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Chapter 9: Problem 26

Given a list of \(n\) positive integers \((n \text { even }),\) divide the list into two sublists such that the difference between the sums of the integers in the two sublists is minimized. Is this problem an \(N P\) complete problem? Is this problem an \(N P\) -hard problem?

Short Answer

Expert verified
Yes, the described task complies with both the definition of NP-complete and NP-hard problems.

Step by step solution

01

Problem Definition

Given a list of \(n\) positive integers (where \(n\) is even), the task is to divide this list into two sublists such that the difference between the sum of integers in these sublists is minimized. This type of problem is known as a partition problem in the field of computer science and mathematics.
02

NP Problem Classification

A problem is considered NP (non-deterministic polynomial time) if its solution can be verified in polynomial time. Given a 'certificate' (proposed solution), we can easily verify if the partitions are valid and if the difference between their sums is minimized by simply calculating the sums and comparing.
03

NP-Completeness Assessment

A problem is NP-Complete if it is in NP and its solution can be obtained in polynomial time from a solution to any other problem in NP. The given problem is a variant of the partition problem, which is known to be NP-Complete. Therefore, this problem can also be solved in polynomial time given a solution to the partition problem. Hence, this problem is NP-Complete.
04

NP-Hardness Judgment

NP-Hard problems are at least as hard as the hardest problems in NP. Any NP problem can be reduced to any NP-Hard problem in polynomial time. Since this problem is NP-Complete, and any NP-Complete problem is by definition NP-Hard, we can conclude that this problem is also NP-Hard.

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Most popular questions from this chapter

For the Sum-of-Subsets problem discussed in Chapter \(5,\) can you develop an approximation algorithm that runs in polynomial time?

Can an algorithm be a polynomial-time algorithm for a problem using one encoding scheme and be an exponential-time algorithm for the same problem using another encoding scheme? Justify your answer.

List three problems that have polynomial-time algorithms. Justify your answer.

Show that a problem is \(N P\) -easy if and only if it reduces to an \(N P\) -complete problem.

Can you develop an approximation algorithm for the CNF-Satisfiability problem by stating it as an optimization problem-that is, by finding a truth assignment of the literals in the expression that makes the maximum possible number of clauses true?
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