|
 3-dimensional matchings. (a) Input T. (b) (c) Solutions.
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 (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, 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 (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). 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 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] The problem is NP-complete even in the special case that k = |X| = |Y| = |Z|.[1][2][3] In this case, a 3-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.[4] 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.[5][6][7] On the positive side, for any constant > 0 there is a polynomial-time (3/2 + )-approximation algorithm for 3-dimensional matching.[5][6] There is a very simple polynomial-time 3-approximation algorithm for 3-dimensional matching: find any maximal 3-dimensional matching.[7] Just like a maximal matching is within factor 2 of a maximum matching,[8] a maximal 3-dimensional matching is within factor 3 of a maximum 3-dimensional matching. See also Notes References
| 
Encyclopedia
top
