpath problem is NP-hard if c ≥ 4, the graph is planar, the triangle inequality holds and the maximum degree is equal to 4.**Corollary4 (Proof):**c c b a f a b d r1,2 = r2,1 = M d c c’ c b b’ a’ a f b a d’ d d r3,4 = r4,3 = M**Some polynomial cases**Theorem 5 Consider Gc with c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality. Then, the minimum symmetric reload s–tpath problem can be solved in polynomial time.**Some polynomial cases**Theorem 5 Consider Gc with c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality. Then, the minimum symmetric reload s–tpath 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. • ∀ ri,j=rj, for colors i and j andri,i=0 toll points s s 0 t auxiliar vertex and edge**NP-completeness**Theorem 6 The minimum asymmetric reload s–ttrail problem is NP-hard if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 4.**NP-completeness (Proof)**• 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) True False Clause graph Variable graph**NP-completeness (Proof)**x3 Reload costs = M**Non-approximation**Theorem 7 (a) In the general case, the minimum asymmetric reload s–ttrail problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of Gc 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 Gc 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 Gc is equal to 3. (b) If , for every i,j the minimum asymmetric reload s–ttrail problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 3.**Non-approximation**Theorem 7 (a) In the general case, the minimum asymmetric reload s–ttrail problem is not approximable at all if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 3. (b) If , for every i,j the minimum asymmetric reload s–ttrail problem is not -approximable for every if c ≥ 3, the triangle inequality holds and the maximum degree of Gc is equal to 3.**A polynomial case**Theorem 8 Consider Gc with c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality. Then, the minimum asymmetric reload s–ttrail problem can be solved in polynomial time.**A polynomial case**Theorem 8 Consider Gc with c=2 colors. Further, suppose that the reload cost matrix R satisfies the triangle inequality. Then, the minimum asymmetric reload s–ttrail problem can be solved in polynomial time. If the triangle ineq. does not hold??**Conclusions and Open Problems**Problem 1 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: Ifthetriangleineq. holds Yes!**Conclusions and Open Problems**Problem 2 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: Ifthetriangleineq. holds Yes!