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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The 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.
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.
Please prove that NP is closed under reversal.
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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- Please answer fast Let (G,g,p) be a cyclic group. Problem P1: Given (g, g^u,g^v), compute g^{u/v} Problem P2: Given (g, g^w, g^z), compute g^{2w/(z+w)} Prove that P1 is reducible to P2.arrow_forwardAlgorithm DEGENERATIONSGiven a rational proper parametrization P(t) = χ1 1(t) χ1 2(t) , χ2 1(t) χ2 2(t)∈ L(t)2,where L is a computable subfield of R, of an affine rational curve C, the algorithm computes DParrow_forwardShow that the 3-CNF satisfiability problem (3-CNF SAT ) is NP-complete.arrow_forward
- Prove the following problem is NPC: Given n sets S1,...,Sn, can we find a set A with size at most k, so that ∀i, Si ∩ A ̸= ∅ ?arrow_forwardProve, by induction, that if S is a finite set with k elements, then P(S), the power set ofS, has 2k elements.arrow_forwardIf QGROUP-ISO is many-one complete for NP(log2n) under polynomial time reductions then CLIQUE is in coNTIME[nO(1), 2O(√n log n)]/poly.I.e. for inputs of length n, CLIQUE has polynomial-size proofs which can be verified in 2O(√n log n) time with the help of a polynomial-size advice.arrow_forward
- Computer Science The 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_forwardHey, Kruskal's algorithm can return different spanning trees for the input Graph G.Show that for every minimal spanning tree T of G, there is an execution of the algorithm that gives T as a result. How can i do that? Thank you in advance!arrow_forwardIf A is NP-complete, and A has a polynomial time algorithm, then there is a polynomial time algorithm to find the largest clique in a directed. A clique in a graph G(V, E) is a subset of vertices V’ such that there is an edge (u,v) in E between every two vertices u and v in V’. For example, the four vertices u2, u3,u5, and u6 form the largest clique in the graph below.arrow_forward
- Given a 3-CNF formula and number k find if there exists a satisfying assignment such that at least kvariables are FALSE. Is this problem NP-complete or not? Why?arrow_forward3 Please show that NP is closed downwards under polynomial-time many-one reductions. (Assume A in NP, and A polynomial-time many-one reduced to B via function f, show that B is in NP as well)arrow_forwardGiven a problem X and Y, if X reduces to Y in polynomial time, and Y is known to be NP-Complete, what can be said about X?arrow_forward
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