The Facets of the Bases Polytope of a Matroid and Two Consequences

Let $M$ to be a matroid defined on a finite set $E$ and $L\subset E$. $L$ is locked in $M$ if $M|L$ and $M^*|(E\backslash L)$ are 2-connected, and $min\{r(L), r^*(E\backslash L)\} \geq 2$. In this paper, we prove that the nontrivial facets of the bases polytope of $M$ are described by the locked subsets. We deduce that finding the maximum--weight basis of $M$ is a polynomial time problem for matroids with a polynomial number of locked subsets. This class of matroids is closed under 2-sums and contains the class of uniform matroids, the V\'amos matroid and all the excluded minors of 2-sums of uniform matroids. We deduce also a matroid oracle for testing uniformity of matroids after one call of this oracle.


Introduction
Sets and their characterisitic vectors will not be distinguished. We refer to Oxley [9] and Schrijver [12] about, respectively, matroids and polyhedra terminolgy and facts. Given a weight function c ∈ R E , the maximum-weight basis problem (MWBP) is the following optimization problem: Maximize{c(B) such that B ∈ B(M)} The corresponding maximum-weight independent problem is clearly (polynomially time) equivalent to MWBP. MWBP is polynomial on |E| and θ, where θ is the complexity of the used matroid oracle [3]. Even if we use the approach introduced by Mayhew [8] by giving the list of bases (for example) in the input, MWBP is polynomial on the size of the input. However, as Robinson and Welsh [11] note, no matter which of the ways to specify a matroid, the size of the input for a matroid problem on an n-element set is O(2 n ). It follows that MWBP is not polynomial in its strict sense, that is on |E|. We prove that MWBP is polynomial on |E| for polynomially locked classes of matroids, i.e., for any matroid M ∈ L k (for a fixed k). This class of polynomially locked matroids is closed under 2-sums and contains the class of uniform matroids, the Vámos matroid and all the excluded minors of 2-sums of uniform matroids. These excluded minors are M(K 4 ), W 3 , Q 6 and P 6 [2]. It follows that this class is larger than 2-sums of uniform matroids. Testing Uniformity of matroids (TUM) is to provide an algorithm in which the matroid is represented by an oracle and which decides whether the given matroid is uniform or not after a number of calls on the oracle which is bounded by a polynomial in the size of the ground set. Jensen and Korte [7] proved that there exists no such algorithm in which the matroid is represented by an independence test oracle (or an oracle polynomially related to an independence test oracle). In this paper, we give a matroid oracle which answers this question. The remainder of the paper is organized as follows: in section 2, we give all facets of the bases polytope, then, in section 3, we deduce two consequences of this characterization. The first one is that MWBP is polynomial (time) for polynomially locked matroids, and the second one is a polynomial time algorithm via a new matroid oracle for testing if a given matroid is uniform or not. In section 4, we describe some polynomially locked classes of matroids, and we conclude in section 5.

Facets of the bases polytope
A description of Q(M) was given by Edmonds [3] as follows.
Later, a minimal description of Q(M) was given also by Edmonds [6] as follows. It is not difficult to see that P (M) is the set of all x ∈ R E satisfying the inequalities (1), (2) and x It seems natural to think that the inequality (2) is a facet of P (M) if and only if A is closed and 2-connected. This is not true because: Proof. It suffices to prove that if X is closed and 2-connected but E\L is not 2connected in the dual then the inequality (2) is not a facet. In fact, there exist A and which implies the inequality (2). So the inequality (2) is redundant and cannot be a facet.
We give now a minimal description of P (M). A part of the proof is inspired from a proof given by Pulleyblank [10] to describe the nontrivial facets of Q(M). Independently, Fujishige [5], and Feichtner and Sturmfels [4], gave a characterization of nontrivial facets of P (M). We give here below a new and complete formulation with a new proof.
Theorem 2.4. A minimal description of P (M) is the set of all x ∈ R E satisfying the equality (3) and the following inequalities: x(P ) ≤ 1 for any parallel closure P ⊆ E (4) x(S) ≥ |S| − 1 for any coparallel closure S ⊆ E x(L) ≤ r(L) for any locked subset L ⊆ E Proof. Without loss of generality, we can suppose that M is without parallel or coparallel elements so the inequalities (5) become as (1) and the inequalities (4) become as follows: x(e) ≤ 1 for any e ∈ E Let C(M) be the cone generated by the incidence vectors of the bases of M. It suffices to prove that the minimal description of C(M) is given by (1) and the following inequalities: x(e) ≤ x(E)/r(E) for any e ∈ E (8) x(L)/r(L) ≤ x(E)/r(E) for any locked subset L ⊆ E It is not difficult to see via induction and operations of deletion and contraction that the inequalities (1) and (8)  Since E\B is a basis in the dual, then E\B ∈ B * (E\X). Claim 3: a j = a k for all j and k of E\L.
Using claim 2, E\L being 2-connected and a similar argument on E\B as in claim 1, we conclude. Claim 4: ax ≥ 0 is a multiple of inequality (10). By claims 1 and 3, ax ≥ 0 becomes: a L x(L) + a E\L x(E\L) ≥ 0. Thus, for B ∈ B(L), we have: But a L = r(L)r(E) and a E\L = r(L) is a solution of this equation, so we conclude.

MWBP and TUM
Since the bases polytope is completly described by the locked structure of the matroid, so a natural matroid oracle follows.
The k-locked oracle Input: a nonegative integer k and a matroid M defined on E. Note that this oracle has time complexity O(|E| k+1 ) because we need to count at most |E| k+1 members of L(M) in order to know that M is not k-locked, even if the memory complexity can be O(|E| + ℓ(M)). Actually this matroid oracle permits to recognize if a given matroid is k-locked or not for a given nonegative integer k (which does not depend on M or |E|). The first consequence of Theorem 2.4 then follows. Proof. Let M be a such matroid. Since M ∈ L k then it can be described by its locked structure in the input of MWBP by using the k-locked oracle. MWBP is equivalent for optimizing on P (M), which is also equivalent to separating on P (M). The k-locked oracle is stronger than the rank and the independence oracles for polynomially locked matroids: We can get the rank of any subset X ⊆ E by choosing the weight function c equal to the characteristic vector of X and optimizing on P (M), which can be done in polynomial time. The obtained optimum value of c is the requested rank. For the independence oracle, we can decide if a subset X ⊆ E is independent or not by choosing the same previous weight function and decide that X is independent if it is included in the optimum basis, and not if else. For testing uniformity of matroids, we need the following result [2]. We can see through the proof of this theorem that ℓ(M) = 0 if M is uniform whatever its connectivity. For disconnected matroids, we have the following result [9].  So we can now characterize uniform matroids as follows.

Conclusion
We have given a complete description of all facets of the bases polytope of a matroid and deduce two consequences. One about MWBP and the second about TUM. Future investigations can be characterizing some or all polynomially locked classes of matroids.