Is the 3 Partition problem NP-complete?

The 3-partition problem is a strongly NP-complete problem in computer science. The problem is to decide whether a given multiset of integers can be partitioned into triplets that all have the same sum.

What are the NP-complete problems?

NP-complete problems are in NP, the set of all decision problems whose solutions can be verified in polynomial time; NP may be equivalently defined as the set of decision problems that can be solved in polynomial time on a non-deterministic Turing machine.

What is partition problem in algorithm?

In number theory and computer science, the partition problem, or number partitioning, is the task of deciding whether a given multiset S of positive integers can be partitioned into two subsets S₁ and S₂ such that the sum of the numbers in S₁ equals the sum of the numbers in S₂.

Which problems related to partition should be resolved?

The restriction of requiring the partition to have equal size, or that all input integers be distinct, is also NP-hard. Product partition is the problem of partitioning a set of integers into two sets with the same product (rather than the same sum). This problem is strongly NP-hard.

How do I know how many partitions a number has?

The number of partitions of n is given by the partition function p(n). So p(4) = 5. The notation λ ⊢ n means that λ is a partition of n….Among the 22 partitions of the number 8, there are 6 that contain only odd parts:

  1. 7 + 1.
  2. 5 + 3.
  3. 5 + 1 + 1 + 1.
  4. 3 + 3 + 1 + 1.
  5. 3 + 1 + 1 + 1 + 1 + 1.
  6. 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1.

Can NP-complete problems be solved in polynomial time?

If an NP-complete problem can be solved in polynomial time then all problems in NP can be solved in polynomial time. If a problem in NP cannot be solved in polynomial time then all problems in NP-complete cannot be solved in polynomial time. Note that an NP-complete problem is one of those hardest problems in NP.

How many NP problems are there?

This list is in no way comprehensive (there are more than 3000 known NP-complete problems). Most of the problems in this list are taken from Garey and Johnson’s seminal book Computers and Intractability: A Guide to the Theory of NP-Completeness, and are here presented in the same order and organization.

How do you prove the partition problem is NP-complete?

Answer: To show that any problem A is NP-Complete, we need to show four things: (1) there is a non-deterministic polynomial-time algorithm that solves A, i.e., A ∈ NP, (2) any NP-Complete problem B can be reduced to A, (3) the reduction of B to A works in polynomial time, (4) the original problem A has a solution if …

Are NP hard problems NP-complete?

NP-Hard problems are as hard as NP-Complete problems. NP-Hard Problem need not be in NP class. A problem is NP-Complete if it is a part of both NP and NP-Hard Problem. A non-deterministic Turing machine can solve NP-Complete problem in polynomial time.

How do you solve partitions?

Partitioning a line segment, AB, into a ratio a/b involves dividing the line segment into a + b equal parts and finding a point that is a equal parts from A and b equal parts from B. When finding a point, P, to partition a line segment, AB, into the ratio a/b, we first find a ratio c = a / (a + b).

Is the set partition problem in NP-complete?

Thus, it can be concluded that the Set partition problem is NP-Complete using the following two propositions: If any problem is in NP, then, given a ‘certificate’, which is a solution to the problem and an instance of the problem (a set S and two partitions A and A’, in this case), it can be proved that the certificate in polynomial time.

How do you prove a problem is NP-complete?

An instance of the Set Partition problem is a set S, and the task is to check whether there exist any two non-overlapping partitions of S having a sum of elements as sum. Since an NP-Complete problem is a problem which is both in NP and NP-hard, the proof for the statement that a problem is NP-Complete consists of two parts:

What is a set partition problem in math?

Set partition problem: Set partition problem partitions an array of numbers into two subsets such that the sum of each of these two subsets is the same. Let S be a set of numbers and A is a subset of numbers with sum S1, then there exists another subset containing the remainder of the elements (S – A) with sum S2, and S1 is equaled to S2.