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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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?
The decision variant of the minimum vertex cover problem is stated as follows. Given an undirected graph G = (V, E) and an integer k. Is there a set V ′ ⊆ V of at most k nodes such that each edge is covered, i.e. e ∩ V ′ ̸= ∅, for all e ∈ E. Show that the decision variant of the minimum vertex cover problem is NP-complete. You may use that the decision variant of the maximum clique problem is NP-complete.
Let R be the reduction from 3-SAT to VERTEX-COVER as discussed in class.
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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- Prove that this choice problem variation is NP-complete; Does the graph G have a spanning tree where the highest cost paid by any vertex is less than the goal cost c?arrow_forwardWe need two methods for topological sorting—a method to determine if avertex has no successors and a method for removing a vertex from a graph.Write the method for determining no successors first.arrow_forwardIs it necessary for every cut edge e in an undirected graph G to be an edge in G's depth-first search tree? Give at least two examples to back up your point. A cut edge is defined as any edge that cuts the graph.)arrow_forward
- Prove 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_forwardQuestion 1 Show that Vertex Cover is still NP-complete even when all vertices in the graph are restricted to have even degree full explain this question and text typing work only thanksarrow_forwardIs this statement about the chromatic number true? Please give some prove and brief explaination: If A and B are graphs on the same vertex-set, then χ(A ∪ B) ⩽ χ(A) + χ(B).arrow_forward
- The graph with edges AB, BC, CD and DA is a complete graph. True or Falsearrow_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_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_forward
- Write the procedure for locating the maximum cliquein the visibility graph Gi of a convex fan Fi under the assumptionthat there exists cross-visibility across every edge connecting a vertex ofC(vi+2, vi−2) to the fan vertex vi.arrow_forwardIn the decision version of the Clique problem you are given a non-empty graph G = (V,E) and a positive integer k and asked whether G has a complete subgraph (or clique) of size k (i.e., k vertices that induce a complete subgraph). Give the pseudocode of a recursive backtracking algorithm for the problem, and analyze your algorithm by providing a worst-case recurrence. Can you guarantee a worst-case running time in O(1.466^n)? Prove your answerarrow_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
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