3-dimensional matching

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3-dimensional matchings. (a) Input T. (b)-(c) Solutions. 3-dimensional-matching.svg
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 (also known as 2-dimensional matching) to 3-partite hypergraphs, which consist of hyperedges each of which contains 3 vertices (instead of edges containing 2 vertices in a usual graph).

Contents

3-dimensional matching, often abbreviated as 3DM, is also the name of a well-known computational problem: finding a largest 3-dimensional matching in a given hypergraph. 3DM is one of the first problems that were proved to be NP-hard.

Definition

Let X, Y, and Z be finite sets, and let T be a subset of X × Y × Z. That is, T consists of triples (x, y, z) 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 (x1, y1, z1) ∈ M and (x2, y2, z2) ∈ 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 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 (x1, y1) ∈ M and (x2, y2) ∈ 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 = (X, Y, T); 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). In case of 2-dimensional matching, we have Y = Z.

Comparison with set packing

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

Decision problem

In computational complexity theory, 3-dimensional matching (3DM) is 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] It is NP-complete even in the special case that k = |X| = |Y| = |Z| and when each element is contained in at most 3 sets, i.e., when we want a perfect matching in a 3-regular hypergraph. [1] [2] [3] In this case, a 3-dimensional 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. [4] The proof is by reduction from 3SAT. Given a 3SAT instance, we construct a 3DM instance as follows: [2] [5]

Special cases

There exist polynomial time algorithms for solving 3DM in dense hypergraphs. [6] [7]

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

There is a very simple polynomial-time 3-approximation algorithm for 3-dimensional matching: find any maximal 3-dimensional matching. [8] Just like a maximal matching is within factor 2 of a maximum matching, [9] a maximal 3-dimensional matching is within factor 3 of a maximum 3-dimensional matching.

For any constant ε > 0 there is a polynomial-time (4/3 + ε)-approximation algorithm for 3-dimensional matching. [10]

However, attaining better approximation factors is probably hard: the problem is APX-complete, that is, it is hard to approximate within some constant. [11] [12] [8]

It is NP-hard to achieve an approximation factor of 95/94 for maximum 3-d matching, and an approximation factor of 48/47 for maximum 4-d matching. The hardness remains even when restricted to instances with exactly two occurrences of each element. [13]

Parallel algorithms

There are various algorithms for 3-d matching in the massively parallel communication model. [14]

See also

Notes

  1. 1 2 Karp (1972).
  2. 1 2 Garey, Michael R.; Johnson, David S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness . Series of Books in the Mathematical Sciences (1st ed.). New York: W. H. Freeman and Company. ISBN   9780716710455. MR   0519066. OCLC   247570676., Section 3.1 and problem SP1 in Appendix A.3.1.
  3. ↑ Korte, Bernhard; Vygen, Jens (2006), Combinatorial Optimization: Theory and Algorithms (3rd ed.), Springer , Section 15.5.
  4. ↑ Papadimitriou & Steiglitz (1998), Section 15.7.
  5. ↑ Demaine, Erik (2016). "16. Complexity: P, NP, NP-completeness, Reductions". YouTube .
  6. ↑ Karpinski, Rucinski & Szymanska (2009)
  7. ↑ Keevash, Knox & Mycroft (2013)
  8. 1 2 Kann (1991)
  9. ↑ Matching (graph theory)#Properties.
  10. ↑ Cygan, Marek (2013). "Improved Approximation for 3-Dimensional Matching via Bounded Pathwidth Local Search". 2013 IEEE 54th Annual Symposium on Foundations of Computer Science. pp. 509–518. arXiv: 1304.1424 . Bibcode:2013arXiv1304.1424C. doi:10.1109/FOCS.2013.61. ISBN   978-0-7695-5135-7. S2CID   14160646.
  11. ↑ Crescenzi et al. (2000).
  12. ↑ Ausiello et al. (2003), problem SP1 in Appendix B.
  13. ↑ Chlebík, Miroslav; Chlebíková, Janka (2006-04-04). "Complexity of approximating bounded variants of optimization problems". Theoretical Computer Science. Foundations of Computation Theory (FCT 2003). 354 (3): 320–338. doi: 10.1016/j.tcs.2005.11.029 . ISSN   0304-3975.
  14. ↑ Hanguir, Oussama; Stein, Clifford (2020-09-21). "Distributed Algorithms for Matching in Hypergraphs". arXiv: 2009.09605 [cs.DS].

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