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 24.4, Problem 7E
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To explain the solving of system of difference constrains by Bellman-Ford
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The third-clique problem is about deciding whether a given graph G = (V, E) has a clique of cardinality at least |V |/3.Show that this problem is NP-complete.
Show that choice problem version is NP-complete; Given a graph G and a goal cost c, is there a spanning tree whose maximum vertex pay is c?
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?
Chapter 24 Solutions
Introduction to Algorithms
Ch. 24.1 - Prob. 1ECh. 24.1 - Prob. 2ECh. 24.1 - Prob. 3ECh. 24.1 - Prob. 4ECh. 24.1 - Prob. 5ECh. 24.1 - Prob. 6ECh. 24.2 - Prob. 1ECh. 24.2 - Prob. 2ECh. 24.2 - Prob. 3ECh. 24.2 - Prob. 4E
Ch. 24.3 - Prob. 1ECh. 24.3 - Prob. 2ECh. 24.3 - Prob. 3ECh. 24.3 - Prob. 4ECh. 24.3 - Prob. 5ECh. 24.3 - Prob. 6ECh. 24.3 - Prob. 7ECh. 24.3 - Prob. 8ECh. 24.3 - Prob. 9ECh. 24.3 - Prob. 10ECh. 24.4 - Prob. 1ECh. 24.4 - Prob. 2ECh. 24.4 - Prob. 3ECh. 24.4 - Prob. 4ECh. 24.4 - Prob. 5ECh. 24.4 - Prob. 6ECh. 24.4 - Prob. 7ECh. 24.4 - Prob. 8ECh. 24.4 - Prob. 9ECh. 24.4 - Prob. 10ECh. 24.4 - Prob. 11ECh. 24.4 - Prob. 12ECh. 24.5 - Prob. 1ECh. 24.5 - Prob. 2ECh. 24.5 - Prob. 3ECh. 24.5 - Prob. 4ECh. 24.5 - Prob. 5ECh. 24.5 - Prob. 6ECh. 24.5 - Prob. 7ECh. 24.5 - Prob. 8ECh. 24 - Prob. 1PCh. 24 - Prob. 2PCh. 24 - Prob. 3PCh. 24 - Prob. 4PCh. 24 - Prob. 5PCh. 24 - Prob. 6P
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- Demonstrate that the decision problem variant is NP-complete; Exists, given a graph G and a cost objective c, a spanning tree where the maximum payment of any vertex does not exceed c?arrow_forwardGive proof that the adapted form of the choice problem has an NP-complete solution. Is there a spanning tree in which the highest possible payment at any vertex does not exceed the target cost, given a graph G and a target cost c?arrow_forwardProvide proof that the modified choice problem has an NP-complete solution; Does a graph G with a target cost c have a spanning tree in which the largest feasible payment at any vertex does not exceed the target cost?arrow_forward
- Provide evidence that the modified version of the choice problem has an NP-complete solution; Exists, given a graph G and a target cost c, a spanning tree in which the highest possible payment at any vertex does not exceed the target cost?arrow_forwardProve that the decision problem variant is NP-complete; Given a graph G and a target cost c does thereexist a spanning tree where the maximum payment of any vertex is no more than c?arrow_forwardexplain a graph problem that has a direct and efficient algorithm using breadth first search, but whose solution is not nearly so straight forward if the algorithm was based on depth first search.arrow_forward
- 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.arrow_forwardShow 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_forwardConsider the following constraint graph. Trace Arc Consistency (AC3) algorithm on this graph. If nothing has changed use "="arrow_forward
- Demonstrate that the variant of the decision problem is NP-complete; Exists, given a graph G and a target cost c, a spanning tree in which the utmost payment of any vertex does not exceed c?arrow_forwardProve 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_forwardImplement The dynamic programming algorithm for the leveled graph problem. pre-cond: G is a weighted directed layered graph, and s and t are nodes. post-cond: optSol is a path with minimum total weight from s tot, and optCost is its weight, and optNum is the number of possible optimal solutions.arrow_forward
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