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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In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share the same color; this is called a vertex coloring. The chromatic number of a graph is the least mumber of colors required to do a coloring of a graph. Example Here in this graph the chromatic number is 3 since we used 3 colors The degree of a vertex v in a graph (without loops) is the number of edges at v. If there are loops at v each loop contributes 2 to the valence of v. A graph is connected if for any pair of vertices u and v one can get from u to v by moving along the edges of the graph. Such routes that move along edges are known by different names: edge progressions, paths, simple paths, walks, trails, circuits, cycles, etc. a. Write down the degree of the 16 vertices in the graph below: 14…
The minimum vertex cover problem is stated as follows: Given an undirected graph G = (V, E) with N vertices and M edges. Find a minimal size subset of vertices X from V such that every edge (u, v) in E is incident on at least one vertex in X. In other words you want to find a minimal subset of vertices that together touch all the edges. For example, the set of vertices X = {a,c} constitutes a minimum vertex cover for the following graph: a---b---c---g d e Formulate the minimum vertex cover problem as a Genetic Algorithm or another form of evolutionary optimization. You may use binary representation, OR any repre- sentation that you think is more appropriate. you should specify: • A fitness function. Give 3 examples of individuals and their fitness values if you are solving the above example. • A set of mutation and/or crossover and/or repair operators. Intelligent operators that are suitable for this particular domain will earn more credit. • A termination criterion for the…
Q2 a Let G be a graph. We say that a set of vertices C form a vertex cover if every edge of G is incident to at least one vertex in C. We say that a set of vertices I form an independent set if no edge in G connects two vertices from I. For example, if G is the graph above, C = [b, d, e, f, g, h, j] is a vertex cover since each of the 20 edges in the graph has at least one endpoint in C, and I = [a, c, i, k] is an independent set because none of these edges appear in the graph: ac, ai, ak, ci, ck, ik. In the example above, notice that each vertex belongs to the vertex cover C or the independent set I. Do you think that this is a coincidence? In the above graph, clearly explain why the maximum size of an independent set is 5. In other words, carefully explain why there does not exist an independent set with 6 or more vertices.
5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…
0 Give me examples of Suns graph Suns : Recall that a graph is chordal if it contains no induced cycles of length at least 4. A sun Sn is a chordal graph on 2n vertices (n3) whose vertex set can be partitioned into sets W = {w1, w2,...,wn} and U = {u1, u2,...,un} such that W is independent and ui is adjacent to wj if and only if i=jor i =j+1 (mod n) If U induces a complete graph then the graph is called the complete sun. 712 (Sn) = n. Proposition: For n>3,
When a vertex Q is connected by an edge to a vertex K, what is the term for the relationship between Q and K? *a) Q and K are "insecure."b) Q and K are "incident."c) Q and K are "adjacent."d) Q and K are "isolated."
Markup the BFS code so that it can determine if the graph g is connected or not. Further modify the BFS code so that that it is possible to determine the number of connected components and their membership (i.e., which vertices belong to which component). // Connectivity Checker and Identifier public static Integer[] ProcessOrderInBFS(Graph g) { int foundCount = 1; Integer[] processOrder = new Integer[g.size(O]; for (int i = 0; i q = new LinkedList(); int currNode = 0; Integer[] adjVector = null; q. add(startVertex); processorder[startVertex] = foundCount; foundCount++; while(!q.isEmpty()) { currNode = q.remove(); adjVector = g.getAdjVector(currNode); for (int i = 0; i < g.size(); i++) { if(adjVector[i] == 1 && processOrder[i] == 0) { q. add(i); processorder[i] = foundCount; foundCount++; } } } return foundCount;
1. Let Sn denote a directed graph with n vertices, numbered from 1 to n, in which there are edges from vertex 1 to all other vertices, and from all other vertices back to vertex 1, but no other edges. Write the Python adjacency list representation for S6.
*Discrete Math In the graph above, let ε = {2, 3}, Let G−ε be the graph that is obtained from G by deleting the edge {2,3}. Let G∗ be the graph that is obtain from G − ε by merging 2 and 3 into a single vertex w. (As in the notes, v is adjacent to w in the new if and only if either {2,v} or {3,v is an edge of G.) (a) Draw G − ε and calculate its chromatic polynomial. (b) Give an example of a vertex coloring that is proper for G − ε, but not for G. (c) Explain, in own words, why no coloring can be proper for G but not proper for G − ε. (d) Draw G∗ and calculate its chromatic polynomial. (e) Verify that, for this example,PG(k) = PG−ε(k) − PG∗ (k).
4-Clique Problem The clique problem is to find cliques in a graph. A clique is a set of vertices that are all adjacent - connected - to each other. A 4-clique is a set of 4 vertices that are all connected to each other. So in this example of the 4-Clique Problem, we have a 7-vertex graph. A brute-force algorithm has searched every possible combination of 4 vertices and found a set that forms a clique: https://en.wikipedia.org/wiki/Clique_problem You should read the Wikipedia page for the Clique Problem (and then read wider if need be) if you need to understand more about it. Note that the Clique Problem is NP-Complete and therefore when the graph size is large a deterministic search is impractical. That makes it an ideal candidate for an evolutionary search. For this assignment you must suppose that you have been tasked to implement the 4-clique problem as an evolutionary algorithm for any graph with any number of vertices (an n-vertex graph). The algorithm succeeds if it finds a…
Throughout, a graph is given as input as an adjacency list. That is, G is a dictionary where the keysare vertices, and for a vertex v,G[v] = [u such that there is an edge going from v to u].In the case that G is undirected, for every edge u − v, v is in G[u] and u is in G[v] 5. Write the full pseudocode for the following problem using BFS/DFS.Input: An undirected graph G that’s not necessarily connected.Output: Is G bipartite? That is, can the vertices of G be partitioned into two sets X, and Y such that thereare no edges within X and Y ?
The Graph Data Structure is made up of nodes and edges. (A Tree Data Structure is a special kind of a Graph Data Structure). A Graph may be represented by an Adjacency Matrix or an Adjacency List. Through this exercise, you should be able to have a better grasp the Adjacency Matrix concept. You are expected to read about the Adjacency Matrix concept as well as the Adjacency List concept. Suppose the vertices A, B, C, D, E, F, G and H of a Graph are mapped to row and column indices(0,1,2,3,4,5,6,and 7) of a matrix (i.e. 2-dimensional array) as shown in the following table. Vertex of Graph Index in the 2-D Array Adjacency Matrix Representation of Graph A B 2 F 6. H 7 Suppose further, that the following is an Adjacency Matrix representing the Graph. 3 4 5. 6. 7 0. 1 1 1 1 01 1 01 1. 3 14 1 1 1 6. 1 Exercise: Show/Draw the Graph that is represented by the above Adjacency matrix. Upload the document that contains your result. (Filename: AdjacencyMatrixExercise.pdf) Notes: -The nodes of the…