In this example, placing a food cart on each of the vertices A and C will result in a profit of|{(A, B), (A, C), (A, D), (C, D), (C, E)}|- |{A, C}|= 5-2= 3.The decision version for the food cart problem asks to find a subset of vertices such that the profit for G,as defined above, is maximized.FoodCart DecInput: G = (V, E), integer k>0Question: Does there exist SCV where |Es|-|S| ≥ k? Here, Es CE denotes the set of all edges(x, y) EE where at least one endpoint of (x, y) is in S, that is x E Sor y E S.1. Define the language LFC corresponding to Food Cartdec2. Show that LFC E NP. For this: Clearly describe what a certificate looks like. Then give apolynomial-time verifier and proof its correctness.3. Define the optimization version Food Cartopt for decision version FoodCart Dec. Consider a graph where vertices represent intersections and edges represent roads. The company FoodTo Go can place a maximum of one food cart at each intersection. A food cart placed at intersection Xhas an income equal to the number of roads at intersection X (ie, the degree of vertex X). If two foodcarts share a road, the income of that road is shared equally among the two.We are interested in calculating the total profit of all food carts placed, ie, the total income minus the cost,where each food cart costs 1 unit and each road provides an income of 1 unit.More formally, the profit for a graph G = (V, E) where the set SC V represents food cart locations, iscalculated by the number of edges covered by the vertices in S minus the size of S (ie. |S).ADCEBIn this example, placing a food cart on each of the vertices A and C will result in a profit ofISCA P (AD(CD) (CFU VACU-5

The design and analysis of algorithms for solving optimization problems involve several steps,…

In this example, placing a food cart on each of the vertices A and C will result in a profit of |{(A, B), (A, C), (A, D), (C, D), (C, E)}|- |{A, C}| = 5-2=3. The decision version for the food cart problem asks to find a subset of vertices such that the profit for G as defined above, is maximized. FoodCart Dec Input: G = (V, E), integer k > 0 Question: Does there exist SCV where |Es|-|S| ≥k? Here, Es CE denotes the set of all edges (x, y) E E where at least one endpoint of (x, y) is in S, that is x E S or y ES. 1. Define the language LFC corresponding to FoodCart dec 2. Show that LFC E NP. For this: Clearly describe what a certificate looks like. Then give a polynomial-time verifier and proof its correctness. 3. Define the optimization version FoodCartopt for decision version Food Cart Dec-

In this example, placing a food cart on each of the vertices A and C will result in a profit of |{(A, B), (A, C), (A, D), (C, D), (C, E)}|- |{A, C}| = 5-2=3. The decision version for the food cart problem asks to find a subset of vertices such that the profit for G as defined above, is maximized. FoodCart Dec Input: G = (V, E), integer k > 0 Question: Does there exist SCV where |Es|-|S| ≥k? Here, Es CE denotes the set of all edges (x, y) E E where at least one endpoint of (x, y) is in S, that is x E S or y ES. 1. Define the language LFC corresponding to FoodCart dec 2. Show that LFC E NP. For this: Clearly describe what a certificate looks like. Then give a polynomial-time verifier and proof its correctness. 3. Define the optimization version FoodCartopt for decision version Food Cart Dec-

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

Problem 1PE

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A weighted, directed graph is a suitable representation to represent the daily airline routes flown by a small airline. The airline have the following daily flights: - Three flights from Cape Town to Johannesburg, - Two flights from Johannesburg to Cape Town. - Four flights from Johannesburg to Durban. - Three flights from Durban to Johannesburg. - One flight from Johannesburg to George. - One flight from George to Johannesburg.Draw the graph and answer the questions that follow. The graph that represents this problem is connected. A. True B. False
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