A branch-and-cut algorithm for the Minimum Labeling Hamiltonian Cycle Problem and two variants

Transcription

1 A branch-and-cut algorithm for the Minimum Labeling Hamiltonian Cycle Problem and two variants Nicolas Jozefowiez 1, Gilbert Laporte 2, Frédéric Semet 3 1. LAAS-CNRS, INSA, Université de Toulouse, Toulouse, France, nicolas.jozefowiez@laas.fr 2. CIRRELT, HEC, Montréal, Canada, gilbert@crt.umontreal.ca 3. LAGIS, Ecole Centrale de Lille, Villeneuve d Ascq, France, frederic.semet@ec-lille.fr 1 / 21

2 The Minimum Labeling Hamiltonian Cycle Problem (MLHCP) Data: Goal: G = (V, E) : an undirected graph C is a set of colors Each e E has a color k C Find a Hamiltonian cycle Minimize the number of colors appearing on the cycle 2 / 21

3 State-of-the-art Minimum labelling hamiltonian cycle problem (Tabu search) [Cerulli, Dell Olmo, Gentili, Raiconi, 2006] Colorful traveling salesman problem (heuristic, GA) [iong, Golden, Wasil, 2007] Traveling salesman problem with labels (approximation algorithm) [Gourvès, Monnot, Telelis, 2008] Minimum labelling spanning tree problem 3 / 21

4 Two variants Valuations on the edges Minimum Labeling Hamiltonian Cycle Problem with Length Constraint (MLHCPLC) Minimize the number of colors appearing on the cycle Impose a bound on the length of the tour Label Constrained Traveling Salesman Problem (LCTSP) Minimize the length of the tour Impose a bound on the number of colors appearing on the tour 4 / 21

5 Integer program for the MLHCP Variables x e = u k = ( 1 if e E is used, 0 otherwise. ( 1 if k C is used, 0 otherwise. Constants and notations e E, δ(e) = k C the color of e k C, ζ(k) = {e E δ(e) = k} S V, ω(s) = {e = (i, j) E i S and j V \ S} 5 / 21

6 Integer program for the MLHCP Objective functions min u k k C Constraints x e = 2 i V e ω({i}) x e 2 S V, 3 S V 3 e ω(s) x e u δ(e) e E x e {0, 1} e E u k {0, 1} k C Valid constraints u k e ζ(k) x e k C 6 / 21

7 Adaptations to the MLHCPLC and LCTSP MLHCPLC : Add constraint P e E cexe ɛ LCTSP : Replace objective with min P e E cexe Add constraint P k C u k ɛ 7 / 21

8 Valid inequalities Proposition. Let T E. If the constraint α ex e β e T is valid for the TSP, then for a label k C, the inequality α ex e βu k e T ζ(k) is valid for the MLHCP, the MLHCPLC, and the LCTSP. 8 / 21

9 Valid inequalities (cont.) Proposition. The constraints e ω({i}) ζ(k) x e 2u k (i V, k C, 3 ω({i}) ζ(k) V 3) are valid for the MLHCP, the MLHCPLC, and the LCTSP. Proposition. The constraints x e ( S 1)u k (k C, S V, 3 S V 3) e E(S) ζ(k) are valid for the MLHCP, the MLHCPLC, and the LCTSP. 0,5 S 1 i 0,5 j 1 1 0,5 0,5 1 0,5 i ,5 0,5 1 0,5 0,5 1 l 0,5 k 1 9 / 21

10 Valid inequalities (cont.) Proposition. Let T E. If the inequality α ex e β e T is valid for the TSP, then the inequality min{ k C e T ζ(k) α e, β}u k β is valid for the MLHCP, the MLHCPLC, and the LCTSP. Proposition. The inequalities γ k (S)u k 2 (S V, 3 S V 3) k C with 8 >< 0 if e ω(s), e ζ(k) γ k (S) = 1 if!e ω(s), e ζ(k) >: 2 otherwise. are valid for the MLHCP, the MLHCPLC, and the LCTSP. 10 / 21

11 Polyhedral results for the MLHCP Lemma A solution for the TSP is also a solution for the MLCHP. Lemma If constraints u k P e ζ(k) xe are added to the model, there is a one to one correspondence between the solutions of the TSP and those of the MLHCP. Theorem. Let k C and i V, if 3 ω({i}) ζ(k) V 3 and ζ(k) \ (ω({i}) ζ(k)), then F1 ki = {(x, u) C MLHCP P e ω({i}) ζ(k) xe 2u k = 0} is a proper face of C MLHCP. Theorem. Let k C and S V, if 3 S V 3 and i S, ω({i}) ζ(k) S 2, then F2 ks = {(x, u) C MLHCP P e E(S) ζ(k) xe ( S 1)u k = 0} is a proper face of C MLHCP. 11 / 21

12 Heuristic for the LCTSP STEP 1: Solve the following mixed integer program (with a time limit) to obtain a solution ( x, ū): min c ex e + m u k e E k C e ω({i}) x e = 2 i V x e u δ(e) e E u k x e k C e ζ(k) u k ɛ k C 0 x e 1 e E u k {0, 1} k C STEP 2: If no feasible solution was found, ɛ ɛ 1. Go to STEP 1 STEP 3: Solve a TSP on G = (V, E) with new costs c e = 1 if ū δ(e) = 1, V max v E c v + c e otherwise (e E). STEP 4: If the tour obtained uses less than ɛ colors, return the solution, otherwise ɛ ɛ 1 and go to STEP / 21

13 Branch-and-cut algorithm Step 1 (Root of the tree) Generate an initial upper bound ub. Define a first subproblem. Insert the subproblem in a list L. Step 2 (Stopping criterion) If L = then STOP, else choose a subproblem from L and remove it from L. Step 3 (Subproblem solution) Solve the subproblem to obtain the lower bound lb. Step 4 (Constraint generation) if lb ub then Go to Step 2. else if the solution is integer then ub lb. Go to Step 2. else if violated constraints are identified then Add them to the model and go to Step 3. else Go to Step 5. end if end if Step 5 (Branching) Branch on a variable and introduce the corresponding subproblems in L. Go to Step / 21

14 Branch-and-cut algorithm : initial subproblem min u k k C x e = 2 i V e ω({i}) x e u δ(e) e E u k x e k C e ζ(k) γ k (S)u k 2 (S V, 3 S V 3) k C x e ( S 1)u k (k C, S = i, j, k V, e S S, u δ(e) = k) e E(S) ζ(k) 0 x e 1 e E 0 u k 1 k C 14 / 21

15 Branch-and-cut algorithm: constraint generation Connectivity constraints x e 2 (S V, 3 S V 3) e ω(s) γ k (S)u k 2 (S V, 3 S V 3) k C with 8 >< 0 if e ω(s), e ζ(k) γ k (S) = 1 if!e ω(s), e ζ(k) >: 2 otherwise. Min-cut problem : G m = (V m, E m), find S V m minimizing P e ω(s m) ce CONCORDE function [Padberg & Rinaldi, 1990] 15 / 21

16 Branch-and-cut algorithm: branching and initial upper bound Branching In priority on the fractional u K with the largest value of ζ(k) Then, on the fractional variable x e closest to 0.5 Initial upper bound MLHCP : none MLHCPLC, LCTSP : heuristic 16 / 21

17 Computational results C Intel Core 2 Duo E Ghz CPU CPLE CONCORDE library TSPLIB for the MHLCPLC and LCTSP An edge (i, j) is given the label C (ija ija ), where A = ( 5 1)/2 17 / 21

18 Computational results: MLHCP V C Opt lb/opt #Nodes #u #x #Cuts Seconds / 21

19 Computational results: LCTSP Inst./ C /% Opt lb/opt #Nodes #u #x #Cuts Seconds kroa100/50/ / kroa100/50/ / kroa100/50/ /37* kroa100/100/ / kroa100/100/ /50* kroa100/100/ /62* krob100/50/ / krob100/50/ /25* krob100/50/ /37* krob100/100/ / krob100/100/ / krob100/100/ / eil101/50/25 931/ eil101/50/50 693/25* eil101/50/75 638/37* eil101/100/25 803/ eil101/100/50 643/50* eil101/100/75 629/66* lin105/50/ /12* lin105/50/ /25* lin105/50/ /37* lin105/100/ / lin105/100/ /50* lin105/100/ /68* / 21

20 Computational results: MLHCPLC Inst./ C /ɛ Opt lb/opt #Nodes #u #x #Cuts Seconds kroa100/50/ / kroa100/50/ /24812* kroa100/50/ /21639* kroa100/100/ / kroa100/100/ / kroa100/100/ / krob100/50/ /35848* krob100/50/ /24912* krob100/50/ /22361* krob100/100/ / krob100/100/ /23095* krob100/100/ / eil101/50/931 12/931* eil101/50/693 25/693* eil101/50/638 37/638* eil101/100/803 25/ eil101/100/643 50/643* eil101/100/629 63/629* lin105/50/ /23803* lin105/50/ /16208* lin105/50/ /14635* lin105/100/ /20180* lin105/100/ /15021* lin105/100/ /14379* / 21

21 Conclusions Variants of the TSP Valid inequalities Polyhedral results on the MLHCP Heuristics for the MLHCPLC and LCTSP Branch-and-cut algorithm 21 / 21