To argue that the solution to instance I of the 0-1 knapsack problem is one of { P 1 , P 1 , ....... , P n } .

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Answer 1

Question

Chapter 35, Problem 7P

a

Program Plan Intro

To argue that the solution to instance I of the 0-1 knapsack problem is one of {P1,P1,.......,Pn} .

b

Program Plan Intro

To prove that an optimal solution Qj can be found to the fractional problem using the greedy algorithm.

c

Program Plan Intro

To prove that that an optimal solution Qj can be always constructed to the fractional problem for instance Ij that includes at most one item fractionally.

c

Program Plan Intro

Toprove that υ(Rj)≥υ(Qj)/2≥υ(Pj)/2 .

d

Program Plan Intro

To give a polynomial-time algorithm that returns a maximum-value solution from the set {R1,R2,......,Rn} and also prove that the algorithm is a polynomial time 2-approximation algorithm for the 0-1 knapsack problem.

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Exercise 1 Function and Structure [30 pts] Please debug the following program and answer the following questions. There is a cycle in a linked list if some node in the list can be reached again by continuously following the next pointer. #include typedef struct node { int value; struct node *next; } node; int 11_has_cycle (node *first) if (first == node *head = { NULL) return 0; first; while (head->next != NULL) { } if (head first) { return 1; } head = head->next; return 0; void test ll_has_cycle () { int i; node nodes [6]; for (i = 0; i < 6; i++) { nodes [i] .next = NULL; nodes [i].value = i; } nodes [0] .next = &nodes [1]; nodes [1] .next = &nodes [2]; nodes [2] .next = &nodes [3]; nodes [3] .next nodes [4] .next &nodes [4]; NULL; nodes [5] .next = &nodes [0]; printf("1. Checking first list for cycles. \n Function 11_has_cycle says it has s cycle\n\n", 11_has_cycle (&nodes [0])?"a":"no"); printf("2. Checking length-zero list for cycles. \n Function 11_has_cycle says it has %s…

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Discrete Mathematics for Computer Engineering

Answer 2

Question

Chapter 35, Problem 7P

a

Program Plan Intro

To argue that the solution to instance I of the 0-1 knapsack problem is one of {P1,P1,.......,Pn} .

b

Program Plan Intro

To prove that an optimal solution Qj can be found to the fractional problem using the greedy algorithm.

c

Program Plan Intro

To prove that that an optimal solution Qj can be always constructed to the fractional problem for instance Ij that includes at most one item fractionally.

c

Program Plan Intro

Toprove that υ(Rj)≥υ(Qj)/2≥υ(Pj)/2 .

d

Program Plan Intro

To give a polynomial-time algorithm that returns a maximum-value solution from the set {R1,R2,......,Rn} and also prove that the algorithm is a polynomial time 2-approximation algorithm for the 0-1 knapsack problem.

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Answer 3

Question

Chapter 35, Problem 7P

a

Program Plan Intro

To argue that the solution to instance I of the 0-1 knapsack problem is one of {P1,P1,.......,Pn} .

b

Program Plan Intro

To prove that an optimal solution Qj can be found to the fractional problem using the greedy algorithm.

c

Program Plan Intro

To prove that that an optimal solution Qj can be always constructed to the fractional problem for instance Ij that includes at most one item fractionally.

c

Program Plan Intro

Toprove that υ(Rj)≥υ(Qj)/2≥υ(Pj)/2 .

d

Program Plan Intro

To give a polynomial-time algorithm that returns a maximum-value solution from the set {R1,R2,......,Rn} and also prove that the algorithm is a polynomial time 2-approximation algorithm for the 0-1 knapsack problem.

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Answer 4

Question

Chapter 35, Problem 7P

a

Program Plan Intro

To argue that the solution to instance I of the 0-1 knapsack problem is one of {P1,P1,.......,Pn} .

b

Program Plan Intro

To prove that an optimal solution Qj can be found to the fractional problem using the greedy algorithm.

c

Program Plan Intro

To prove that that an optimal solution Qj can be always constructed to the fractional problem for instance Ij that includes at most one item fractionally.

c

Program Plan Intro

Toprove that υ(Rj)≥υ(Qj)/2≥υ(Pj)/2 .

d

Program Plan Intro

To give a polynomial-time algorithm that returns a maximum-value solution from the set {R1,R2,......,Rn} and also prove that the algorithm is a polynomial time 2-approximation algorithm for the 0-1 knapsack problem.

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Answer 5

Chapter 35, Problem 7P

a

Program Plan Intro

To argue that the solution to instance I of the 0-1 knapsack problem is one of {P1,P1,.......,Pn} .

b

Program Plan Intro

To prove that an optimal solution Qj can be found to the fractional problem using the greedy algorithm.

c

Program Plan Intro

To prove that that an optimal solution Qj can be always constructed to the fractional problem for instance Ij that includes at most one item fractionally.

c

Program Plan Intro

Toprove that υ(Rj)≥υ(Qj)/2≥υ(Pj)/2 .

d

Program Plan Intro

To give a polynomial-time algorithm that returns a maximum-value solution from the set {R1,R2,......,Rn} and also prove that the algorithm is a polynomial time 2-approximation algorithm for the 0-1 knapsack problem.

Answer 6

Chapter 35, Problem 7P

a

Program Plan Intro

To argue that the solution to instance I of the 0-1 knapsack problem is one of {P1,P1,.......,Pn} .

Answer 7

Chapter 35, Problem 7P

a

Program Plan Intro

To argue that the solution to instance I of the 0-1 knapsack problem is one of {P1,P1,.......,Pn} .

Answer 8

b

Program Plan Intro

To prove that an optimal solution Qj can be found to the fractional problem using the greedy algorithm.

Answer 9

b

Program Plan Intro

To prove that an optimal solution Qj can be found to the fractional problem using the greedy algorithm.

Answer 10

c

Program Plan Intro

To prove that that an optimal solution Qj can be always constructed to the fractional problem for instance Ij that includes at most one item fractionally.

Answer 11

c

Program Plan Intro

To prove that that an optimal solution Qj can be always constructed to the fractional problem for instance Ij that includes at most one item fractionally.

Answer 12

c

Program Plan Intro

Toprove that υ(Rj)≥υ(Qj)/2≥υ(Pj)/2 .

Answer 13

c

Program Plan Intro

Toprove that υ(Rj)≥υ(Qj)/2≥υ(Pj)/2 .

Answer 14

d

Program Plan Intro

To give a polynomial-time algorithm that returns a maximum-value solution from the set {R1,R2,......,Rn} and also prove that the algorithm is a polynomial time 2-approximation algorithm for the 0-1 knapsack problem.

Answer 15

d

Program Plan Intro

To give a polynomial-time algorithm that returns a maximum-value solution from the set {R1,R2,......,Rn} and also prove that the algorithm is a polynomial time 2-approximation algorithm for the 0-1 knapsack problem.

Answer 16

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Answer 17

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Answer 18

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Answer 19

Ch. 35.1 - Prob. 1E Ch. 35.1 - Prob. 2E Ch. 35.1 - Prob. 3E Ch. 35.1 - Prob. 4E Ch. 35.1 - Prob. 5E Ch. 35.2 - Prob. 1E Ch. 35.2 - Prob. 2E Ch. 35.2 - Prob. 3E Ch. 35.2 - Prob. 4E Ch. 35.2 - Prob. 5E

To argue that the solution to instance I of the 0-1 knapsack problem is one of { P 1 , P 1 , ....... , P n } .

To argue that the solution to instance I of the 0-1 knapsack problem is one of {P1,P1,.......,Pn} .

To prove that an optimal solution Qj can be found to the fractional problem using the greedy algorithm.

To prove that that an optimal solution Qj can be always constructed to the fractional problem for instance Ij that includes at most one item fractionally.

Toprove that υ(Rj)≥υ(Qj)/2≥υ(Pj)/2 .

To give a polynomial-time algorithm that returns a maximum-value solution from the set {R1,R2,......,Rn} and also prove that the algorithm is a polynomial time 2-approximation algorithm for the 0-1 knapsack problem.

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Chapter 35 Solutions