To help jog your memory, here are some definitions: Vertex Cover: given an undirected unweighted graph G = (V, E), a vertex cover Cy of G is a subset of vertices such that for every edge e = (u, v) € E, at least one of u or v must be in the vertex cover Cy. Set Cover: given a universe of elements U and a collection of sets S = {S₁, ..., Sm}, a set cover is any (sub) collection C's whose union equals U. In the minimum vertex cover problem, we are given an undirected unweighted graph G = (V, E), and are asked to find the smallest vertex cover. For example, in the following graph, {A, E, C, D} is a vertex cover, but not a minimum vertex cover. The minimum vertex covers are {B, E, C} and {A, E, C}. A B E Q F D Then, recall in the minimum set cover problem, we are given a set U and a collection S = {S₁, ..., Sm} of subsets of U, and are asked to find the smallest set cover. For example, given U := {a, b, c, d}, S₁ = {a, b,c}, S₂ = {b,c}, and S3 := {c, d), a solution to the problem is Cs = {S₁, S3}. Give an efficient reduction from the minimum vertex cover problem to the min- imum set cover problem. Briefly justify the correctness of your reduction (i.e. 1-2 sentences).
To help jog your memory, here are some definitions: Vertex Cover: given an undirected unweighted graph G = (V, E), a vertex cover Cy of G is a subset of vertices such that for every edge e = (u, v) € E, at least one of u or v must be in the vertex cover Cy. Set Cover: given a universe of elements U and a collection of sets S = {S₁, ..., Sm}, a set cover is any (sub) collection C's whose union equals U. In the minimum vertex cover problem, we are given an undirected unweighted graph G = (V, E), and are asked to find the smallest vertex cover. For example, in the following graph, {A, E, C, D} is a vertex cover, but not a minimum vertex cover. The minimum vertex covers are {B, E, C} and {A, E, C}. A B E Q F D Then, recall in the minimum set cover problem, we are given a set U and a collection S = {S₁, ..., Sm} of subsets of U, and are asked to find the smallest set cover. For example, given U := {a, b, c, d}, S₁ = {a, b,c}, S₂ = {b,c}, and S3 := {c, d), a solution to the problem is Cs = {S₁, S3}. Give an efficient reduction from the minimum vertex cover problem to the min- imum set cover problem. Briefly justify the correctness of your reduction (i.e. 1-2 sentences).
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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