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Question 1 Consider the following greedy algorithm for knapsack packing. (a) Sort items in non-increasing order of v/s (b) Greedily add items until we hit an item a, that is too big (-1 $L > B) (c) Pick the better of {a1, a2,...,a1-1} and {a}. Your task is the following. (a) Show that the value of the solution found by the greedy algorithm is at least half of the (unknown) optimal value as the number of items n tends to infinity. (2-Approximation) (b) There exists an instance (set of items) such that the optimal value reached by the greedy algorithm is half of the value reached by the optimal algo- rithm as the size of knapsack goes to infinity. (tightness)

Question 1 Consider the following greedy algorithm for knapsack packing. (a) Sort items in non-increasing order of v/s (b) Greedily add items until we hit an item a, that is too big (-1 $L > B) (c) Pick the better of {a1, a2,...,a1-1} and {a}. Your task is the following. (a) Show that the value of the solution found by the greedy algorithm is at least half of the (unknown) optimal value as the number of items n tends to infinity. (2-Approximation) (b) There exists an instance (set of items) such that the optimal value reached by the greedy algorithm is half of the value reached by the optimal algo- rithm as the size of knapsack goes to infinity. (tightness)

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Question 1 Consider the following greedy algorithm for knapsack packing. (a) Sort items in non-increasing order of v/s (b) Greedily add items until we hit an item a, that is too big (-1 SL > B) (c) Pick the better of {a1, a2, ...,aL-1} and {a}. Your task is the following. (a) Show that the value of the solution found by the greedy algorithm is at least half of the (unknown) optimal value as the number of items n tends to infinity. (2-Approximation) (b) There exists an instance (set of itens) such that the optimal value reached by the greedy algorithm is half of the value reached by the optimal algo- rithm as the size of knapsack goes to infinity. (tightness)
Transcribed Image Text:Question 1 Consider the following greedy algorithm for knapsack packing. (a) Sort items in non-increasing order of v/s (b) Greedily add items until we hit an item a, that is too big (-1 SL > B) (c) Pick the better of {a1, a2, ...,aL-1} and {a}. Your task is the following. (a) Show that the value of the solution found by the greedy algorithm is at least half of the (unknown) optimal value as the number of items n tends to infinity. (2-Approximation) (b) There exists an instance (set of itens) such that the optimal value reached by the greedy algorithm is half of the value reached by the optimal algo- rithm as the size of knapsack goes to infinity. (tightness)
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