A branch-and-cut algorithm for the Minimum Labeling Hamiltonian - - PowerPoint PPT Presentation

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

Nicolas Jozefowiez1, Gilbert Laporte2, Fr´ ed´ eric Semet3

• 1. LAAS-CNRS, INSA, Universit´

e de Toulouse, Toulouse, France, nicolas.jozefowiez@laas.fr

• 2. CIRRELT, HEC, Montr´

eal, Canada, gilbert@crt.umontreal.ca

• 3. LAGIS, Ecole Centrale de Lille, Villeneuve d’Ascq, France, frederic.semet@ec-lille.fr

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The Minimum Labeling Hamiltonian Cycle Problem (MLHCP)

Data: G = (V, E) : an undirected graph C is a set of colors Each e ∈ E has a color k ∈ C Goal: Find a Hamiltonian cycle Minimize the number of colors appearing on the cycle

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State-of-the-art

Minimum labelling hamiltonian cycle problem (Tabu search) [Cerulli, Dell’Olmo, Gentili, Raiconi, 2006] Colorful traveling salesman problem (heuristic, GA) [Xiong, Golden, Wasil, 2007] Traveling salesman problem with labels (approximation algorithm) [Gourv` es, Monnot, Telelis, 2008] Minimum labelling spanning tree problem

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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

Integer program for the MLHCP

Variables xe = ( 1 if e ∈ E is used,

• therwise.

uk = ( 1 if k ∈ C is used,

• therwise.

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}

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Integer program for the MLHCP

Objective functions min X

k∈C

uk Constraints X

e∈ω({i})

xe = 2 ∀i ∈ V X

e∈ω(S)

xe ≥ 2 ∀S ⊂ V, 3 ≤ |S| ≤ |V | − 3 xe ≤ uδ(e) ∀e ∈ E xe ∈ {0, 1} ∀e ∈ E uk ∈ {0, 1} ∀k ∈ C Valid constraints uk ≤ X

e∈ζ(k)

xe ∀k ∈ C

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Adaptations to the MLHCPLC and LCTSP

MLHCPLC : Add constraint P

e∈E cexe ≤ ǫ

LCTSP : Replace objective with min P

e∈E cexe

Add constraint P

k∈C uk ≤ ǫ 7 / 21

Valid inequalities

• Proposition. Let T ⊆ E. If the constraint

X

e∈T

αexe ≤ β is valid for the TSP, then for a label k ∈ C, the inequality X

e∈T ∩ζ(k)

αexe ≤ βuk is valid for the MLHCP, the MLHCPLC, and the LCTSP.

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Valid inequalities (cont.)

• Proposition. The constraints

X

e∈ω({i})∩ζ(k)

xe ≤ 2uk (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∈E(S)∩ζ(k)

xe ≤ (|S| − 1)uk (k ∈ C, S ⊂ V, 3 ≤ |S| ≤ |V | − 3) are valid for the MLHCP, the MLHCPLC, and the LCTSP.

1 i 0,5 0,5 0,5 0,5 0,5 0,5 1 1 1 S 0,5 0,5 1 1 1 0,5 0,5 l k j i 1 1 1

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Valid inequalities (cont.)

• Proposition. Let T ⊆ E. If the inequality

X

e∈T

αexe ≥ β is valid for the TSP, then the inequality X

k∈C

min{ X

e∈T ∩ζ(k)

αe, β}uk ≥ β is valid for the MLHCP, the MLHCPLC, and the LCTSP.

• Proposition. The inequalities

X

k∈C

γk(S)uk ≥ 2 (S ⊂ V, 3 ≤ |S| ≤ |V | − 3) with γk(S) = 8 > < > : if ∄e ∈ ω(S), e ∈ ζ(k) 1 if ∃!e ∈ ω(S), e ∈ ζ(k) 2

• therwise.

are valid for the MLHCP, the MLHCPLC, and the LCTSP.

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Polyhedral results for the MLHCP

Lemma A solution for the TSP is also a solution for the MLCHP. Lemma If constraints uk ≤ P

e∈ζ(k) xe are added to the model, there is a one to

• ne 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 F ki

1

= {(x, u) ∈ CMLHCP| P

e∈ω({i})∩ζ(k) xe − 2uk = 0} is a proper face of

CMLHCP.

• Theorem. Let k ∈ C and S ⊂ V , if 3 ≤ |S| ≤ |V | − 3 and

∀i ∈ S, |ω({i}) ∩ ζ(k)| ≥ |S|

2 , then

F kS

2

= {(x, u) ∈ CMLHCP| P

e∈E(S)∩ζ(k) xe − (|S| − 1)uk = 0} is a proper face of

CMLHCP.

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Heuristic for the LCTSP

STEP 1: Solve the following mixed integer program (with a time limit) to obtain a solution (¯ x, ¯ u): min X

e∈E

cexe + m X

k∈C

uk X

e∈ω({i})

xe = 2 ∀i ∈ V xe ≤ uδ(e) ∀e ∈ E uk ≤ X

e∈ζ(k)

xe ∀k ∈ C X

k∈C

uk ≤ ǫ 0 ≤ xe ≤ 1 ∀e ∈ E uk ∈ {0, 1} ∀k ∈ C STEP 2: If no feasible solution was found, ǫ ← ǫ − 1. Go to STEP 1 STEP 3: Solve a TSP on G′ = (V, E) with new costs c′

e = 1 if ¯

uδ(e) = 1, |V | maxv∈E cv + ce otherwise (e ∈ E). STEP 4: If the tour obtained uses less than ǫ colors, return the solution, otherwise ǫ ← ǫ − 1 and go to STEP 1.

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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 2. 13 / 21

Branch-and-cut algorithm : initial subproblem

min X

k∈C

uk X

e∈ω({i})

xe = 2 ∀i ∈ V xe ≤ uδ(e) ∀e ∈ E uk ≤ X

e∈ζ(k)

xe ∀k ∈ C X

k∈C

γk(S)uk ≥ 2 (S ⊂ V, 3 ≤ |S| ≤ |V | − 3) X

e∈E(S)∩ζ(k)

xe ≤ (|S| − 1)uk (k ∈ C, S = i, j, k ⊂ V, ∀e ∈ S × S, uδ(e) = k) 0 ≤ xe ≤ 1 ∀e ∈ E 0 ≤ uk ≤ 1 ∀k ∈ C

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Branch-and-cut algorithm: constraint generation

Connectivity constraints X

e∈ω(S)

xe ≥ 2 (S ⊂ V, 3 ≤ |S| ≤ |V | − 3) X

k∈C

γk(S)uk ≥ 2 (S ⊂ V, 3 ≤ |S| ≤ |V | − 3) with γk(S) = 8 > < > : if ∄e ∈ ω(S), e ∈ ζ(k) 1 if ∃!e ∈ ω(S), e ∈ ζ(k) 2

• therwise.

Min-cut problem : Gm = (Vm, Em), find S ⊂ Vm minimizing P

e∈ω(Sm) ce

CONCORDE function [Padberg & Rinaldi, 1990]

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Branch-and-cut algorithm: branching and initial upper bound

Branching In priority on the fractional uK with the largest value of |ζ(k) Then, on the fractional variable xe closest to 0.5 Initial upper bound MLHCP : none MLHCPLC, LCTSP : heuristic

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Computational results

C Intel Core 2 Duo E6550 2.33Ghz CPU CPLEX 11.1.2 CONCORDE library TSPLIB for the MHLCPLC and LCTSP An edge (i, j) is given the label ⌊|C|(ijA − ⌊ijA⌋)⌋, where A = ( √ 5 − 1)/2

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Computational results: MLHCP

|V | |C| Opt lb/Opt #Nodes #u #x #Cuts Seconds 50 50 4 0.552 113 56 33 19 50 100 5 0.691 37 18 2 4 50 150 6 0.749 41 20 4 3 50 200 7 0.783 3427 1711 2 75 295 100 50 3 0.572 13 6 26 100 100 4 0.561 143 68 3 7 290 100 150 5 0.587 973 486 6 2036 100 200 6 0.602 6587 3289 4 87 12583 150 50 2 0.826 5 1 1 7 46 150 100 3 0.667 11 5 9 91 150 150 4 0.626 81 40 2 524 150 200 5 0.580 1379 688 1 16 8872 200 50 2 0.812 3 1 4 152 200 100 3 0.643 31 3 12 17 289 200 150 4 0.564 285 140 2 12 4105 200 200 4 0.617 133 59 7 40 2630

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Computational results: LCTSP

Inst./|C|/% Opt lb/Opt #Nodes #u #x #Cuts Seconds kroA100/50/25 36423/12 0.901 11623 5815 1139 3600 kroA100/50/50 24812/25 0.945 43106 21246 315 664 3600 kroA100/50/75 21639/37* 0.984 72289 21890 14254 221 909 kroA100/100/25 30056/25 0.933 20689 10350 957 3600 kroA100/100/50 21833/50* 0.989 79747 38602 1271 290 2100 kroA100/100/75 21282/62* 0.984 206817 78684 24724 212 1860 kroB100/50/25 35882/12 0.908 12028 5986 31 1246 3600 kroB100/50/50 24912/25* 0.976 14701 7255 95 447 1192 kroB100/50/75 22361/37* 0.988 23503 6769 4982 207 379 kroB100/100/25 30764/25 0.943 25196 12600 5 778 3600 kroB100/100/50 23095/50 0.982 126400 61772 1436 344 3600 kroB100/100/75 22211/68 0.983 256317 98252 29918 236 3600 eil101/50/25 931/12 0.899 13084 6503 43 1158 3600 eil101/50/50 693/25* 0.968 23581 11759 31 1104 2174 eil101/50/75 638/37* 0.994 431 214 1 88 12 eil101/100/25 803/25 0.936 27371 13661 31 897 3600 eil101/100/50 643/50* 0.994 1059 528 1 125 30 eil101/100/75 629/66* 0.998 12459 5598 631 208 227 lin105/50/25 23803/12* 0.921 2153 1071 5 470 558 lin105/50/50 16208/25* 0.980 1885 902 40 228 136 lin105/50/75 14635/37* 0.999 7 3 37 1 lin105/100/25 20420/25 0.946 23169 11583 5 772 3600 lin105/100/50 15021/50* 0.994 10971 5440 45 135 248 lin105/100/75 14379/68* 0.999 3 1 33 19 / 21

Computational results: MLHCPLC

Inst./|C|/ǫ Opt lb/Opt #Nodes #u #x #Cuts Seconds kroA100/50/36423 13/34895 0.762 10122 5064 1159 3600 kroA100/50/24812 25/24812* 0.852 12911 6450 5 503 1045 kroA100/50/21639 37/21639* 0.923 22683 6760 4581 217 386 kroA100/100/30056 27/30034 0.804 24399 12179 29 992 3600 kroA100/100/21833 59/21794 0.808 135467 58987 8773 314 3600 kroA100/100/21282 62/21282 0.863 137787 67469 1442 360 3600 kroB100/50/35882 12/35848* 0.821 1357 678 240 278 kroB100/50/24912 25/24912* 0.928 3047 1513 10 259 237 kroB100/50/22361 37/22361* 0.936 31159 13152 2427 279 710 kroB100/100/30764 27/30634 0.819 27412 13703 11 955 3600 kroB100/100/23095 50/23095* 0.938 42055 20489 538 395 1221 kroB100/100/22211 67/22141 0.820 147998 71429 2583 412 3600 eil101/50/931 12/931* 0.796 1597 798 138 326 eil101/50/693 25/693* 0.858 7801 3828 72 706 587 eil101/50/638 37/638* 0.940 865 428 4 139 56 eil101/100/803 25/803 0.838 25375 12665 27 1070 3600 eil101/100/643 50/643* 0.956 901 448 2 108 58 eil101/100/629 63/629* 0.921 139 69 47 4 lin105/50/23803 12/23803* 0.860 1569 777 7 431 380 lin105/50/16208 25/16208* 0.935 299 147 2 123 79 lin105/50/14635 37/14635* 0.991 3 1 33 2 lin105/100/20420 25/20180* 0.900 1507 753 164 211 lin105/100/15021 50/15021* 0.972 529 260 4 114 39 lin105/100/14379 68/14379* 0.985 3 1 33 1 20 / 21

Conclusions

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

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