EXACT ALGORITHMS FOR THE ATSP

Transcription

1 EXACT ALGORITHMS FOR THE ATSP Branch-and-Bound Algorithms: Little-Murty-Sweeney-Karel (Operations Research, ); Bellmore-Malone (Operations Research, ); Garfinkel (Operations Research, ); Smith-Srinivasan-Thompson (Annals Discrete Math., ); Carpaneto-T. (Management Science, 80); Balas-Christofides (Mathematical Programming, 8); Pekny-Miller (Oper. Res. Letters, 8); Fischetti-T. (Mathematical Programming, ); Pekny-Miller (Mathematical Programming, ); Carpaneto-Dell Amico-T. (ACM Trans. Math. Software, );

2 EXACT ALGORITHMS FOR THE ATSP Branch-and-Cut Algorithms: Fischetti-T. (Management Science, ); Fischetti-Lodi-T. (LNCS Springer, 00); Surveys: Balas-T. (The Traveling Salesman Problem, Lawler et al. eds, Wiley, 8); Fischetti-Lodi-T.(The TSP and its Variations, Gutin- Punnen eds, Kluwer, 00); Roberti-T. (EURO Journal Transp. and Logistics, 0);

3 CLASSICAL LOWER BOUNDS ASSIGNMENT PROBLEM (AP) RELAXATION (O(n ) time) s.t. min Σ Σ c ij ij i V j V Σ ij = j V i V Σ ij = j V i V Σ Σ ij S - S V, S Connectivity Constraints i S j S ij {0, } i V, j V ij 0 (LP Relaation) i V, j V * The AP solution is given by a family of subtours (partial circuits)

4 Eample: AP relaation of ATSP n = V = 8 8 Optimal assignment

5 Eample: AP relaation of ATSP n = V = Optimal assignment Lower bound = v(ap) = ( + + ) + ( + + ) = Optimal solution Cost =

6 r SHORTEST SPANNING ARBORESCENCE RELAXATION s.t. min Σ Σ c ij ij i V j V Σ ij = i V out-degree constraints j V Σ ij = i V j V in-degree constraints Σ Σ ij S V, r S (for a fied r V) i S j V\S ij {0, } i V, j V

7 r SHORTEST SPANNING ARBORESCENCE RELAXATION Partition the Arc Set A into two subsets: (i,j) (i,r) i V, j V \ r i V s.t. min Σ Σ c ij ij i V j V\r + Σ c ir ir i V Σ ij = j V \ r i V Σ ir = i V in-degree constraints Σ Σ ij S V, r S (j V \ r ) i S j V\S ij {0, } ir {0, } i V, j V \ r i V The Relaed Problem can be split into two independent problems

8 r - SHORTEST SPANNING ARBORESCENCE RELAXATION Problem concerning the arc set: (i,j) i V, j V \ r s.t. min Σ Σ c ij ij i V j V\r Σ ij = j V \ r in-degree constraints i V Σ Σ ij S V, r S connectivity constraints from r i S j V\S ij {0, } ij 0 (LP Relaation) i V, j V \ r i V, j V \ r The problem calls for an r-ssa: (O(n ) time) Shortest (Minimum Cost) Spanning Arborescence rooted at verte r (involving n arcs) 8

9 r SHORTEST SPANNING ARBORESCENCE RELAXATION Problem concerning the arc set: (i,r) i V s.t. min Σ c ir ir i V Σ ir = i V ir {0, } in-degree constraint for verte r i V ir 0 (LP Relaation) i V The problem calls for an r-mca: Minimum Cost Arc entering verte r (O(n) time) Lower Bound = v(r-ssar) = v(r-ssa) + v(r-mca)

10 0 Eample: r-ssa relaation of ATSP (r = ) Optimal -SSA, v(-ssa) = = 8 Optimal -MCA, v(-mca) =, Lower bound = 8 + =, Optimal solution Cost = 8

11 r SHORTEST SPANNING ARBORESCENCE RELAXATION Different choices of the root verte r generally produce different values of the corresponding lower bounds. The r Shortest Spanning Anti Arborescence Relaation can be used as well (connectivity toward r). The lower bounds can be strengthened through Lagrangian Relaation of the out-degree constraints, and Subgradient Optimization Procedures (to determine good Lagrangian multipliers).

12 Eample: r-ssa relaation of ATSP (r = ) Optimal -SSA, v(-ssa) = = Optimal -MCA, v(-mca) =, Lower bound = + =, Optimal solution Cost = 8

13 Eample: r-anti-ssa relaation of ATSP (r = ) Optimal -Anti-SSA, v(-anti-ssa) = = Optimal -Anti-MCA, v(-anti-mca) =, Lower bound = + =, Optimal solution Cost = 8

14 LAGRANGIAN RELAXATION OF THE r - SHORTEST SPANNING ARBORESCENCE RELAXATION (u i : Lagrangian multiplier associated with the i-th out-degree constraint) s.t. min Σ Σ c ij ij i V j V u i Σ ij = i V out-degree constraints j V Σ ij = i V j V in-degree constraints Σ Σ ij S V, r S (for a fied r V) i S j V\S ij {0, } i V, j V

15 (u i : Lagrangian multiplier associated with the i-th out-degree constraint) s.t. LAGRANGIAN RELAXATION OF THE r SHORTEST SPANNING ARBORESCENCE RELAXATION z(u) = min Σ Σ c ij ij + Σ u i ( Σ ij - ) i V j V i V j V u i Σ ij = i V out-degree constraints j V Σ ij = i V j V in-degree constraints Σ Σ ij S V, r S (for a fied r V) i S j V\S ij {0, } i V, j V

16 LAGRANGIAN RELAXATION OF THE r SHORTEST SPANNING ARBORESCENCE RELAXATION z(u) = min Σ Σ c ij ij + Σ u i ( Σ ij - ) i V j V i V j V = - Σ u i + min Σ Σ (c ij + u i ) ij i V i V j V = - Σ u i + min Σ Σ c ij ij (where c ij = c ij + u i ) i V i V j V c ij is the Lagrangian cost of arc (i,j) ( with (i,j) A) LAGRANGIAN DUAL PROBLEM Find the optimal Lagrangian multiplier vector u* such that: z(u*) = ma {z(u)} u (with z(u) nondifferentiable function ) Difficult problem (heuristic solution through Subgradient Optimization Procedures)

17 SUBGRADIENT OPTIMIZATION PROCEDURE (Held-Karp, Operations Research, for STSP) Lagrangian cost: c ij = c ij + u i (i,j) A * Initially: u i := 0 i V ; LB := 0 At each iteration: * Solve the r-ssa Relaation with respect to the current Lagrangian costs c ij : ( ij ) is the corresponding optimal solution * LB := ma (LB, v( ij )) * Subgradient vector (s i ): s i := Σ ij - i V j V * If s i = 0 for all i V then STOP (( ij ) is the opt. sol. of ATSP) * u i := u i + a s i i V where a := b (UB LB) / Σ s i ( with 0 < b ) (a > 0) i V Stopping criteria: ma number of iterations, slow increase of LB,

18 SUBGRADIENT OPTIMIZATION PROCEDURE: UPDATING OF THE LAGRANGIAN MULTIPLIERS * c ij = c ij + u i (with (i,j) A) * u i := u i + a s i i V (with a positive) * s i := Σ ij - i V j V * d i = outdegree of verte i (i V) (d i = 0 if s i = -; d i = if s i = 0; d i if s i ) * the Lagrangian cost c ij of arc (i,j) is decreased if d i = 0. unchanged if d i =, increased if d i. Note that: a Σ s i = a ( Σ Σ ij - n) = a (n n) = 0 i V i V j V Σ u i = 0 (if initially u i = 0 i V ) i V 8

19 GREEDY ALGORITHM NEAREST NEIGHBOR FOR ATSP. Choose any verte h as initial verte of the current path. Set i:= h, V := V \ {i} (set of the unvisited vertices).. Determine the unvisited verte k nearest to verte i (k : c ik = min {c ij : j V }).. Insert verte k just after verte i in the current path (V := V \ {k}); set i:= k; If V (at least one verte is unvisited) return to STEP.. Complete the Hamiltonian circuit with arc (i, h); STOP. (Upper Bound computation) v Time compleity: O(n ). v Different choices of the initial verte lead to different solutions.

20 0 Eample: Nearest Neighbor Heuristic Algorithm (Upper Bound computation) 8 UB = = Initial verte

21 Eample: r-ssa relaation of ATSP (r = ) Subgradient Procedure (b = 0., UB = ) 8 LB = 8 + = (u) = (0, 0, 0, 0, 0, 0) (d) = (,, 0,,, ), (s) = (, 0, -, 0, 0, 0) a := b (UB LB) / Σ s i = 0. ( ) / = (u) = (u) + a (s) i V (u) = (, 0, -, 0, 0, 0)

22 Eample: r-ssa relaation of ATSP (r = ) 8 Subgradient Procedure (a = ) (u) = (, 0, -, 0, 0, 0) (c ) = (c) + (u)

23 Eample: r-ssa relaation of ATSP (r = ) 8 Subgradient Procedure (a =, LB = ) (d) = (,, 0,,, ) LB = =

24 The AP Relaation can be strengthened by combining the AP substructure with the SSA substructures: Restricted Lagrangian Relaation (Balas-Christofides, 8): ) Lagrangian Relaation of a subset of Subtour Elimination Constraints (SECs); ) Solve the corresponding AP problem (w.r.t. the current Lagrangian costs); ) Determine good Lagrangian multipliers (through a subgradient optimization procedure) and additional SECs, to be inserted into the current Lagrangian relaation and iterate on Steps ) and ).

25 Additive Bounding Procedure (Fischetti-T., Math.Progr.) ) Solve the AP Relaation through a primal-dual algorithm (O(n ) time) LB := v(ap), c ij := reduced cost of arc (i,j); (O(n ) time): ) Solve the r-ssa Relaation w.r.t. costs c ij LB := LB + v(r-ssar), c ij := new reduced cost of arc (i,j); ) Solve the r-anti-ssa Relaation w.r.t. costs c (O(n ) time): ij LB := LB + v(r-anti-ssar), c ij := new reduced cost of arc (i,j); Iterate Steps ) and ) with different root nodes r. (globally O(n ) time)

26 Eample: Additive Bounding Procedure r-ssa Relaation (r = ) LB = (c ) = AP reduced costs Optimal assignment v(-ssar) = = LB = LB + v(-ssar) = + =

27 Eample: Additive Bounding Procedure r-anti-ssa Relaation (r = ) LB = (c ) = -SSAR reduced costs v(-anti-ssar) = = LB = LB + v(-anti-ssar) = + =

28 BRANCH-AND-BOUND ALGORITHM FOR ATSP At each node of the decision tree solve the AP-Relaation of the corresponding subproblem (LB= v(ap)). If LB Z* fathom the node (Z* = cost of the best solution). If the AP solution contains no subtour (feasible solution), update Z* (and the best solution) and fathom the node. Otherwise: SUBTOUR-ELIMINATION BRANCHING SCHEME: - Select the subtour S with the minimum number h of not imposed arcs. BASED ON THE AP RELAXATION (Carpaneto-T., Man. Sc. 80) - Generate h descendent nodes so as to forbid subtour S for each of them (by imposing and ecluding proper arc subsets). 8

29 BRANCHING TREE FOR ATSP AP(k) solution level... arc (8, ) imposed X8, = k X, = 0... level X, = 0 X,8 = 0 X, = X, = 0 X, = X, = X,8 = X, = X,8 = k k k k

30 BRANCH-AND-BOUND ALGORITHM by Carpaneto-Dell Amico-T. (ACM TOMS, ) At the root node of the decision tree: - solve the AP Relaation; - apply the Patch Heuristic Algorithm (Karp-Steele, 8) to determine an initial tour of cost Z*; - apply a Reduction Procedure (based on the AP reduced costs c ij ) to fi to zero as many variables as possible (transformation of the complete graph into a sparse one): if v(ap) + c ij Z* set ij = 0 (i.e. remove arc (i,j) from A). At each node, the corresponding AP Relaation is computed (by using effective parametric techniques) in O(n ) time. 0

31 BRANCH-AND-CUT ALGORITHMS (Padberg - Rinaldi for STSP, 8-) At each node, a Lower Bound is obtained by solving an LP Relaation of the corresponding subproblem, containing only a subset of the constraints (degree constraints, some connectivity constraints, ). The LP Relaation is iteratively tightened by adding valid inequalities that are violated by the current optimal LP solution (* ij ). These inequalities are identified by solving the Separation Problem: - given a solution (* ij ), find a member a b of a given family F of valid inequalities for ATSP, such that a * > b holds ( maimum of d = a * - b, with d > 0 ). * Eact or heuristic Separation Procedures can be used.

32 SEPARATION PROBLEM FOR THE CONNECTIVITY CONSTRAINTS Given an optimal LP solution (* ij ) such that: Σ * ih = Σ * hj = i V i V h V does eist a verte subset S ( S V, r S) such that: Σ Σ * ij <? i S j V\S

33 EXACT SEPARATION PROCEDURE FOR THE CONNECTIVITY CONSTRAINTS For each verte t V \ {r}, define the NETWORK G* = (V,A), source = r, sink = t capacity of arc (i,j) = * ij CAPACITY of cut (S, V \ S) with r S and t V \ S = Σ Σ i S j V\S * ij

34 Eample NETWORK G* = (V,A) capacity (*) 8 S r t CAPACITY of cut (S, V \ S) = 0 Only degree constraints imposed

35 EXACT SEPARATION PROCEDURE FOR THE CONNECTIVITY CONSTRAINTS Minimum Capacity Cut from r to t = Maimum Flow from r to t Determine the MAXIMUM FLOW from r to t: capacity of cut (S*, V \ S*) * If Maimum Flow < then constraint Σ Σ ij is violated i S* j V\S* (most violated connectivity constraint with r S and t V \ S) O(n ) time for each verte t (global time O(n ) )

36 Eample NETWORK G* = (V,A) capacity (*) S* r 8 Minimum Capacity Cut from r to t = (S*, V\S*) Maimum Flow from r to t = 0: Σ Σ = 0 t i S* j V\S* ij

37 BRANCH-AND-CUT ALGORITHM (Fischetti-T., Man. SC. ) Separation procedures for the identification of the following valid inequalities: - Connectivity constraints (eact alg.); - Comb inequalities (heur. alg.); - D k + and D k - inequalities (Groetschel-Padberg, 8) (eact and heur. alg.); - ODD CAT inequalities (Balas, 8) (heur. alg.)

38 PRICING PROCEDURE A pricing procedure is applied to decrease the number of variables considered in the LP Relaation: ) Select a subset T of variables (other variables set to zero); ) Solve the LP Relaation by considering only the variables in T; ) Compute the reduced cost of the variables not in T; ) Add to T (a subset of) the variables with negative reduced cost and return to Step ); if no negative reduced cost is found then STOP (the current solution is the optimal solution of the original LP Relaation). An effective Pricing Scheme, based on the optimal solution of an associated Assignment Problem, is used to decrease the number of variables added to T in Step ). 8

39 BRANCHING SCHEME (Fischetti-Lodi-T., 00) Select a fractional variable * ij close to 0. and having been persistently fractional in the last optimal LP solutions. Generate descendent nodes by imposing: ij = and ij =0, respectively.

40 TRANSFORMATION OF ATSP INTO STSP Any ATSP instance with n vertices can be transformed into an equivalent STSP instance with n nodes (Jonker-Volgenant, 8; Junger-Reinelt-Rinaldi, ; Kumar-Li, 00). Very effective Branch-and-Cut algorithms for STSP have been proposed: - Padberg-Rinaldi (Oper. Res. Letters 8, SIAM Review ) - Groetschel-Holland (Math. Progr. ) - Applegate-Biby-Chvatal-Cook (Doc. Math., J. DMV 8, Concorde Code, Princeton Univ. Press 00). * Concorde Code Most effective code for STSP 0

41 TRANSFORMATION OF ATSP INTO STSP Given a directed graph G = (V,A) with n vertices and m arcs, build an undirected graph G` = (V`,E) with n nodes and (m + n) edges: - for each verte i V consider two nodes : i and (n + i); - for each verte i V consider an edge (i, n + i) with cost 0; - for each arc (i, j) A consider an edge (n + i, j) with cost c ij + M where M is a sufficiently large positive value Cost of ATSP = Cost of STSP - n M

42 TRANSFORMATION OF ATSP INTO STSP arc (i, j): edge (n + i, j) 8 0 +M 0 +M +M M 0 +M +M n = 0 0+M

43 TRANSFORMATION OF ATSP INTO STSP 8 0 Optimal solution Cost = +M 0 +M 0 8+M +M 0 +M 0+M 8 0 +M Cost = + M