Delta-matroid

In mathematics, a delta-matroid or Δ-matroid is a family of sets obeying an exchange axiom generalizing an axiom of matroids. A non-empty family of sets is a delta-matroid if, for every two sets and in the family, and for every element in their symmetric difference , there exists an such that is in the family. For the basis sets of a matroid, the corresponding exchange axiom requires in addition that and , ensuring that and have the same cardinality. For a delta-matroid, either of the two elements may belong to either of the two sets, and it is also allowed for the two elements to be equal.[1] An alternative an equivalent definition is that a family of sets forms a delta-matroid when the convex hull of its indicator vectors (the analogue of a matroid polytope) has the property that every edge length is either one or the square root of two.

Delta-matroids were defined by André Bouchet in 1987.[2] Algorithms for matroid intersection and the matroid parity problem can be extended to some cases of delta-matroids.[3][4]

Delta-matroids have also been used to study constraint satisfaction problems.[5] As a special case, an even delta-matroid is a delta-matroid in which all sets have an even number of elements. If a constraint satisfaction problem has a Boolean variable on each edge of a planar graph, and if the variables of the edges incident to each vertex of the graph are constrained to belong to an even delta-matroid (possibly a different even delta-matroid for each vertex), then the problem can be solved in polynomial time. This result plays a key role in a characterization of the planar Boolean constraint satisfaction problems that can be solved in polynomial time.[6]

References
  1. Chun, Carolyn (July 13, 2016), "Delta-matroids: Origins", The Matroid Union
  2. Bouchet, André (1987), "Greedy algorithm and symmetric matroids", Mathematical Programming, 38 (2): 147–159, doi:10.1007/BF02604639, MR 0904585
  3. Bouchet, André; Jackson, Bill (2000), "Parity systems and the delta-matroid intersection problem", Electronic Journal of Combinatorics, 7: R14:1–R14:22, MR 1741336
  4. Geelen, James F.; Iwata, Satoru; Murota, Kazuo (2003), "The linear delta-matroid parity problem", Journal of Combinatorial Theory, Series B, 88 (2): 377–398, doi:10.1016/S0095-8956(03)00039-X, MR 1983366
  5. Feder, Tomás; Ford, Daniel (2006), "Classification of bipartite Boolean constraint satisfaction through delta-matroid intersection", SIAM Journal on Discrete Mathematics, 20 (2): 372–394, CiteSeerX 10.1.1.124.8355, doi:10.1137/S0895480104445009, MR 2257268
  6. Kazda, Alexandr; Kolmogorov, Vladimir; Rolínek, Michal (December 2018), "Even delta-matroids and the complexity of planar Boolean CSPs", ACM Transactions on Algorithms, 15 (2): 22:1–22:33, arXiv:1602.03124, doi:10.1145/3230649
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