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CONCEPTION DE RESEAUX AVEC CONTRAINTES DE BORNES

CONCEPTION DE RESEAUX AVEC CONTRAINTES DE BORNES. A. Ridha Mahjoub LIMOS, Université Blaise Pascal, Clermont-Ferrand, France. Introduction. Survivability The ability to restore network service in the event of a catastrophic failure. Goal

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CONCEPTION DE RESEAUX AVEC CONTRAINTES DE BORNES

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  1. CONCEPTION DE RESEAUX AVEC CONTRAINTES DE BORNES A. Ridha Mahjoub LIMOS, Université Blaise Pascal, Clermont-Ferrand, France

  2. Introduction Survivability The ability to restore network service in the event of a catastrophic failure. Goal Satisfy some connectivity requirements in the network. Motivation Design of optical communication networks.

  3. Introduction Bounded lengths Motivation To have effective rerouting. Two rerouting strategies Local rerouting: Each edge must belong to a bounded cycle (ring). SONET/SDH networks

  4. Introduction

  5. Introduction End-to-end rerouting: the paths between the terminals should not exceed a certain length (a certain number of hops) (hop-constrained paths). ATM networks, INTERNET

  6. Contents 1. Design of survivable networks with bounded rings 1.1) Node case 1.2) Edge case 2. Design of survivable networks with bounded paths 2.1) A general model 2.2) Special cases and related problems 3. Formulation for L≤ 3 4. The 2-edge connected hop-constrained network design problem 4.1. Complexity 4.2. Polyhedral results and Branch&Cut algorithm 4.4. Formulation for L=4 5. Open questions and concluding remarks

  7. 1.Bounded rings 1.1. Node case 1. Bounded rings 1.1. 2-node connected graphs Fortz, Labbé, Maffioli (1999) Fortz, Labbé (2002) The problem: Given a graph G=(V,E) with weights and lenghts associated with the edges, and a constant B, determine a minimum 2- node connected spanning subgraph such that each edge belongs to a cycle of length no more than B.

  8. 1.Bounded rings 1.1. Node case Extended formulations Valid inequalities Separation algorithms Lower bounds on the optimal value Cutting plane algorithms Cyclomatic inequalities(unitary lengths case) Let (V1,…,Vp) be a partition of V. Then the inequality is valid for the problem. (V1,…,Vp) is the set of edges between the elements of the partition.

  9. 1.Bounded rings 1.1. Node case To separate the cyclomatic inequalities approximatly Fortz, Labbé and Maffioli considered the inequalities x((V1,...,Vp)) ≥ a(p-1), with where n is the number of nodes in the garph.

  10. 1.Bounded rings 1.1. Node case Consider the constraints x((V1,...,Vp)) ≥ p-1. These arecalled partition inequalities. They arise as valid inequalities in many connectivity problems. They are valid for the problem, when considered on G\v, vV. The separation problem for these inequalities reduce to |E| min cut problems Cunningham (1985) . It can also be reduced to|V| min cut problemsBarahona (1992). Both algorithms provide the most violated inequality if there is any.

  11. 1.Bounded rings 1.2. Edge case 1.2. 2-edge connected graphs Fortz, M., McCormick, Pesneau (2003) The problem: Given a graph G=(V,E) with weights on the edges, and an integer B, determine a minimum 2-edge connected spanning subgraph such that each edge belongs to a cycle of length no more than B.

  12. 1.Bounded rings 1.2. Edge case Formulation Valid inequalities d(W) is called a cut cut inequalities

  13. 1.Bounded rings 1.2. Edge case Let π=(V0,V1,...,Vp) be a partition of V such that p≥B. Let e  (V0,Vp) cycle inequalities

  14. 1.Bounded rings 1.2. Edge case Le problème est équivalent au programme: min Subject to x(d (W)) ≥ 2for all (W) for all partition and e 0 ≤ x(e) ≤ 1 for all e  E, x(e){0,1} for all e  E.

  15. 1.Bounded rings 1.2. Edge case If we add the constraints x(dG-v(W)) ≥ 1,for allW V\{v}, vV we obtain a formulation for the 2-node case.

  16. 1.Bounded rings 1.2. Edge case Separation of cycle inequalities - If the solution is in 0-1, the separation can be done easily The minimum L-st-path cut problem LetG=(V,E)be a graph. Lets,t VandL a fixed integer. We callL-st-path cutany edge setCthatintersects every st-path of length ≤ L. Given weigts on the edges,the minimum st-L-path cut problem is to find an L-st-path cut of minimum weight.

  17. 1.Bounded rings 1.2. Edge case Lemma:The separation problem for cycle inequalities reduces to the minimum (B-1)-st-path cut problem. Let e=st an edge of G. Let C be a (B-1)-st-path cut.We have that G\C does not contain a cycle of lenght ≤ B. Ifx is a solution and Cis minimum (B-1)-st-path cut, then - if x(C) < x(e), then there is a violated cycle inequality. - if not, then there is no violated cycle inequalities.

  18. s t 1.Bounded rings 1.2. Edge case Solving the minimum st-L-path cut problem when L ≤ 3 L ≤ 2 It suffices to calculate a min cut separating s and t in the graph Induced by the st-paths of length ≤ 2.

  19. 1.Bounded rings 1.2. Edge case L=3

  20. 1.Bounded rings 1.2. Edge case Theorem :The minimum st-L-path cut problem for L≤3 can be solved in polynomial time. Corollary :The separation problem for the cycle inequalities for B≤4 can be solved in polynomial time. Theorem :(Baier, Erlobach, Hall, Schilling, Skutella (2006)) The minimum st-L-path cut problem is NP-hard for L≥4. (Reduction from the vertex cover problem)

  21. 2. Bounded paths 2.1. General model 2. Bounded paths 2.1. A general model Given a graph with weights on the edges, a set D of terminal- pairs (origine-destinations), two intgers K, L, find a minimum weight subgraph such that between each pair of terminals in D there are at least K edge-disjoint paths of length (in number of edges (hops)) no more than L. The hop-constrained network design problem (HCNDP)

  22. 2. Bounded paths 2.1. General model The HCNDP is NP-hard in general NP-hard even for L=2 (Dahl (1998)) Polynomially solvable when |D|=1

  23. 2. Bounded paths 2.2. Special cases 2.2. Special cases and related problems 2.2.1. |D|=1, K=1, L fixed The minimum hop-constrained path problem Description of the associated polyhedron for L≤ 3. Dahl (1999) Formulation in the natural space of variables Valid inequalities Description of the associated polytope when L=2,3. Dahl & Gouveia (2004)

  24. 2. Bounded paths 2.2. Special cases Description of the assiciated polyhedron for all L. Nguyen (2003) Description of the polyhedron of the directed st-walks having exactly L arcs. Coullard, Gamble, Liu (1994) Description of the polytope of the directed st-walks having no more than L=4arcs. Extended formulation for the underlaying problem Dahl, Foldnes, Gouveia (2004)

  25. 2. Bounded paths 2.2. Special cases 2.2.2. K=1, L fixed, D is rooted The minimum hop constrained spanning tree problem Determine a minimum spanning tree such that the number of links between a root node and any node in the tree does not exceed a bound L. (NP-hard (even for L=2)) Multicommodity flow formulations Hop-indexed formulation Lagrangean relaxations Gouveia (1996,1998)

  26. 2. Bounded paths 2.2. Special cases Other Lagrangean relaxations Gouveia & Requejo (2001) Descriptionof the associated polytopeon a wheel when L=2 Dahl (1998) Minimum spanning trees with bounded diameter Integer programming formulation. Gouveia & Magnanti (2000) other modeling approach when the diameter is odd. Gouveia, Magnanti & Requejo (2004)

  27. 2. Bounded paths 2.2. Special cases 2.2.3. K=1 (and L arbitrary) Extended formulation, Lagrangean relaxation Balakrishnan, Altinkemer (1992) Multicommodity flow formulation and heuristics Pirkul, Sony (2003) 2.2.4. K=1, L=2 Formulation of the problem in the natural space of variables Valid inequalities Greedy approximation algorithms Cutting plane algorithm Dahl, Johannessen (2004)

  28. 2. Bounded paths 2.2. Special cases Length constrained 2-connected graphs Ben Ameur (1998, 2000) Classes of length constrained 2-connected graphs Lower bounds on the number of edges Valid inequalities for the 2-connected polytope with length constraints

  29. 3. Formulation, L≤3 3.1. Valid inequalities 3. Formulation for L≤ 3 3.1. Valid inequalities s t st-cut inequalities

  30. The L-st-path-cut inequalities (Dahl (1999)) Let V0,V1,...,VL+1be a partition of V such that sV0 and tVL+1. where (s,t) D. Suppose K=1. V V V V 1 2 L+1 3 V 0 T 3. Formulation, L≤3 3.1. Valid inequalities s t

  31. The L-st-path-cut inequalities (Dahl (1999)) Let V0,V1,...,VL+1be a partition of V such that sV0 and tVL+1. where (s,t) D. Suppose K=1. V V V V 1 2 L+1 3 V 0 3. Formulation, L≤3 3.1. Valid inequalities s t

  32. The L-st-path-cut inequalities (Dahl (1999)) Let V0,V1,...,VL+1be a partition of V such that sV0 and tVL+1. where (s,t) D. Suppose K=1. V V V V 3 L+1 1 2 x(T)≥ 1 s t (L-st-path-cut inequality) V 0 T 3. Formulation, L≤3 3.1. Valid inequalities

  33. 3. Formulation, L≤3 3.1. Valid inequalities If at least K paths are required between s and t, then x(T) ≥ K is valid for the corresponding polytope. The separation problem for the L-st-path cut inequalities can be solved in polynomial time, if L≤3. Fortz, M., McCormick, Pesneau (2006)

  34. min Subject to x(d(W)) ≥ Kfor allst-cut(W), for all (s,t) D 0 ≤ x(e) ≤ 1 for all e  E, x(e){0,1} for all e  E. 3. Formulation, L≤3 3.1. Valid inequalities Theorem:(Huygens, M., Pesneau (2004)) For L≤3, the HCNDP is equivalent to the following integer program x(T) ≥ K for all L-path-cut T, for all (s,t) D The linear relaxation of this program, when L≤3, can be solved in polynomial time by the ellipsoid method.

  35. 3. Formulation, L≤3 3.1. Valid inequalities Remark: The formulation given above is not valid for L≥4. L=4 s t Further inequalities are needed to formulate the problem for L≥4

  36. 4. The 2-edge case 4.1. Complexity 4. The Two edge connected hop-constrained network design problem (THNDP) That is the case whenK = 2 4.1. Complexity The THNDP is NP-hard in general (the 2-edge connected subgraph problem is a special case). Even more: Theorem:The THNDP is NP-hard when: - D is rooted - L≥2, and fixed - all edge weights are 1. Huygens, Labbé, M., Pesneau(2005)

  37. u2 V2 s w2 v2 z2 u u1 u1 v w w1 w1 v1 z V1 z1 4. The 2-edge case 4.1. Complexity Proof: (Outline) Reduction from the dominating set problem L=3 G’=(V’,E’) G=(V,E)

  38. 4. The 2-edge case 4.1. Complexity • Lemma:A minimum cardinality solution S* to the rooted THNDP • in G’, w.r.t. s and the nodes of V2, can be chosen so that: • S* contains all the paths between s and V2, • S* contains exactly |V| paths between V1 and V2 that cover • all the nodes of V2. Thus the rooted THNDP in G’ reduces to finding a minimum cardianlity subset of V1 that covers all the nodes of V2. This subset corresponds to a dominating set in G.

  39. u2 V2 s w2 v2 z2 u u1 u1 v w w1 w1 v1 z V1 z1 4. The 2-edge case 4.1. Complexity Proof: (Outline) Reduction from the dominating set problem L=3 G’=(V’,E’) G=(V,E)

  40. 4. The 2-edge case 4.1. Complexity However, If the graph is complete and all edge weights are equal to 1, the rooted THNDP can be solved in polynomial time for evey fixed L≥2. (The algorithm is linear.)

  41. 4. The 2-edge case 4.2. Polyhedralresults 4.2. Polyhedral results 4.2.1. THNDP polytope when L=2, 3, |D|=1. Theorem:(Huygens, M., Pesneau (2004)) If D={(s,t)} and L=2,3, then the THNDP polytope is given by the inequalities x(d (W )) ≥ 2for allst-cut(W), x(T) ≥ 2 for allL-st-path-cutT, 0 ≤ x(e) ≤ 1for alle  E.

  42. 4. The 2-edge case 4.2. Polyhedralresults Theorem:(Dahl, Huygens, M. Pesneau (2005)) If D={(s,t)}, L=2, and K arbitrary, then the HCNDP polytope is given by the inequalities x (d (W)) ≥ Kfor allst-cut (W), x (T) ≥ K for allL-st-path-cutT, 0 ≤ x (e) ≤ 1for alle  E.

  43. V V V V 3 1 2 4 F V 0 4. The 2-edge case 4.2. Polyhedralresults 4.2.2. Valid inequalities (Huygens, Labbé, M., Pesneau (2005)) a) Double cut inequalities L=3 s1 t2 t1 e F=E\([V2,V3][V3,V4]{e})

  44. V V V V 1 2 4 3 V 0 4. The 2-edge case 4.2. Polyhedralresults 4.2.1. Valid inequalities (Huygens, Labbé, M., Pesneau (2005)) a) Double cut inequalities L=3 s1 t2 t1 e

  45. V V V V 1 2 4 3 F V 0 x(F) ≥ 3 Double cut inequality 4. The 2-edge case 4.2. Polyhedralresults 4.2.1. Valid inequalities (Huygens, Labbé, M., Pesneau (2005)) a) Double cut inequalities L=3 s1 t2 t1 e F=E\([V2,V3][V3,V4]{e})

  46. 4. The 2-edge case 4.2. Polyhedralresults b) Rooted-partition inequalities Theorem:Let U={t1,…,tp} be a subset of p destination nodes relatively to node s. Let (V0,V1,…,Vp) be a partition of the node set V such that sV0, tiVi, for all i=1,…,p. Then the inequality t1 V1 x((V0,V1,…,Vp)) ≥ p+p/L V0 t2 V2 is valid. s V3 rooted-partition inequality t3 t4 Theorem: If L=2, a rooted-partition inequality defines a facet only if p is odd and Vi={ti} for i=1,…,p. tp V4 Vp

  47. 4. The 2-edge case 4.2. Polyhedralresults Theorem:The separation problem for the rooted-partition inequalities when L=2, p odd, and Vi={ti} for i=1,…,p, can be solved in polynomial time. Proof:(Outline) Byreduction to the minimization of a submodular function.

  48. 4. The 2-edge case 4.3. Branch&Cut 4.3. Branch&Cut algorithm (Huygens, Labbé, M., Pesneau (2005)) THNDP,L=2,3. Used constraints: trivial inequalities st-cut inequalities L-st-path-cutinequalities double cutinequalities rooted-partition inequalities (and other inequalities)

  49. 4. The 2-edge case 4.3. Branch&Cut Some computational results - Random and real instances, - Max runtime: 5 hours. - The double cut and rooted partition inequalities are separated heuristically.

  50. 4. The 2-edge case 4.3. Branch&Cut Results for random instances for L=2, 3 and rooted demands

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