On the Kth best base of a matroid

https://doi.org/10.1016/j.orl.2007.05.007Get rights and content

Abstract

Given a weighted matroid M and a positive integer K, the Kth best base of M problem is to find K distinct minimum (or maximum) bases regarding the weight function. This problem is NP-hard. We prove that it is polynomial for 2-sums of uniform matroids and a fixed number of k-sums of series parallel graphs, M(K4), W3, Q6 and P6.

Introduction

We refer to Bondy and Murty's book [1] about Graph theory, to Oxley's book [6] about Matroid theory and to Schrijver's book [7] about Polyhedra. Given a matroid M with a weight function w on its elements set E and a positive integer number K, the Kth best base problem is to find K distinct bases B1,B2,,BK such that there is not a base with a weight less than their weights. This problem is NP-hard even in the graphical case, the Kth best spanning tree (KBST) [4]. There is a pseudo-polynomial algorithm for the problem and if K is fixed then it is polynomial time solvable [5]. We give a polyhedral approach to the problem and deduce a large class of matroids (respectively, graphs) for which KBBM (respectively, KBST) is polynomial time solvable. Let us give some notations and definitions. Given a matroid M and its dual M*, a subset of elements HE is called a locked subset if H is a proper subset, i.e. HE, H is M-2-connected, EH is M*-2-connected and r(H)max{2,2+r(E)-|EH|}. We denote by L(M) the class of locked subsets of M. Locked subsets were introduced by Chaourar [2] in a slight different manner. Using the definition of the dual rank, we have |L(M)|=|L(M*)|. A matroid (respectively, a graph) is polynomially locked if |L(M)| (respectively, |L(M(G))|, where M(G) is the cycle matroid of G) is polynomial on the size of |E|. Particular cases of polynomially locked matroids are uniform matroids and series parallel graphs. For an element eE, we denote by P(e) (respectively, S(e)) the largest subset containing parallel (respectively, coparallel) elements to e after adding e to it. The class of parallel (respectively, coparallel) closures is denoted by P(M)={P(e)E such that eE} (respectively, S(M)={S(e)E such that eE}). Note that P(M*)=S(M), P(M)=S(M*), and also |P(M)|+|S(M)|2|E|.

Locked subsets, parallel and coparallel closures will minimize the matroid bases polytope in terms of constraints. It is known that the matroid bases polytope P(M) [3] is given by the following constraints, where xRE:x(E)=r(E),x(e)0foranyelementeE,x(e)1foranyelementeE,x(H)r(H)foranysubsetHofE.We will introduce a technique based on neighbor extreme points of this polytope. Let x be an extreme point of a polytope. We denote by N(x) the set of neighbor extreme points of x.

Given two matroids M1, M2 and an element {e}=E(M1)E(M2) where M(Ee)=M1(M2, the e-sum or the 2-sum of M1 and M2, denoted by M1eM2, or more usually M12M2, is the matroid M such that E(M)=E(M1)E(M2), r(M)=r(M1)+r(M2)-1 where e is identified and all the remaining elements stay with the same.

Given two graphs G1=(V1,E1), G2=(V2,E2) and a clique H=E1E2, the H-sum of G1 and G2, denoted by G1HG2, or more usually G1|H|G2, is the graph G=(V1V2,E1E2) where nodes and edges of H are identified and all the remaining nodes and edges stay the same. Particular cases of this operation are 2-sum, denoted 2, if H is one edge, and 3-sum, denoted 3, if H is a triangle. Inversely, if H is a disconnecting clique of G (with possible parallel edges), then G=G1HG2 taking appropriate G1 and G2.

The remainder of the paper is organized in the following manner. In Section 2, we prove that KBBM is polynomial time solvable for polynomially locked matroids. In Section 3, we discuss some operations for preserving polynomial lockedness and deduce a large class of matroids and graphs constructed by k-sums for which KBBM is polynomial time solvable. Section 4 is reserved for conclusions.

Section snippets

KBBM is polynomial time solvable for polynomially locked matroids

Without loss of generality, we can suppose that considered matroids are 2-connected.

Theorem 1

The matroid bases polytope is given by constraint (1) and the following constraints:x(S(e))|S(e)|-1foranyelementS(e)S(M),x(P(e))1foranyelementP(e)P(M),x(H)r(H)foranylockedsubsetHofM.

Proof

We are going to prove that any other constraint of type (2), (3) or (4) is redundant.

Constraints (2) and (3): Let eE. We have three cases: either e has no parallel or series elements, or it has parallel elements but no series

Some operations preserving polynomial lockedness

We need the following property of locked subsets in order to simplify proofs.

Lemma 2

Any locked subset is closed.

Proof

Let L be a locked subset in M and eEL. Using the dual rank, we haver*(EL)=|EL|-r(E)+r(L)andr*(E(L{e}))=|EL|-1-r(E)+r(L{e}).However, EL={e}E(L{e}) is u M*-2-connected. Hence, r*(EL)=r*(E(L{e})). Using (7) and (8), we have|EL|-r(E)+r(L)=|EL|-1-r(E)+r(L{e}).Thus r(L{e})=r(L)+1and we are done.  

Now we are ready for the first operation preserving polynomially lockedness.

Theorem 3

The

Conclusion

Future perspectives for this work is to define appropriate k-sums for matroids and to prove that polynomial lockedness is closed under such k-sums.

Acknowledgment

I wish to thank an anonymous referee for valuable indications and remarks on the presentation of the results.

References (7)

  • B. Chaourar

    On greedy bases packings in matroids

    Eur. J. Combin.

    (2002)
  • J.A. Bondy et al.

    Graph Theroy and Applications

    (1982)
  • J. Edmonds

    Matroids and the greedy algorithm

    Math. Program.

    (1971)
There are more references available in the full text version of this article.

Cited by (0)

View full text