# 3-dimensional matching

3-dimensional matchings. (a) Input T. (b)–(c) Solutions.

In the mathematical discipline of graph theory, a 3-dimensional matching is a generalization of bipartite matching (a.k.a. 2-dimensional matching) to 3-uniform hypergraphs. Finding a largest 3-dimensional matching is a well-known NP-hard problem in computational complexity theory.

## Definition

Let X, Y, and Z be finite, disjoint sets, and let T be a subset of X × Y × Z. That is, T consists of triples (xyz) such that x ∈ X, y ∈ Y, and z ∈ Z. Now M ⊆ T is a 3-dimensional matching if the following holds: for any two distinct triples (x1y1z1) ∈ M and (x2y2z2) ∈ M, we have x1 ≠x2, y1 ≠y2, and z1 ≠z2.

### Example

The figure on the right illustrates 3-dimensional matchings. The set X is marked with red dots, Y is marked with blue dots, and Z is marked with green dots. Figure (a) shows the set T (gray areas). Figure (b) shows a 3-dimensional matching M with |M| = 2, and Figure (c) shows a 3-dimensional matching M with |M| = 3.

The matching M illustrated in Figure (c) is a maximum 3-dimensional matching, i.e., it maximises |M|. The matching illustrated in Figures (b)–(c) are maximal 3-dimensional matchings, i.e., they cannot be extended by adding more elements from T.

### Comparison with bipartite matching

A 2-dimensional matching can be defined in a completely analogous manner. Let X and Y be finite, disjoint sets, and let T be a subset of X × Y. Now M ⊆ T is a 2-dimensional matching if the following holds: for any two distinct pairs (x1y1) ∈ M and (x2y2) ∈ M, we have x1 ≠x2 and y1 ≠y2.

In the case of 2-dimensional matching, the set T can be interpreted as the set of edges in a bipartite graph G = (XYT); each edge in T connects a vertex in X to a vertex in Y. A 2-dimensional matching is then a matching in the graph G, that is, a set of pairwise non-adjacent edges.

Hence 3-dimensional matchings can be interpreted as a generalization of matchings to hypergraphs: the sets X, Y, and Z contain the vertices, each element of T is a hyperedge, and the set M consists of pairwise non-adjacent edges (edges that do not have a common vertex).

### Comparison with set packing

A 3-dimensional matching is a special case of a set packing: we can interpret each element (xyz) of T as a subset {xyz} of X ∪ Y ∪ Z; then a 3-dimensional matching M consists of pairwise disjoint subsets.

## Decision problem

In computational complexity theory, 3-dimensional matching is also the name of the following decision problem: given a set T and an integer k, decide whether there exists a 3-dimensional matching M ⊆ T with |M| ≥ k.

This decision problem is known to be NP-complete; it is one of Karp's 21 NP-complete problems.[1] There exist though polynomial time algorithms for that problem for dense hypergraphs.[2][3]

The problem is NP-complete even in the special case that k = |X| = |Y| = |Z|.[1][4][5] In this case, a 3-dimensional (dominating) matching is not only a set packing but also an exact cover: the set M covers each element of X, Y, and Z exactly once.[6]

## Optimization problem

A maximum 3-dimensional matching is a largest 3-dimensional matching. In computational complexity theory, this is also the name of the following optimization problem: given a set T, find a 3-dimensional matching M ⊆ T that maximizes |M|.

Since the decision problem described above is NP-complete, this optimization problem is NP-hard, and hence it seems that there is no polynomial-time algorithm for finding a maximum 3-dimensional matching. However, there are efficient polynomial-time algorithms for finding a maximum bipartite matching (maximum 2-dimensional matching), for example, the Hopcroft–Karp algorithm.

### Approximation algorithms

The problem is APX-complete, that is, it is hard to approximate within some constant.[7][8][9] On the positive side, for any constant ε > 0 there is a polynomial-time (3/2 + ε)-approximation algorithm for 3-dimensional matching.[7][8]

There is a very simple polynomial-time 3-approximation algorithm for 3-dimensional matching: find any maximal 3-dimensional matching.[9] Just like a maximal matching is within factor 2 of a maximum matching,[10] a maximal 3-dimensional matching is within factor 3 of a maximum 3-dimensional matching.

## Notes

1. ^ a b
2. ^
3. ^
4. ^ Garey & Johnson (1979), Section 3.1 and problem SP1 in Appendix A.3.1.
5. ^ Korte & Vygen (2006), Section 15.5.
6. ^ Papadimitriou & Steiglitz (1998), Section 15.7.
7. ^ a b
8. ^ a b Ausiello et al. (2003), problem SP1 in Appendix B.
9. ^ a b
10. ^

## References

• Ausiello, Giorgio; Crescenzi, Pierluigi; Gambosi, Giorgio; Kann, Viggo; Marchetti-Spaccamela, Alberto; Protasi, Marco (2003), Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties, Springer.
• Crescenzi, Pierluigi; Kann, Viggo; Halldórsson, Magnús; Karpinski, Marek; Woeginger, Gerhard (2000), "Maximum 3-dimensional matching", A Compendium of NP Optimization Problems.
• .
• Kann, Viggo (1991), "Maximum bounded 3-dimensional matching is MAX SNP-complete", Information Processing Letters 37 (1): 27–35, doi:10.1016/0020-0190(91)90246-E.
• Karp, Richard M. (1972), "Reducibility among combinatorial problems", in Miller, Raymond E.; Thatcher, James W., Complexity of Computer Computations, Plenum, pp. 85–103.
• Karpinski, Marek; Rucinski, Andrzej; Szymanska, Edyta (2009), "The Complexity of Perfect Matching Problems on Dense Hypergraphs", ISAAC '09 Proceedings of the 20th International Symposium on Algorithms: 626–636, doi:10.1007/978-3-642-10631-6_64.
• Keevash, Peter; Knox, Fiachra; Mycroft, Richard (2013), "Polynomial-Time perfect matchings in dense hypergraphs", STOC '13 Proceedings of the forty-fifth annual ACM symposium: 311–320, doi:10.1145/2488608.2488647.
• Korte, Bernhard; Vygen, Jens (2006), Combinatorial Optimization: Theory and Algorithms (3rd ed.), Springer.
• Papadimitriou, Christos H.; Steiglitz, Kenneth (1998), Combinatorial Optimization: Algorithms and Complexity, Dover Publications.