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Problema de upgrading de delay mınimo dearvore geradora mınima
Francisco Glaubos
Universidade Federal do Rio de Janeiro
2 de outubro de 2016
Formulacao: Variaveis
I xij
I 1 se a aresta i , j e utilizada;I 0 c.c.
I uiI 1 se o vertice i sofreu upgrade;I 0 c.c.
I Di ,j,l : (i , j) ∈ TI 1, se a aresta (i , j) recebeu upgrade do tipo l ;I 0 c.c.
Formulacao: Restricoes
n∑i
n∑j
xij = n − 1 (1)
n∑i∈S
n∑j∈S
xij ≤ |S| − 1 ∀(i , j) ∈ E , ∀S ⊆ V (2)
2∑l=0
Dijl = xij ∀(i , j) ∈ E (3)
Formulacao: Restricoes
Dij0 ≤ 1− ui ∀(i , j) ∈ E (4)
Dij0 ≤ 1− uj ∀(i , j) ∈ E (5)
ui + uj ≥ Dij1 ∀(i , j) ∈ E (6)
Dij1 + ui + uj ≤ 2 ∀(i , j) ∈ E (7)
Dij1 ≥ ui − uj − (1− xij) ∀(i , j) ∈ E (8)
Dij1 ≥ uj − ui − (1− xij) ∀(i , j) ∈ E (9)
Formulacao: Restricoes e F.O.
Dij2 ≤ ui ∀(i , j) ∈ E (10)
Dij2 ≤ uj ∀(i , j) ∈ E (11)
Dij2 ≥ ui + uj − 1− (1− xij) ∀(i , j) ∈ E (12)
∑i∈V
ui ci ≤ B (13)
F.O.:Minn∑i
n∑j
2∑l=0
Dijl Cijl (14)
Estrategias
Branch & BoundI Geracao de todas as SECsI heuristica primal
Branch & CutI Cut setI Cover cutI Clique cut
Relaxacao LagrangeanaI Subproblema: AGM + mochilaI Heurıstica Lagrangeana
Branch & Bound: Heurıstica primal
I Utilizada para encontrar um limite primal (cutoff )I Algoritmo:
1. E ′ ← E2. Para cada (i , i) ∈ E ′ :
I E′[i ].custo ← dij0
3. Kruskal(E ′ )
Relaxacao Lagrangeana: subproblema
I Restricoes da AGM + mochilaI NP-Completude mantida
I Qualidade do limite dual X Velocidade
Relaxacao Lagrangeana: heurıstica lagrangeana
1. E ′ ← E2. Para cada (i , i) ∈ E ′ :
I E ′ [i ].custo ← dij0
3. Kruskal(E ′)4. Para cada u ∈ V :
4.1 Se ui > 0, tenta fazer upgrade em{(i , j) : (i , j) ∈ E ′
, (i , j) ∈ δ(ui )}
Resultados Computacionais
Ambiente de execucaoI Linguagem C++ e compilador g + + versao 4.6.3.I Computador com Intel R©CoreTM i7 CPU 650 @ 3.20GHz, 8Gb
de memoria RAM.
ParametrosI Cutoff: heurıstica primalI Branching Priority em ui
I Reducao do espaco de buscaI Se ui = 0 entao D2
ij = 0 ∀(i , j) ∈ δ+(i)I Desativados: PreSolve, Cuts e Heuristics.I Threads = 1
Branch & Bound
Tabela: Instancias DInstancias B LP Tlp(s) # nos T (s) Sol Best B.
D103911.6 66.61 0.00 3 0.02 73 7358 23.21 0.00 2 0.03 33 33
104.4 17.00 0.00 0 0.01 21 21
D50498329.4 - - - 0.00 365 -1647 - - - 0.00 365 -
2964.6 - - - 0.00 365 -
D1009781349.5 - - - 0.00 897 -6747.5 - - - 0.00 897 -
12145.5 - - - 0.00 897 -
Tabela: Instancias FInstancias B LP Tlp(s) # nos T(s) Sol Best B.
F104417 26.11 0.01 3 0.05 31 3187 9.00 0.01 2 0.04 9 9
157 9.00 0.01 0 0.02 9 9
F5015081 - - - 0.01 543 -
405 - - - 0.01 543 -729 - - - 0.01 543 -
F1004401780 - - - 0.01 687 -8902 - - - 0.01 687 -
16023 - - - 0.01 687 -
Branch & Bound
Tabela: Instancias HInstancias B LP Tlp(s) # nos T(s) Sol Best B.
H104560 28.56 0.01 5 0.05 34 34
302 9.33 0.01 875 0.53 21 21543 9 0.01 1028 0.40 12 12
H501225601 - - - 0.01 343 -
3009 - - - 0.01 343 -5417 - - - 0.01 343 -
H10049501232.1 - - - 0.03 693 -6160.5 - - - 0.03 693 -
11088.9 - - - 0.03 693 -
Tabela: Instancias VInstancias B LP Tlp(s) # nos T(s) Sol Best B.
v104530,49 103.78 0.00 7 0.03 113.49 113.49
152,49 91.90 0.00 26 0.04 93.28 93.28274,48 91.57 0.00 0 0.01 92.06 92.06
v501057605.33 - - - 0.01 2636 -
3026.69 - - - 0.01 2636 -5448.04 - - - 0.01 2636 -
v10041092545.34 - - - 0.01 2636 -
12726.73 - - - 0.01 2636 -22908.11 - - - 0.01 2636 -
Obs.: Branch & Bound
I Instancias mais faceis: D
I Instancias mais difıceis: H
I Nao foi possıvel carregar o modelo para 50 e 100 nos
Branch & CutTabela: Instancias D
Instancias B LP Tlp(s) # nos T(s) Sol Best B. UserCuts
LazyConst.
# mincut
T(s)min cuts
D103911.6 66.61 0.00 5 0.04 73 73 2 11 16 0.0058 23.18 0.00 9 0.03 33 33 0 3 9 0.00
104.4 17 0.00 0 0.03 21 21 1 4 6 0.00
D50498329.4 113.14 0.04 45 0.38 157 157 20 29 41 0.001647 49 0.02 90 0.73 55 55 7 37 59 0.00
2964.6 49 0.01 4 0.28 56 56 2 49 52 0.00
D1009781349.5 226.03 0.09 232 2.80 230 230 86 223 230 0.0046747.5 111.04 0.04 1719 12.65 113 113 132 764 1333 0.02
12145.5 99 0.03 10 0.55 100 100 0 59 62 0.001
Tabela: Instancias V
Instancias B LP Tlp(s) # nos T(s) Sol Best B. UserCuts
LazyConst.
# mincut
T(s)min cuts
v104521.72 102.20 0.01 5 0.02 113.48 113.48 2 3 6 0.00
108.63 92.40 0.00 29 0.02 93.28 93.28 12 13 30 0.00195.54 92.06 0.00 0 0.01 92.06 92.06 0 1 1 0.00
v501057610.42 2591.68 0.03 477 1.20 2633.37 2633.37 73 140 344 0.004
3052.13 2475.94 0.01 39 0.31 2477.07 2477.07 10 11 38 0.005493.84 2470.78 0.01 0 0.05 2479.83 5493.84 0 1 2 0.00
v10041092260.66 10018.26 0.30 979 45.12 10035.93 10035.93 58 169 791 0.02
11303.32 9926.81 0.23 115 9.52 9927.26 9927.26 27 40 118 0.00420345.97 9919.54 0.08 0 0.30 9919.54 9919.54 0 1 1 0.30
Branch & Cut
Tabela: Instancias HInstancias B LP Tlp(s) # nos T(s) Sol Best B. User
CutsLazy
Const.# min
cutT(s)
min cuts
H104560 27.55 0.000 6 0.01 34 34 1 7 13 0.000
302 9 0.000 129 0,05 21 21 16 17 82 0.000543 9 0.000 16 0,02 12 12 3 12 18 0.000
H501225601 57.06 0.210 27538 1800 160 154 11 182 19727 0.843
3009 49 0.050 23996 1800 50 49 3159 14365 15519 0.4825417 49 0.080 551 84.09 49 49 73 1577 2180 0.076
H10049501232.1 99 1.770 2277 1800 - 99 0 43 3182 0.2566160.5 99 0.290 2279 1800 117 101 87 3266 5048 0.332
11088.9 99 0.590 658 318.06 100 100 0 0 2128 0.172
Tabela: Instancias FInstancias B LP Tlp(s) # nos T(s) Sol Best B. User
CutsLazy
Const.# min
cutT(s)
min cuts
F104417 17.6 0.00 1 0.03 31 31 2 3 5 0.0087 9 0.00 0 0.02 9 9 1 1 4 0.00
157 9 0.00 0 0.02 9 9 0 2 3 0.00
F5015081 177.56 0.02 152 0.31 256 256 19 91 94 0.31
405 83 0.00 109 0.18 91 91 15 64 66 0.002729 72 0.00 0 0.03 72 72 0 1 1 0.00
F1004401780 251.80 0.07 785 2.58 276 276 86 166 546 0.018902 109.61 0.03 28 0.25 111 111 1 14 29 0.25
16023 99 0.02 0 0.08 99 99 0 1 1 0.00
Obs.: Branch & Cut
I Instancias mais difıceis: HI Tempo maximo excedido em 4 instancias
I Instancias mais faceis: F
I Algoritmo de separacao bastante rapido
Relaxacao Lagrangeana
Tabela: Instancias DInstancias B Valor RL T(s) # It. Sol
D103911.6 73.099 4.749 600 85.00058 32.293 7.609 1000 47.000
104.4 20.980 3.620 1000 33.000
D50498329.4 112.481 2341.125 600 296.0001647 49.706 202.452 600 203.000
2964.6 54.211 138.581 600 182.000
D1009781349.5 156.891 1823.339 16 703.0006747.5 110.385 1810.866 84 521.000
12145.5 104.687 1801.475 683 461.000
Tabela: Instancias FInstancias B Valor RL T(s) # It. Sol
F104417 27.864 8.026 1000 51.00087 10.764 4.971 1000 22.000
157 9.556 3.525 1000 15.000
F5015081 219.021 45.261 1000 399.000
405 108.857 33.132 1000 238.000729 94.390 38.768 1000 176.000
F1004401780 166.344 1883.311 75 612.0008902 144.839 1840.280 119 399.000
16023 133.793 732.628 1000 288.000
Tabela: Instancias HInstancias B Valor RL T(s) # It. Sol
H104560 32.085 11.831 1000 59.000
302 9.878 10.626 1000 29.000543 9.941 4.375 1000 18.000
H501225601 51.397 1804.633 190 260.000
3009 51.775 310.026 1000 169.0005417 51.259 221.766 1000 107.000
H10049501232.1 97.551 3245.636 6 520.0006160.5 101.718 1818.515 294 314.000
11088.9 102.655 1800.726 471 216.000
Tabela: Instancias VInstancias B Valor RL T(s) # It. Sol
v104521.72 110.116 8.875 1000 120.000
108.63 96.827 17.744 1000 108.000195.54 94.503 7.069 1000 104.000
v501057610.42 2595.953 499.286 1000 2896.000
3052.13 2493.225 239.044 1000 2660.0005493.84 2478.924 68.401 1000 2559.000
v10041092260.66 10016.663 1803.096 598 10482.000
11303.32 9929.867 1596.772 1000 10210.00020345.97 9921.301 509.582 1000 10083.000