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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
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.
-
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
