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E The knapsack problem consists of a set of n items with weights a₁,..., an. Each item i € {1,..., n} has a value ci. We have a knapsack that can hold a total weight of at most W. The goal is to find the subset of items with maximum total value that can fit into the knapsack. One can formulate an integer program for the knapsack problem: max C1x1 C2x2 + . . . Спхп subject to a1x1a2x2 + ... Anxn < W Xi xi > 0 Vi = {1,..., n} xi ≤ 1 E Vi = {1, • • " ‚n} Xi E Z Vi Є {1,..., n} We have the family of cutting planes known as "cover inequalities”. Let C C {1, ..., n} be a subset such that Σiec ai > W. Then C is called a cover. C is a minimal cover if C \ {j} is not a cover for every j = C. For every minimal cover C, a feasible solution can pick at most |C| – 1 items from the set C. Thus, the following inequality is valid: Σiec xi ≤ |C| − 1 for every minimal cover C. Show how to obtain the cover inequalities as CG cuts starting from the system above. -

E The knapsack problem consists of a set of n items with weights a₁,..., an. Each item i € {1,..., n} has a value ci. We have a knapsack that can hold a total weight of at most W. The goal is to find the subset of items with maximum total value that can fit into the knapsack. One can formulate an integer program for the knapsack problem: max C1x1 C2x2 + . . . Спхп subject to a1x1a2x2 + ... Anxn < W Xi xi > 0 Vi = {1,..., n} xi ≤ 1 E Vi = {1, • • " ‚n} Xi E Z Vi Є {1,..., n} We have the family of cutting planes known as "cover inequalities”. Let C C {1, ..., n} be a subset such that Σiec ai > W. Then C is called a cover. C is a minimal cover if C \ {j} is not a cover for every j = C. For every minimal cover C, a feasible solution can pick at most |C| – 1 items from the set C. Thus, the following inequality is valid: Σiec xi ≤ |C| − 1 for every minimal cover C. Show how to obtain the cover inequalities as CG cuts starting from the system above. -

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The knapsack problem

E The knapsack problem consists of a set of n items with weights a₁,..., an. Each item i € {1,..., n} has a value ci. We have a knapsack that can hold a total weight of at most W. The goal is to find the subset of items with maximum total value that can fit into the knapsack. One can formulate an integer program for the knapsack problem: max C1x1 C2x2 + . . . Спхп subject to a1x1a2x2 + ... Anxn < W Xi xi > 0 Vi = {1,..., n} xi ≤ 1 E Vi = {1, • • " ‚n} Xi E Z Vi Є {1,..., n} We have the family of cutting planes known as "cover inequalities”. Let C C {1, ..., n} be a subset such that Σiec ai > W. Then C is called a cover. C is a minimal cover if C \ {j} is not a cover for every j = C. For every minimal cover C, a feasible solution can pick at most |C| – 1 items from the set C. Thus, the following inequality is valid: Σiec xi ≤ |C| − 1 for every minimal cover C. Show how to obtain the cover inequalities as CG cuts starting from the system above. -
Transcribed Image Text:E The knapsack problem consists of a set of n items with weights a₁,..., an. Each item i € {1,..., n} has a value ci. We have a knapsack that can hold a total weight of at most W. The goal is to find the subset of items with maximum total value that can fit into the knapsack. One can formulate an integer program for the knapsack problem: max C1x1 C2x2 + . . . Спхп subject to a1x1a2x2 + ... Anxn < W Xi xi > 0 Vi = {1,..., n} xi ≤ 1 E Vi = {1, • • " ‚n} Xi E Z Vi Є {1,..., n} We have the family of cutting planes known as "cover inequalities”. Let C C {1, ..., n} be a subset such that Σiec ai > W. Then C is called a cover. C is a minimal cover if C \ {j} is not a cover for every j = C. For every minimal cover C, a feasible solution can pick at most |C| – 1 items from the set C. Thus, the following inequality is valid: Σiec xi ≤ |C| − 1 for every minimal cover C. Show how to obtain the cover inequalities as CG cuts starting from the system above. -
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