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{{Short description|Strongly NP-complete problem in computer science}}
The '''3-partition problem''' is a [[Strong NP-completeness|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, ''S''<sub>1</sub>, ''S''<sub>2</sub>,
* 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''.
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==Variants==
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 ''S<sub>u</sub>'' of the unrestricted variant, construct a new instance of the restricted version {{math|''S<sub>r</sub>'' ≔ {s + 2{{hsp|''T''}} {{!}} s ∈ ''S<sub>u</sub>''}}}. Every solution of ''S<sub>u</sub>'' corresponds to a solution of ''S<sub>r</sub>'' but with a sum of 7{{hsp|''T''}} instead of ''T'', and every element of ''S<sub>r</sub>'' is in {{nowrap|[2{{hsp|''T''}}, 3{{hsp|''T''}}]}} which is contained in {{nowrap|{{pars|7{{hsp|''T''}}/4, 7{{hsp|''T''}}/2}}}}.
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.<ref>{{Cite journal|last1=Hulett|first1=Heather|last2=Will|first2=Todd G.|last3=Woeginger|first3=Gerhard J.|date=2008-09-01|title=Multigraph realizations of degree sequences: Maximization is easy, minimization is hard|url=http://www.sciencedirect.com/science/article/pii/S0167637708000552|journal=Operations Research Letters|language=en|volume=36|issue=5|pages=594–596|doi=10.1016/j.orl.2008.05.004|issn=0167-6377|url-access=subscription}}</ref>
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 ''S<sub>r</sub>'' of the restricted variant, with 3''m'' items summing up to ''mT'', construct a new instance of the unrestricted variant {{math|''S<sub>u</sub>'' ≔ {s + 2''T'' {{!}} s ∈ ''S<sub>r</sub>''}}}, with 3m items summing up to 7mT, and with target sum 7{{hsp|''T''}}. Every solution of ''S<sub>r</sub>'' naturally corresponds to a solution of ''S<sub>u</sub>''. Conversely, in every solution of ''S<sub>u</sub>'', since the target sum is 7{{hsp|''T''}} and each element is in {{nowrap|{{pars|7{{hsp|''T''}}/4, 7{{hsp|''T''}}/2}}}}, there must be exactly 3 elements per set, so it corresponds to a solution of ''S<sub>r</sub>''.
The '''ABC-partition problem''' (also called '''[[Numerical 3-dimensional matching|numerical 3-d matching]])''' is a variant in which, instead of a set ''S'' with 3{{hsp|''m''}} integers, there are three sets ''A'', ''B'', ''C'' with ''m'' integers in each. The sum of numbers in all sets is {{tmath|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''.<ref>{{Cite web|last=Demaine|first=Erik|date=2015|title=MIT OpenCourseWare - Hardness made Easy 2 - 3-Partition I|url=https://www.youtube.com/watch?v=ZaSMm2xvatw |archive-url=https://ghostarchive.org/varchive/youtube/20211214/ZaSMm2xvatw |archive-date=2021-12-14 |url-status=live|website=Youtube}}{{cbignore}}</ref>
The '''4-partition problem''' is a variant in which ''S'' contains ''n'' = 4{{hsp|''m''}} integers, the sum of all integers is {{tmath|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-
▲The '''4-partition problem''' is a variant in which ''S'' contains ''n'' = 4{{hsp|''m''}} integers, the sum of all integers is {{tmath|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.
== Proofs ==
Garey and Johnson (1975) originally proved 3-Partition to be NP-complete, by a reduction from [[3-dimensional matching]].<ref>{{cite journal|author=[[Michael Garey|Garey, Michael R.]] and [[David S. Johnson]]|year=1975|title=Complexity
=== Reduction from 3d-matching to ABCD-partition ===
We are given an instance of E of 3d-matching, containing some ''m'' triplets {w<sub>i</sub>,x<sub>j</sub>,y<sub>k</sub>}, where the vertices are w<sub>1</sub>,...,w<sub>q</sub> and x<sub>1</sub>,...,x<sub>q</sub> and y<sub>1</sub>,...,y<sub>q</sub>. We construct an instance of ABCD-partition with 4*''m'' elements, as follows (where r := 32q):
* For each triplet t = {w<sub>i</sub>,x<sub>j</sub>,y<sub>k</sub>} in E, the set A contains an element u<sub>t</sub> = 10r<sup>4</sup>-kr<sup>3</sup>-jr<sup>2</sup>-ir.
* For each triplet t = {w<sub>i</sub>,x<sub>j</sub>,y<sub>k</sub>} in E, the set B contains w<sub>it</sub>, C contains x<sub>jt</sub>, and D contains y<sub>kt</sub>. So for each of w<sub>i</sub>, x<sub>j</sub>, y<sub>k</sub>, there may be many corresponding elements in B, C, D - one for each triplet in which they appear. We consider one of these elements (denoted by "1") as the "real" one, and the others as "dummy" ones. The element sizes are as follows:
** w<sub>i</sub>[1] = 10r<sup>4</sup>+ir; w<sub>i</sub>[2
** x<sub>j</sub>[1] = 10r<sup>4</sup>+jr<sup>2</sup>; x<sub>j</sub>[2
** y<sub>k</sub>[1] = 10r<sup>4</sup>+kr<sup>3</sup>; y<sub>k</sub>[2
* The sum of every three "real" elements or every three "dummy" elements is 30r<sup>4</sup>+ir+jr<sup>2</sup>+kr<sup>3</sup>; and
* The threshold for the ABCD-partition instance is T=40r<sup>4</sup>. Note that the size of each element is in (T/3,T/5).
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== Applications ==
The NP-hardness of 3-partition was used to prove the NP-hardness [[rectangle packing]], as well as of [[Tetris#Computational complexity|Tetris]]<ref>{{Cite journal|date=2002-10-28|title=Tetris is hard, even to approximate|url=http://dx.doi.org/10.1038/news021021-9|journal=Nature|doi=10.1038/news021021-9|issn=0028-0836|url-access=subscription}}</ref><ref>{{Cite journal|last1=BREUKELAAR|first1=RON|last2=DEMAINE|first2=ERIK D.|last3=HOHENBERGER|first3=SUSAN|last4=HOOGEBOOM|first4=HENDRIK JAN|last5=KOSTERS|first5=WALTER A.|last6=LIBEN-NOWELL|first6=DAVID|title=Tetris is Hard, Even to Approximate|date=2004-04-01|url=http://dx.doi.org/10.1142/s0218195904001354|journal=International Journal of Computational Geometry & Applications|volume=14|issue=1n02|pages=41–68|doi=10.1142/s0218195904001354|issn=0218-1959|arxiv=cs/0210020|s2cid=1177 }}</ref> and some other puzzles,<ref>{{Cite journal|last1=Demaine|first1=Erik D.|last2=Demaine|first2=Martin L.|date=2007-06-01|title=Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity|url=http://dx.doi.org/10.1007/s00373-007-0713-4|journal=Graphs and Combinatorics|volume=23|issue=S1|pages=195–208|doi=10.1007/s00373-007-0713-4|s2cid=17190810 |issn=0911-0119|url-access=subscription}}</ref> and some [[Job scheduling|job scheduling problems]].<ref>{{Cite journal|last1=Bernstein|first1=D.|last2=Rodeh|first2=M.|last3=Gertner|first3=I.|date=1989|title=On the complexity of scheduling problems for parallel/pipelined machines|url=http://dx.doi.org/10.1109/12.29469|journal=IEEE Transactions on Computers|volume=38|issue=9|pages=1308–1313|doi=10.1109/12.29469|issn=0018-9340|url-access=subscription}}</ref>
==References==
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