On the th best base of a matroid
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 with a weight function on its elements set and a positive integer number , the th best base problem is to find distinct bases 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 th best spanning tree (KBST) [4]. There is a pseudo-polynomial algorithm for the problem and if 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 and its dual , a subset of elements is called a locked subset if is a proper subset, i.e. , is -2-connected, is -2-connected and . We denote by the class of locked subsets of . Locked subsets were introduced by Chaourar [2] in a slight different manner. Using the definition of the dual rank, we have . A matroid (respectively, a graph) is polynomially locked if (respectively, , where is the cycle matroid of ) is polynomial on the size of . Particular cases of polynomially locked matroids are uniform matroids and series parallel graphs. For an element , we denote by (respectively, ) the largest subset containing parallel (respectively, coparallel) elements to after adding to it. The class of parallel (respectively, coparallel) closures is denoted by such that (respectively, such that ). Note that , , and also .
Locked subsets, parallel and coparallel closures will minimize the matroid bases polytope in terms of constraints. It is known that the matroid bases polytope [3] is given by the following constraints, where :We will introduce a technique based on neighbor extreme points of this polytope. Let be an extreme point of a polytope. We denote by the set of neighbor extreme points of .
Given two matroids , and an element where , the -sum or the 2-sum of and , denoted by , or more usually , is the matroid such that , where is identified and all the remaining elements stay with the same.
Given two graphs , and a clique , the -sum of and , denoted by , or more usually , is the graph where nodes and edges of are identified and all the remaining nodes and edges stay the same. Particular cases of this operation are 2-sum, denoted , if is one edge, and 3-sum, denoted , if is a triangle. Inversely, if is a disconnecting clique of (with possible parallel edges), then taking appropriate and .
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 -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: Proof We are going to prove that any other constraint of type (2), (3) or (4) is redundant. Constraints (2) and (3): Let . We have three cases: either 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 be a locked subset in and . Using the dual rank, we haveandHowever, is u -2-connected. Hence, . Using (7) and (8), we haveThus and 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 -sums for matroids and to prove that polynomial lockedness is closed under such -sums.
Acknowledgment
I wish to thank an anonymous referee for valuable indications and remarks on the presentation of the results.
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