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 34.5, Problem 8E
Program Plan Intro
To prove that half 3-CNF satisfiablity problem is NP complete.
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Show that the following problem belongs to NP class:
we are given a set S of integer numbers and an integer number t. Does S have a subset such that sum of its elements is t?
Note: Data Structures and Algorithm problem
(a) Stingy SAT is the following problem: given a set of clauses (each a disjunction of
literals) and an integer k, find a satisfying assignment in which at most k variables
are true, if such an assignment exists. Prove that stingy SAT is NP-hard.
(b) The Double SAT problem asks whether a given satisfiability problem has at least two
different satisfying assignments. For example, the problem {{V1, V2}, {V1, V2}, {V1, V2}}
is satisfiable, but has only one solution (v₁ = F, v₂ = T). In contrast, {{V1, V2}, {V1, V2}}
has exactly two solutions. Show that Double-SAT is NP-hard.
We are given a 3-CNF formula with n variables and m clauses, where m is even, in the half 3-CNF satisfiability issue. We want to know if there is a true variable assignment where precisely half of the clauses evaluate to 0 and exactly half of the clauses evaluate to 1. prove the NP-completeness of the partial 3-CNF satisfiability issue.
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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- Long chain of friends: You are given a list of people, and statements of the form “x knows y”. You are asked to find, is there a chain of k distinct people, such as x1 knows x2, x2 knows x3, and xk-1 knows xk. Prove that this problem is NP-complete by using one of the known NP-complete problems (CLIQUE, 3-SAT, Hamiltonian Path, Hamiltonian Cycle, Independent Set, etc.)arrow_forwardA 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.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
- In this group of problems, you are given the predicate P(x), where the domain of x is the set of natural numbers.arrow_forwardLet P be the problem of checking whether a given number is prime or not. Whatis the complementary problem of P. Say Q be the complementary problem, provethat Q is Co-NP or not.arrow_forwardL = {f in SAT | the number of satisfying assignments of f > 1/2 |f| } Show that L is NP-completearrow_forward
- Show that the 3-CNF satisfiability problem (3-CNF SAT ) is NP-complete.arrow_forwardProve Theorem:If any NP-complete problem is polynomial-time solvable, then P D NP. Equivalently, if any problem in NP is not polynomial-time solvable, then no NP-complete problem is polynomial-time solvable.arrow_forwardA student wants to prove by induction that a predicate P holds for certain nonnegative integers. They have proven that for all integers n ≥ 0 that P(n) → P(3n). Suppose the student has proven P(3). Which of the following propositions can they infer? (The domain for any quantifiers appearing in the answer choices is the natural numbers.) O Vn, P(3+3) O P(n) does not hold for ʼn < 0 P(n) for n = 6, 9, 12,... Vn ≥ 1, P(3¹)arrow_forward
- Problem 4: Let S = {s1, s2, . . . , sn} and T = {t1, t2, . . . , tm}, n ≤ m, be two sets of integers. (i) Describe a deterministic algorithm that checks whether S is a subset of T. What is the running time of your algorithm?arrow_forwardGiven 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_forwardQuestion 2: Consider the 0/1 knapsack problem. Given Nobjects where each object is specified by a weight and a profit, you are to put the objects in a bag of capacity C such that the sum of weights of the items in the bag does not exceed Cand the profits of the items is maximized. Note that you cannot use an item type more than once. a. Using dynamic programming, write an algorithm that finds the maximum total value according to the above constraints. b. What is the complexity of your algorithm? c. Show the dynamic programming table for the following data: W= { 2 ,7 , 1} , P={ 3 ,15 , 2 } and C=8.arrow_forward
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