In the derivation of the spectral graph partitioning algorithm, we relax a combinatorial optimization problem to a continuous optimization problem. This relaxation has the following effects. The combinatorial problem requires an ex- act bisection of the graph, but the continuous al- gorithm can produce (after rounding) partitions that aren't perfectly balanced The combinatorial problem cannot be modi- fied to accommodate vertices that have different masses, whereas the continuous problem can The combinatorial problem requires finding eigenvectors, whereas the continuous problem re- quires only matrix multiplication ■The combinatorial problem is NP-hard, but the continuous problem can be solved in polyno- mial time

In the derivation of the spectral graph partitioning algorithm, we relax a combinatorial optimization problem to a continuous optimization problem. This relaxation has the following effects. The combinatorial problem requires an ex- act bisection of the graph, but the continuous al- gorithm can produce (after rounding) partitions that aren't perfectly balanced The combinatorial problem cannot be modi- fied to accommodate vertices that have different masses, whereas the continuous problem can The combinatorial problem requires finding eigenvectors, whereas the continuous problem re- quires only matrix multiplication ■The combinatorial problem is NP-hard, but the continuous problem can be solved in polyno- mial time

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. Explain how to prove that a problem is NP-Complete based onreduction from a known NP-Complete problem.b. An independent set of a graph G = (V, E) is a subset V’  V of verticessuch that each edge in E is incident on at most one vertex in V’. Theindependent-set problem is to find a maximum-size independent set in G.Formulate a related decision problem.c. Prove that this decision problem is NP-complete. (Hint: Reduce fromthe clique problem or from the vertex cover problem.)
Please show written work with answer!! An independent set in a graph is a set of vertices no two of which are adjacent to each other. A clique is a complete subgraph of a given graph. This means that there is an edge between any two nodes in the subgraph. The maximal clique is the complete subgraph of a given graph which contains the maximum number of nodes. We know that independent set is a NP complete problem. Transform the independent set to max clique (in polynomial time) to show that Max Clique is also NP
An independent set of a graph G = (V, E) is a subset V’ Í V of vertices such that each edge in E is incident on at most one vertex in V’. The independent-set problem is to find a maximum-size independent set in G . Analyse the complexity of the approximation algorithm for solving the independent-set problem
Given a directed graph with positive edge lengths and two distinct vertices uand v in the graph, the “all-pairs uv-constrained shortest path problem” is the problemof computing for each pair of vertices i and j the length of the shortest path from i toj that goes through the vertex u or through the vertex v. If no such path exists, theanswer is ∞. Describe an algorithm that takes a graph G = (V, E) and vertices u and v asinput parameters and computes values L(i, j) that represent the length of uv-constrainedshortest path from i to j for all 1 ≤ i, j ≤ |V|, i ! = u, j ! = u, i != v, j ! = v. Provide clearpseudocode solution. Prove your algorithm correct. Your algorithm must have runningtime in O(|V| ^2).
Which of the following problems can be solved in O(n3) time, where n is the number of vertices in the input graph? (Choose all that apply.) a) Given a directed graph G=(V,E) with nonnegative edge lengths, compute the maximum length of a shortest path between any pair of vertices (that is, maxv,w∈Vdist(v,w)). b) Given a directed acyclic graph with real-valued edge lengths, compute the length of a longest path between any pair of vertices. c) Given a directed graph with nonnegative edge lengths, compute the length of a longest cycle-free path between any pair of vertices. d) Given a directed graph with real-valued edge lengths, compute the length of a longest cycle-free path between any pair of vertices.
Given the adjacency matrix of an undirected simple graph G = (V, E) mapped in a natural fashion onto a mesh of size n2, in Θ(n) time a directed breadth-first spanning forest T = (V, A) can be created. As a byproduct, the undirected breadth first spanning forest edge set EA can also be created, where EA consists of the edges of A and the edges of A directed in the opposite direction.give proof of theorem.
An independent set of a graph G = (V, E) is a subset V’ is subset of V of verticessuch that each edge in E is incident on at most one vertex in V’. Theindependent-set problem is to find a maximum-size independent set in G. Question: Prove that this decision problem is NP-complete. (Hint: Reduce fromthe clique problem or from the vertex cover problem.)
Let G = (V,E) be a bipartite graph, but this time it is a weighted graph. The weight of acomplete matching is the sum of the weights of its edges. We are interested in finding aminimum-weight complete matching in G.a) Give a legitimate C for a branch-and-bound (B&B) algorithm that finds a minimum-weightcomplete matching in G, and prove that your C is valid. Your C cannot be just the cost sofar.b) Using your C, apply B&B to find a minimum-weight complete matching in the followingweighted bipartite graph G: A = (1,2,3), B = (4,5,6,7)E = (((1,4), 3]. [(1,5), 4]. [(1,7). 15]. ((2,4), 1]. |(2,5), 8]. [(2,6), 3]. |(3,4), 3]. [(3,5), 9]. [ (3,6). 51).Show the solution tree. the C of every tree node generated, and the optimal solution. Also,mark the order in which each node in the solution tree is visited.
Please show written work with answer please!! A vertex cover of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. Finding the minimum size vertex cover set is a NP problem, find an greedy approximation algorithm to find a close to optimum solution in polynomial time.
Recall the Clique problem: given a graph G and a value k, check whether G has a set S of k vertices that's a clique. A clique is a subset of vertices S such that for all u, v € S, uv is an edge of G. The goal of this problem is to establish the NP-hardness of Clique by reducing VertexCover, which is itself an NP-hard problem, to Clique. Recall that a vertex cover is a set of vertices S such that every edge uv has at least one endpoint (u or v) in S, and the VertexCover problem is given a graph H and a value 1, check whether H has a vertex cover of size at most 1. Note that all these problems are already phrased as decision problems, and you only need to show the NP-Hardness of Clique. In other words, we will only solve the reduction part in this problem, and you DO NOT need to show that Clique is in NP. Q4.1 Let S be a subset of vertices in G, and let C be the complement graph of G (where uv is an edge in C if and only if uv is not an edge in G). Prove that for any subset of vertices…
Consider the d-Independent Set problem:Input: an undirected graph G = (V,E) such that every vertex has degree less or equal than d.Output: The largest Independent Set. Describe a polynomial time algorithm Athat approximates the optimal solution by a factor α(d). Your mustwrite the explicit value of α, which may depend on d. Describe your algorithm in words (no pseudocode) andprove the approximation ratio α you are obtaining. Briefly explain why your algorithm runs in polytime.
Given: graph G, find the smallest integer k such that the vertex set V of G contains a set A consisting of k elements satisfying the condition: for each edge of G at least one of its ends is in A. The size of the problem is the number n of vertices in G. Please help answer problems 3 & 4 from the given information. 3. Find an instance for which the suggested greedy algorithm gives an erroneous answer. 4. Suggest a (straightforward) algorithm which solves the problem correctly.