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
Program Plan Intro
Toprove: P is a subset of NP (
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Show that the 3-CNF satisfiability problem (3-CNF SAT ) is NP-complete.
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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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- 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_forwardPlease prove that NP is closed under reversal.arrow_forwardGiven A and B are sets, prove that (A ⋃ B) ⋃ C = A ⋃ (B ⋃ C) by showing (A ⋃ B) ⋃ C ⊆ A ⋃ (B ⋃ C) and A ⋃ (B ⋃ C) ⊆ (A ⋃ B) ⋃ C. (You cannot use the Associative Laws for Sets since this is what you are proving.)arrow_forward
- L = {f in SAT | the number of satisfying assignments of f > 1/2 |f| } Show that L is NP-completearrow_forwardShow that if A and B are sets with A ⊆ B then A ∪ B = B. (Hint: Show that x ∈ A ∪ B implies x ∈ B and vice versa.)arrow_forwardIt was claimed that:(a, b) ≤ (c, d) ⇔ (a < c) ∨ (a = c ∧ b ≤ d) defines a well-ordering on N x N. Show that this is actually the case.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_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_forwardSuppose A and B are sets, and A is uncountable. Prove that A∪B is uncountable. Theorem 1: Any subset of a countable set is countable.arrow_forward
- If B is NP-complete and B ∈ P, then P = NP.arrow_forwardSuppose we have the following sets: P, Q, and R. Prove or disprove that if for all x, (x ∈ P) → ((x ∈ Q) → (x ∈ R)), then P ⋂ Q ⊆ Rarrow_forward· Tonsider a set S of 4 elements. Prove that we can choose any two subsets óf S of size 3, say A and B, and construct a matroid M = (S, 1) such that A and B are the only maximum independent sets in M.arrow_forward
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