Structural and decomposition results for binet matrices, bidirected graphs and signed-graphic matroids.

In this thesis we deal with binet matrices and the class of signed-graphic matroids which is the class of matroids represented over R by binet matrices. The thesis is divided in three parts. In the first part, we provide the vast majority of the notions used throughout the thesis and some results regarding the class of binet matrices. In this part, we focus on the class of linear and integer programming problems in which the constraint matrix is binet and provide methods and algorithms which solve these problems efficiently. The main new result is that the existing combinatorial methods can not solve the {lcub}0, 1/2{rcub}-separation problem (special case of the well known separation problem) with integral binet matrices. The main new results of the whole thesis are provided in the next two parts. In the second part, we present a polynomial time algorithm to construct a bidirected graph for any totally unimodular matrix B by finding node-edge incidence matrices Q and S such that QB = S. Seymour's famous decomposition theorem for regular matroids states that any totally unimodular matrix can be constructed through a series of composition operations called k-sums starting from network matrices and their transposes and two compact representation matrices B1 and B2 of a certain ten element matroid. Given that B1 and B2 are binet matrices, we examine the k-sums of network and binet matrices (k = 1,2, 3). It is shown that the k-sum of a network and a binet matrix is a binet matrix, but binet matrices are not closed under this operation for k = 2, 3. A new class of matrices is introduced, the so-called tour matrices, which generalises network and totally unimodular matrices. For any such matrix there exists a bidirected graph such that the columns represent a collection of closed tours in the graph. It is shown that tour matrices are closed under 1-, 2- and 3-sum as well as under elementary operations on their rows and columns. Given the constructive proofs of the above results regarding the k-sum operations and existing recognition algorithms for network and binet matrices, an algorithm is presented which constructs a bidirected graph for any totally unimodular matrix. In the third part of this thesis we deal with the frame matroid of a signed graph, or simply the signed-graphic matroid. Several new results are provided in this last part of the thesis. Specifically, given a signed graph, we provide methods to find representation matrices of the associated signed-graphic matroid over GF(2), GF(3) and R. Furthermore, two new matroid recognition algorithms are presented in this last part. The first one determines whether a binary matroid is signed-graphic or not and the second one determines whether a (general) matroid is binary signed-graphic or not. Finally, one of the most important new results of this thesis is the decomposition theory for the class of binary signed-graphic matroids which is provided in the last chapter. In order to achieve this result, we employed Tutte's theory of bridges. The proposed decomposition differs from previous decomposition results on matroids that have appeared in the literature in the sense that it is not based on k-sums, but rather on the operation of deletion of a cocircuit. Specifically, it is shown that certain minors resulting from the deletion of a cocircuit of a binary matroid will be graphic matroids except for one that will be signed-graphic if and only if the matroid is signed-graphic. The decomposition theory for binary signed-graphic matroids is a joint work with G. Appa and L. Pitsoulis

Regular Matroids with Graphic Cocircuits

We introduce the notion of graphic cocircuits and show that a large class of regular matroids with graphic cocircuits belongs to the class of signed-graphic matroids. Moreover, we provide an algorithm which determines whether a cographic matroid with graphic cocircuits is signed-graphic or not

A Bilevel Linear Programming Model for Developing a Subsidy Policy to Minimize the Environmental Impact of the Agricultural Sector

The agro-food industry, while critical for establishing food security, is the most environmentally impactful industry since it causes biodiversity loss and the conversion of natural land to farms and pastures, requires pesticide and fertilizer use as well as high water consumption, and leads to greenhouse gas emissions as well as soil and environmental degradation. This impact can be mitigated through proper policy design. Environmental policy in agriculture, however, is inherently complex, due to the conflict between actors in the system, namely policy makers and farmers. This article introduces a bilevel linear programming (BLP) approach for the development of subsidy policies with the upper-level objective being the minimization of the environmental impact of the agricultural sector. Both levels of the model are formulated as linear programs and by considering the Water-Energy-Food-Climate Nexus, a general-purpose model is introduced. The methodology of the model formulation is spelled out. Finally, different approaches for fine tuning the BLP model are discussed in order to adjust it to each case study’s needs, and the model is applied to the case study of the region of Thessaly, Greece

On the representability of totally unimodular matrices on bidirected graphs

We present a polynomial time algorithm to construct a bidirected graph for any totally unimodular matrix B by finding node-edge incidence matrices Q and S such that QB=S. Seymour’s famous decomposition theorem for regular matroids states that any totally unimodular (TU) matrix can be constructed through a series of composition operations called k-sums starting from network matrices and their transposes and two compact representation matrices B1,B2 of a certain ten element matroid. Given that B1,B2 are binet matrices we examine the k-sums of network and binet matrices. It is shown that thek-sum of a network and a binet matrix is a binet matrix, but binet matrices are not closed under this operation for k=2,3. A new class of matrices is introduced, the so-called tour matrices, which generalise network, binet and totally unimodular matrices. For any such matrix there exists a bidirected graph such that the columns represent a collection of closed tours in the graph. It is shown that tour matrices are closed under k-sums, as well as under pivoting and other elementary operations on their rows and columns. Given the constructive proofs of the above results regarding the k-sum operation and existing recognition algorithms for network and binet matrices, an algorithm is presented which constructs a bidirected graph for any TU matrix

Optimization with binet matrices

This paper deals with linear and integer programming problems in which the constraint matrix is a binet matrix. Linear programs can be solved with the generalized network simplex method, while integer programs are converted to a matching problem. It is also proved that an integral binet matrix has strong Chvátal rank 1