Matroid Duality

In this section we construct the dual matroid M^∗ of a given matroid M. We stress that the notion of duality of matroids differs from the duality known from linear algebra: the dual matroid of a finite vector space isnot the matroid formed by the dual space. Matroid duality has an interesting meaning in graph theory; see Result5.3.5below. The following construction of the dual matroid is due to Whitney [Whi35].

Theorem 5.3.1 Let M= (E,S)be a matroid.Put M^∗= (E,S^∗), where S^∗={J⊂E:J⊂E\B for some basisB of M}.

ThenM^∗ is a matroid as well, and the rank functionρ^∗ of M^∗ is given by ρ^∗(A) =|A|+ρ(E\A)−ρ(E). (5.1) Proof Obviously,S^∗is closed under inclusion. By Theorem5.2.1, it suffices to verify the following condition for each subsetAofE: all maximal subsets of Awhich are independent with respect toS^∗have the cardinalityρ^∗(A) given in (5.1). Thus letJ be such a subset ofA. Then there exists a basisB ofM withJ= (E\B)∩A; moreover,J is maximal with respect to this property.

This means thatB is chosen such that A\J=A\((E\B)∩A) =A∩B is minimal with respect to inclusion. HenceK:= (E\A)∩B is maximal with respect to inclusion. Thus K is a basis ofE\A in the matroid M and has cardinalityρ(E\A). Therefore the minimal subsetsA∩Ball have cardinality

|B| − |K|=|B| −ρ(E\A) =ρ(E)−ρ(E\A);

144 5 The Greedy Algorithm

and all maximal subsetsJ∈S^∗ ofAhave cardinality

|J|=|A| − |A\J|=|A| − |A∩B|=|A|+ρ(E\A)−ρ(E).

The matroidM^∗ constructed in Theorem5.3.1is called thedual matroid of M. The bases of M^∗ are the cobases of M; the circuits of M^∗ are the cocircuitsofM. According to Exercise5.2.11, the independent sets ofM^∗are precisely the complements of generating sets ofM. This implies the following result.

Corollary 5.3.2 Let M= (E,S)be a matroid.Then the independent sets of M^∗ are the complements of the generating sets ofM.In particular,the bases ofM^∗ are the complements of the bases of M.Hence (M^∗)^∗=M.

Example 5.3.3 LetM=M(G) be the matroid corresponding to a connected graph G. Then the bases of M are the spanning trees of G, and the bases ofM^∗ are thecotrees, that is, the complements of the spanning trees. More generally, a set S is independent in M^∗ if and only if its complement S contains a spanning tree of G, that is, if and only if S is connected. By definition, the circuits of a matroid are the minimal dependent sets. Thus the circuits ofM are the cycles inG, and the circuits of M^∗are the minimal sets C for which the complement C is not connected. In other words, the circuits of M^∗ are the simple cocycles of G—all those cocycles which are minimal with respect to inclusion.

In the general case, if G has p connected components, n vertices, and m edges, then M(G) has rankn−pand M(G)^∗ has rank m−(n−p), by Theorem 4.2.4.

Exercise 5.3.4 Let C be a circuit of a matroid M = (E,S). Show that E\C is a hyperplane of the dual matroid M^∗. Hint: Use condition (c) in Exercise5.2.11 to show thatE\C is a closed set, and then determine the rank ofE\Cin M^∗.

We now state an important theorem due to Whitney [Whi33] which clar-ifies the role of matroid duality in graph theory; a proof can be found in [Wel76] or [Oxl92].

Result 5.3.5 A graphGis planar if and only if the dual matroid M(G)^∗ is graphic.

Remark 5.3.6 While a proof of Result5.3.5is beyond the scope of this book, let us at least give a rough idea how the dual matroid of a planar graphG can be seen to be graphic; to simplify matters, we shall assume that each edge lies in a cycle. SupposeGis drawn in the plane. Construct a multigraph G^∗= (V^∗, E^∗) whose vertices correspond to the faces of G, by selecting a

Fig. 5.1 A geometric dual ofK5\e

pointvF in the interior of each faceF; two such points are connected by as many edges as the corresponding faces share inG.

More precisely, assume that the boundaries of two faces F and F share exactly k edges, say e1, . . . , ek. Then the corresponding vertices vF and vF

are joined bykedgese₁, . . . , e_k drawn in such a way that the edgee_i crosses the edgeeibut no other edge of the given drawing ofG. This results in a plane multigraphG^∗, and one may showM(G)^∗∼=M(G^∗). The planar multigraph G^∗ is usually called ageometric dualofG.⁴See Fig.5.1for an example of the construction just described, usingG=K5\e, which was shown to be planar in Exercise 1.5.6; here we actually obtain a graph. The reader might find it instructive to draw a few more examples, for instance usingG=K3,3\e (where a multigraph arises).

Exercise 5.3.7 Let M = (E,S) be a matroid, and let A and A^∗ be two disjoint subsets ofE. If A is independent inM and if A^∗ is independent in M^∗, then there are basesB andB^∗ ofM andM^∗, respectively, withA⊂B, A^∗⊂B^∗, andB∩B^∗=∅. Hint: Note ρ(E) =ρ(E\A^∗).

Exercise 5.3.8 LetM= (E,S) be a matroid. A subsetX ofE is a basis of M if and only ifX has nonempty intersection with each cocircuit ofM and is minimal with respect to this property.

Exercise 5.3.9 Let C be a circuit and C^∗ a cocircuit of the matroid M. Prove|C∩C^∗| = 1. Hint: Use Exercise5.3.7for an indirect proof.

This result plays an important role in characterizing a pair (M, M^∗) of dual matroids by the properties of their circuits and cocircuits; see [Min66].

Exercise 5.3.10 Let x and y be two distinct elements of a circuit C in a matroidM. Then there exists a cocircuitC^∗inM such thatC∩C^∗={x, y}.

4If edges not lying in a cycle—that is, edges belonging to just one faceF—are allowed, one has to associate such edges with loops in the construction ofG^∗. It should also be noted that a planar graph may admit essentially different plane embeddings and, hence, nonisomorphic geometric duals.

146 5 The Greedy Algorithm

Hint: Complete C\ {x} to a basis B of M and consider B^∗∪ {y}, where B^∗=E\B is a cobasis.

We will return to matroids several times throughout this book. For a thor-ough study of matroid theory we recommend the book by Welsh [Wel76], which is still a standard reference. We also mention the monographs [Tut71], [Rec89], and [Oxl92]; of these, Oxley’s book is of particular interest as it also includes applications of matroids. A series of monographs concerning matroid theory was edited by White [Whi86, Whi87, Whi92].

5.4 The Greedy Algorithm as an Approximation