subpack-glpk
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[9] | 1 | /* TODD, a class of hard instances of zero-one knapsack problems */ |
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| 2 | |
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| 3 | /* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */ |
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| 4 | |
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| 5 | /* Chvatal describes a class of instances of zero-one knapsack problems |
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| 6 | due to Todd. He shows that a wide class of algorithms - including all |
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| 7 | based on branch and bound or dynamic programming - find it difficult |
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| 8 | to solve problems in the Todd class. More exactly, the time required |
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| 9 | by these algorithms to solve instances of problems that belong to the |
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| 10 | Todd class grows as an exponential function of the problem size. |
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| 11 | |
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| 12 | Reference: |
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| 13 | Chvatal V. (1980), Hard knapsack problems, Op. Res. 28, 1402-1411. */ |
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| 14 | |
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| 15 | param n > 0 integer; |
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| 16 | |
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| 17 | param log2_n := log(n) / log(2); |
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| 18 | |
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| 19 | param k := floor(log2_n); |
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| 20 | |
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| 21 | param a{j in 1..n} := 2 ** (k + n + 1) + 2 ** (k + n + 1 - j) + 1; |
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| 22 | |
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| 23 | param b := 0.5 * floor(sum{j in 1..n} a[j]); |
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| 24 | |
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| 25 | var x{1..n} binary; |
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| 26 | |
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| 27 | maximize obj: sum{j in 1..n} a[j] * x[j]; |
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| 28 | |
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| 29 | s.t. cap: sum{j in 1..n} a[j] * x[j] <= b; |
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| 30 | |
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| 31 | data; |
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| 32 | |
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| 33 | param n := 15; |
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| 34 | /* change this parameter to choose a particular instance */ |
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| 35 | |
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| 36 | end; |
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