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 34.3, Problem 7E
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To show that L is complete for NP if and only if
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P is the set of problems that can be solved in polynomial time.
More formally, P is the set of decision problems (e.g. given a graph G, does this graph G contain an odd cycle) for
which there exists a polynomial-time algorithm to correctly output the answer to that problem.
What is NP? Consider these five options.
A. NP is the set of problems that cannot be solved in polynomial time.
B. NP is the set of problems whose answer can be found in polynomial time.
C. NP is the set of problems whose answer cannot be found in polynomial time.
D. NP is the set of problems that can be verified in polynomial time.
E. NP is the set of problems that cannot be verified in polynomial time.
Determine which option is correct. Answer either A, B, C, D, or E.
P is the set of problems that can be solved in polynomial time.
More formally, P is the set of decision problems (e.g. given a graph G, does this graph G
contain an odd cycle) for which there exists a polynomial-time algorithm to correctly output the
answer to that problem.
What is NP? Consider these five options and determine which option is correct.
O NP is the set of problems that cannot be solved in polynomial time.
NP is the set of problems whose answer can be found in polynomial time.
O NP is the set of problems whose answer cannot be found in polynomial time.
O NP is the set of problems that can be verified in polynomial time.
O NP is the set of problems that cannot be verified in polynomial time.
A graduate student is working on a problem X. After working on it for several days she is unable to find a polynomial-time solution to the problem. Therefore, she attempts to prove that he problem is NP-complete. To prove that X is NP-complete she first designs a decision version of the problem. She then proves that the decision version is in NP. Next, she chooses SUBSET-SUM, a well-known NP-complete problem and reduces her problem to SUBSET-SUM (i.e., she proves X £p SUBSET-SUM). Is her approach correct? Explain your answer.
Chapter 34 Solutions
Introduction to Algorithms
Ch. 34.1 - Prob. 1ECh. 34.1 - Prob. 2ECh. 34.1 - Prob. 3ECh. 34.1 - Prob. 4ECh. 34.1 - Prob. 5ECh. 34.1 - Prob. 6ECh. 34.2 - Prob. 1ECh. 34.2 - Prob. 2ECh. 34.2 - Prob. 3ECh. 34.2 - Prob. 4E
Ch. 34.2 - Prob. 5ECh. 34.2 - Prob. 6ECh. 34.2 - Prob. 7ECh. 34.2 - Prob. 8ECh. 34.2 - Prob. 9ECh. 34.2 - Prob. 10ECh. 34.2 - Prob. 11ECh. 34.3 - Prob. 1ECh. 34.3 - Prob. 2ECh. 34.3 - Prob. 3ECh. 34.3 - Prob. 4ECh. 34.3 - Prob. 5ECh. 34.3 - Prob. 6ECh. 34.3 - Prob. 7ECh. 34.3 - Prob. 8ECh. 34.4 - Prob. 1ECh. 34.4 - Prob. 2ECh. 34.4 - Prob. 3ECh. 34.4 - Prob. 4ECh. 34.4 - Prob. 5ECh. 34.4 - Prob. 6ECh. 34.4 - Prob. 7ECh. 34.5 - Prob. 1ECh. 34.5 - Prob. 2ECh. 34.5 - Prob. 3ECh. 34.5 - Prob. 4ECh. 34.5 - Prob. 5ECh. 34.5 - Prob. 6ECh. 34.5 - Prob. 7ECh. 34.5 - Prob. 8ECh. 34 - Prob. 1PCh. 34 - Prob. 2PCh. 34 - Prob. 3PCh. 34 - Prob. 4P
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- True/False. Give a short explanation. i. Let T be a tree constructed by Dijkstra's algorithm for a weighted connected graph G. T is a spanning tree of G? ii. Let T be a tree constructed by Dijkstra's algorithm for a weighted connected graph G. T is a minimum spanning tree of G? iii. If an NP-complete problem can be solved in linear time, then all NP-complete problems can be solved in linear time. iv. If P # NP, there could be a polynomial-time algorithm for SAT.arrow_forwardL = {f in SAT | the number of satisfying assignments of f > 1/2 |f| } Show that L is NP-completearrow_forwardShow that the 3-CNF satisfiability problem (3-CNF SAT ) is NP-complete.arrow_forward
- Please help me with this practice problem in python : Implement two-level iterative method B = B_{TL} for graph Laplacian matrices. We want the symmetric B. Components: Given a graph, construct its graph Laplacian matrix. Then using Luby's algorithm, construct the P matrix that ensures a prescribed coarsening factor, e.g., 2, 4, or 8 times smaller number of coarse vertices. Since the graph Laplacian matrix is singular (it has the constants in its nullspace), to make it invertible, make its last row and columns zero, but keep the diagonal as it were (nonzero). The resulting modified graph Laplacian matrix A is invertible and s.p.d.. Form the coarse matrix A_c = P^TAP. To implement symmetric two-level cycle use one of the following M and M^T: (i) M is forward Gauss-Seidel, M^T - backward Gauss-Seidel (both corresponding to A) (ii) M = M^T - the ell_1 smoother. Compare the performance (convergence properties in terms of number of iterations) of B w.r.t. just using the smoother M…arrow_forwardConsider 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.arrow_forwardThe sum-of-subsets problem is the following: Given a sequence a1 , a2 ,..., an of integers, and an integer M, is there a subset J of {1,2,...,n} such that i∈J ai = M? Show that this problem is NP-complete by constructing a reduction from the exact cover problem.arrow_forward
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