Question
In: Computer Science
Prove that the 0/1 KNAPSACK problem is NP-Hard. (One way to prove this is to prove...
Prove that the 0/1 KNAPSACK problem is NP-Hard. (One way to prove this is to prove the decision version of 0/1 KNAPSACK problem is NP-Complete. In this problem, we use PARTITION problem as the source problem.)
(a) Give the decision version of the O/1 KNAPSACK problem, and name it as DK.
(b) Show that DK is NP-complete (by reducing PARTITION problem to DK).
(c) Explain why showing DK, the decision version of the O/1 KNAPSACK problem, is NP-Complete is good enough to show that the O/1 KNAPSACK problem is NP-hard.
Solutions
Expert Solution
The following pages consists of the information showing how
Knapsack problem is NP-Hard.
venereology answered 1 year agoRelated Solutions
What is NP? What is P? What is NP-complete? What is NP- hard?
What is NP? What is P? What is NP-complete? What is NP- hard? Give brief definitions. Give an example of an NP- complete problem. Is P equal to NP?
Assuming that SAT is NP-hard, prove that both SAT and 3-SAT (which is SAT restricted to...
Assuming that SAT is NP-hard, prove that both SAT and 3-SAT
(which is SAT restricted to only having clauses containing at most
three literals) are NP-complete.
In C++ or Java Write the Greedy programming Algorithm for the 0-1 Knapsack problem. (a)...
In C++ or Java Write the Greedy programming Algorithm for the
0-1 Knapsack problem.
(a) No fraction allowed
Max Weight W= 9
item 1: profit $20, weight 2, prof/weight=$10
item 2: profit $30, weight 5, prof/weight=$6
item 3: profit $35, weight 7, prof/weight=$5
item 4: profit $12, weight 3, prof/weight=$4
item 5: profit $3, weight 1, prof/weight=$3
Show that the following problem is NP-hard. Input: A boolean function in CNF such that every...
Show that the following problem is NP-hard.
Input: A boolean function in CNF such that every clause has at
most three literals and every variable appears in at most three
clauses.
Output: An assignment that evaluates the given function
TRUE.
Recall the dynamic programming algorithm we saw in class for solving the 0/1 knapsack problem for...
Recall the dynamic programming algorithm we saw in class for
solving the 0/1 knapsack problem for n objects with a knapsack
capacity of K. In particular, we characterized our recurrence
OPT(j, W) to be following quantity:
OPT(j, W) := The maximum profit that can be obtained when
selecting from objects 1, 2, . . . , j with a knapsack capacity of
W , where (after filling in our dynamic programming table), we
return the value stored at OPT(n, K)...
What are the different algorithms we may use to solve the 0/1 Knapsack problem? Compare and...
What are the different algorithms we may use to solve the 0/1
Knapsack problem? Compare and contrast the different methods.
discuss the differences and similarities (implementation and performance) between the solution for 0-1 knapsack problem using...
discuss the differences and similarities (implementation and
performance) between the solution for 0-1 knapsack problem using
Backtracking versus Branch and Bound. Do not just define the
algorithms compare the performance and implementation of the 0-1
knapsack problem
The Knapsack problem is an optimization problem that asks to fill a knapsack with maximum possible...
The Knapsack problem is an optimization problem that asks to
fill a knapsack with maximum possible value. Using greedy paradigm,
we can solve this problem easily. Your task is the following:
(a) Write the pseudo-code of a greedy solution for knapsack
problem.
(b) Give is the time complexity of your solution in part
(a).
(c) Implement part (a) in C programming language.
The Knapsack problem is an optimization problem that asks to fill a knapsack with maximum possible...
The Knapsack problem is an optimization problem that asks to
fill a knapsack with maximum possible value. Using greedy paradigm,
we can solve this problem easily. Your task is the following:
(a) Write the pseudo-code of a greedy solution for knapsack
problem.
(b) Give is the time complexity of your solution in part
(a).
(c) Implement part (a) in C programming language.
What is P-Problem? What is NP-Problem? What is NP-Complete problem?
What is P-Problem?
What is NP-Problem?
What is NP-Complete problem?
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