3-partition problem

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. More precisely:

Input: a multiset S containing n positive integer elements. Conditions: S must be partitionable into m triplets, S1, S2, , Sm, where n = 3m. These triplets partition S in the sense that they are disjoint and they cover S. The target value T is computed by taking the sum of all elements in S, then divided by m.
Output: whether or not there exists a partition of S such that, for all triplets, the sum of the elements in each triplet equals T.

The 3 partition problem remains strongly NP complete under the restriction that every integer in S is strictly between T/4 and T/2.

The set can be partitioned into the four sets , each of which sums to T = 90. The set can be partitioned into the two sets each of which sum to T = 15. (every integer in S is strictly between T/4 and T/2): , thus m=2, and T=15. There is feasible 3 partition (every integer in S is strictly between T/4 and T/2): , thus m=2, and T=15. There is no feasible solution.

The 3 partition problem remains NP complete even when the integers in S are bounded above by a polynomial in n. In other words, the problem remains NP complete even when representing the numbers in the input instance in unary. i. e., 3 partition is NP complete in the strong sense or strongly NP complete. This property, and 3 partition in general, is useful in many reductions where numbers are naturally represented in unary.

The 3 partition problem is similar to the partition problem, in which the goal is to partition S into two subsets with equal sum, and the multiway number partitioning, in which the goal is to partition S into k subsets with equal sum, where k is a fixed parameter. In 3 Partition the goal is to partition S into m = n/3 subsets, not just a fixed number of subsets, with equal sum. Partition is "easier" than 3 Partition: while 3 Partition is strongly NP hard, Partition is only weakly NP hard it is hard only when the numbers are encoded in non unary system, and have value exponential in n. When the values are polynomial in n, Partition can be solved in polynomial time using the pseudopolynomial time number partitioning algorithm.

In the unrestricted input variant, the inputs can be arbitrary integers; in the restricted input variant, the inputs must be in (T/4, T/2). The restricted version is as hard as the unrestricted version: given an instance Su of the unrestricted variant, construct a new instance of the restricted version }. Every solution of Su corresponds to a solution of Sr but with a sum of 7 instead of T, and every element of Sr is in which is contained in

In the distinct input variant, the inputs must be in (T/4, T/2), and in addition, they must all be distinct integers. It, too, is as hard as the unrestricted version. In the unrestricted output variant, the m output subsets can be of arbitrary size not necessarily 3 (but they still need to have the same sum T). The restricted output variant can be reduced to the unrestricted variant: given an instance Sr of the restricted variant, with 3m items summing up to mT, construct a new instance of the unrestricted variant }, with 3m items summing up to 7mT, and with target sum 7. Every solution of Sr naturally corresponds to a solution of Su. Conversely, in every solution of Su, since the target sum is 7 and each element is in , there must be exactly 3 elements per set, so it corresponds to a solution of Sr. The ABC partition problem (also called numerical 3 d matching) is a variant in which, instead of a set S with 3 integers, there are three sets A, B, C with m integers in each. The sum of numbers in all sets is m T. The goal is to construct m triplets, each of which contains one element from A, one from B and one from C, such that the sum of each triplet is T.

The 4 partition problem is a variant in which S contains n = 4 integers, the sum of all integers is m T, and the goal is to partition it into m quadruplets, all with a sum of T. It can be assumed that each integer is strictly between T/5 and T/3. Similarly, ABCD parititon is a variant of 4 partition in which each there are 4 input sets and each quadruplet should contain one element from each set.

Garey and Johnson (1975) originally proved 3 Partition to be NP complete, by a reduction from 3 dimensional matching. The classic reference by Garey and Johnson (1979) describes an NP completeness proof, reducing from 3 dimensional matching to 4 partition to 3 partition. Logically, the reduction can be partitioned into several steps.

The NP hardness of 3 partition was used to prove the NP hardness rectangle packing, as well as of Tetris and some other puzzles, and some job scheduling problems.