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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. -

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 28EQ
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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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