Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 35.3, Problem 2E
Program Plan Intro
To show that the decision version of the set-covering problem is NP-complete by reducing it from the vertex-cover problem.
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Illustrate the approximation algorithm for vertex-
cover problem on the below graph. Show steps.
DENSE-SUBGRAPH: Given a graph G and two integers m and n, does G have a set
of m vertices with at least n edges between them? Prove that DENSE-SUBGRAPH is NP-complete.
Suppose the clique problem is polynomially reducible to the vertex-cover problem.
If the clique problem is in P, then the vertex-cover problem is also in P.
True
False
All the NP-complete problems are known to be reducible to one another.
True
False
Chapter 35 Solutions
Introduction to Algorithms
Ch. 35.1 - Prob. 1ECh. 35.1 - Prob. 2ECh. 35.1 - Prob. 3ECh. 35.1 - Prob. 4ECh. 35.1 - Prob. 5ECh. 35.2 - Prob. 1ECh. 35.2 - Prob. 2ECh. 35.2 - Prob. 3ECh. 35.2 - Prob. 4ECh. 35.2 - Prob. 5E
Ch. 35.3 - Prob. 1ECh. 35.3 - Prob. 2ECh. 35.3 - Prob. 3ECh. 35.3 - Prob. 4ECh. 35.3 - Prob. 5ECh. 35.4 - Prob. 1ECh. 35.4 - Prob. 2ECh. 35.4 - Prob. 3ECh. 35.4 - Prob. 4ECh. 35.5 - Prob. 1ECh. 35.5 - Prob. 2ECh. 35.5 - Prob. 3ECh. 35.5 - Prob. 4ECh. 35.5 - Prob. 5ECh. 35 - Prob. 1PCh. 35 - Prob. 2PCh. 35 - Prob. 3PCh. 35 - Prob. 4PCh. 35 - Prob. 5PCh. 35 - Prob. 6PCh. 35 - Prob. 7P
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- Write the linear program (variable list, objective function and constraints) for the vertex cover problem .of fallowing graph A B D E H Farrow_forwardBellman-Ford should be changed so that it only visits a vertex v if its SPT parent edgeTo[v] is not already in the waiting list. Cherkassky, Goldberg, and Radzik reported that this heuristic was practical. Show that the worst-case running time is proportional to EV and that it correctly computes the shortest paths.arrow_forwardThe graph-coloring problem is usually stated as the vertex-coloring problem: assign the smallest number of colors to vertices of a given graph so that no two adjacent vertices are the same color. Consider the edge-coloring problem: assign the smallest number of colors possible to edges of a given graph so that no two edges with the same end point are the same color. Explain how the edge-coloring problem can be polynomial reduced to a vertex-coloring problem. Give an example.arrow_forward
- 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…arrow_forwardIt is well-known that planar graphs are 4-colorable. However finding a vertex cover on planar graphs is NP-hard. Design an approximation algorithm to solve the vertex cover problem on planar graph. Prove your algorithm approximation ratio.arrow_forwardA Vertex Cover of an undirected graph G is a subset of the nodes of G,such that every edge of G touches one of the selected nodes.The VERTEX-COVER problem is to decide if a graph G has a vertex cover of size k.VERTEX-COVER = { <G,k> | G is an undirected graph with a k-node vertex cover }The VC3 problem is a special case of the VERTEX-COVER problem where the value of k is fixed at 3.VERTEX-COVER 3 = { <G> | G is an undirected graph with a 3-node vertex cover }Use parts a-b below to show that Vertex-Cover 3 is in the class P.a. Give a high-level description of a decider for VC3.A high-level description describes an algorithmwithout giving details about how the machine manages its tape or head.b. Show that the decider in part a runs in deterministic polynomial time.arrow_forward
- Show the choice issue variant is NP-complete; Does a graph G have a spanning tree with a target cost c and a vertex's maximum payment?arrow_forwardLet G be a graph with n vertices. If the maximum size of an independent set in G is k, clearly explain why the minimum size of a vertex cover in G is n - k.arrow_forwardProve the choice problem variant is NP-complete; Exists a spanning tree with a goal cost c for a graph G and a vertex's maximum payment?arrow_forward
- Prove that the following problem is NP-complete: Given a graph G, and an integer k, find whether or not graph G has a spanning degree where the maximum degree of any node is k. (Hint: Show a reduction from one of the following known NP-complete problems: Vertex Cover, Ham Path or SAT.)arrow_forwardCOMPLETE-SUBGRAPH problem is defined as follows: Given a graph G = (V, E) and an integer k, output yes if and only if there is a subset of vertices S ⊆ V such that |S| = k, and every pair of vertices in S are adjacent (there is an edge between any pair of vertices). How do I show that COMPLETE-SUBGRAPH problem is in NP? How do I show that COMPLETE-SUBGRAPH problem is NP-Complete? (Hint 1: INDEPENDENT-SET problem is a NP-Complete problem.) (Hint 2: You can also use other NP-Complete problems to prove NP-Complete of COMPLETE-SUBGRAPH.)arrow_forwardThe correct statements are: 3-SAT is polynomially reducible to the INDEPENDENT SET language. 3-SAT is polynomially reducible to the VERTEX-COVER language. 3-SAT is polynomially reducible to the EDGE-COVER language. 3-SAT is polynomially reducible to the TSP-DECIDE language.arrow_forward
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