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1. Making the arrangements. My garden features a number of flower beds. As a person of particular tastes, I have a rating system for how lovely a pairing of flowers is. I encode these ratings in a huge table: Rose Daisy Tulip Sunflower Rose Daisy Tulip 3 -2 5 Sunflower 0 1 -4 (You may assume that the table can include any (finite) number of types of flowers; loveliness ratings are integers; and that flower names are arbitrary and can be invented.) The loveliness of a flowerbed is the total rating of all pairs of flower types in the flowerbed. The LOVELY-FLOWERBED problem is to determine, given my preferences as a table in the form described above and an integer f, whether I can have a flowerbed whose total loveliness rating is at least l. (a) After ruining my garden path, Alfie suggests that the following greedy approach will solve the LOVELY-FLOWERBED problem: Choose the first flower as the one that has the highest total loveliness with all other flower types. Choose the next flower as the one with the highest loveliness rating with any flower already in the flowerbed. Repeat until a flowerbed with loveliness is achieved, or you run out of flowers. Is Alfie right or wrong? Rigorously justify your answer. (b) Prove that LOVELY-FLOWERBED is complete for NP.

1. Making the arrangements. My garden features a number of flower beds. As a person of particular tastes, I have a rating system for how lovely a pairing of flowers is. I encode these ratings in a huge table: Rose Daisy Tulip Sunflower Rose Daisy Tulip 3 -2 5 Sunflower 0 1 -4 (You may assume that the table can include any (finite) number of types of flowers; loveliness ratings are integers; and that flower names are arbitrary and can be invented.) The loveliness of a flowerbed is the total rating of all pairs of flower types in the flowerbed. The LOVELY-FLOWERBED problem is to determine, given my preferences as a table in the form described above and an integer f, whether I can have a flowerbed whose total loveliness rating is at least l. (a) After ruining my garden path, Alfie suggests that the following greedy approach will solve the LOVELY-FLOWERBED problem: Choose the first flower as the one that has the highest total loveliness with all other flower types. Choose the next flower as the one with the highest loveliness rating with any flower already in the flowerbed. Repeat until a flowerbed with loveliness is achieved, or you run out of flowers. Is Alfie right or wrong? Rigorously justify your answer. (b) Prove that LOVELY-FLOWERBED is complete for NP.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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1. Making the arrangements. My garden features a number of flower beds. As a person of particular tastes, I have a rating system for how lovely a pairing of flowers is. I encode these ratings in a huge table: Rose Daisy Tulip Sunflower Rose Daisy Tulip 3 -2 5 - - Sunflower 0 1 -4 (You may assume that the table can include any (finite) number of types of flowers; loveliness ratings are integers; and that flower names are arbitrary and can be invented.) The loveliness of a flowerbed is the total rating of all pairs of flower types in the flowerbed. The LOVELY-FLOWERBED problem is to determine, given my preferences as a table in the form described above and an integer , whether I can have a flowerbed whose total loveliness rating is at least l. (a) After ruining my garden path, Alfie suggests that the following greedy approach will solve the LOVELY-FLOWERBED problem: Choose the first flower as the one that has the highest total loveliness with all other flower types. Choose the next flower as the one with the highest loveliness rating with any flower already in the flowerbed. Repeat until a flowerbed with loveliness is achieved, or you run out of flowers. Is Alfie right or wrong? Rigorously justify your answer. (b) Prove that LOVELY-FLOWERBED is complete for NP.
Transcribed Image Text:1. Making the arrangements. My garden features a number of flower beds. As a person of particular tastes, I have a rating system for how lovely a pairing of flowers is. I encode these ratings in a huge table: Rose Daisy Tulip Sunflower Rose Daisy Tulip 3 -2 5 - - Sunflower 0 1 -4 (You may assume that the table can include any (finite) number of types of flowers; loveliness ratings are integers; and that flower names are arbitrary and can be invented.) The loveliness of a flowerbed is the total rating of all pairs of flower types in the flowerbed. The LOVELY-FLOWERBED problem is to determine, given my preferences as a table in the form described above and an integer , whether I can have a flowerbed whose total loveliness rating is at least l. (a) After ruining my garden path, Alfie suggests that the following greedy approach will solve the LOVELY-FLOWERBED problem: Choose the first flower as the one that has the highest total loveliness with all other flower types. Choose the next flower as the one with the highest loveliness rating with any flower already in the flowerbed. Repeat until a flowerbed with loveliness is achieved, or you run out of flowers. Is Alfie right or wrong? Rigorously justify your answer. (b) Prove that LOVELY-FLOWERBED is complete for NP.
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