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 34.2, Problem 9E
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Toprove: P is a subset of NP (
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if P = NP, then show that SAT P.
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.
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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- 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.arrow_forwardProve NP-completeness. S and t are integers. T-sum S subsets? Algorithm and Data Structure Issuesarrow_forwardProve 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_forward
- Make a maximal covering problem and formulate it. And Convert that problem into the set covering problem including formulation. What happens?arrow_forwardPlease prove that NP is closed under reversal.arrow_forwardL = {f in SAT | the number of satisfying assignments of f > 1/2 |f| } Show that L is NP-completearrow_forward
- show that SAT ∈ P iff P = NParrow_forwardLet P be a simple polygon. If W is a finite minimal witness set forP, then no element of W lies in int(P).Proofarrow_forwardLet CLIQUES= { <G, k> | Ghas at least two k-cliquesthat differby at least one node}. a. Show that CLIQUES ∊ NP by writing either a verifier or an NDTM. b. Show that CLIQUES is NP-complete by reduction from CLIQUE.arrow_forward
- demonstrate that SAT P if P = NP.arrow_forwardComputer 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_forward2 Prove NP = PSPACE if TQBF ∈ NP . Prove P = PSPACE if TQBF ∈ P.arrow_forward
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