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In this thesis we discuss possible generalizations of totally unimodular and network matrices. Our purpose is to introduce new classes of matrices that preserve the advantageous properties of these well-known matrices. In particular, our focus is on the polyhedral consequences of totally unimodular matrices, namely we look for matrices that can ensure vertices that are scalable to an integral vector by an integer k. We argue that simply generalizing the determinantal structure of totally unimodular matrices does not suffice to achieve this goal and one has to extend the range of values the inverses of submatrices can contain. To this end, we define k-regular matrices. We show that k-regularity is a proper generalization of total unimodularity in polyhedral terms, as it guarantees the scalability of vertices. Moreover, we prove that the k-regularity of a matrix is necessary and sufficient for substituting mod-k cuts for rank-1 Chvatal-Gomory cuts. In the second part of the thesis we introduce binet matrices, an extension of network matrices to bidirected graphs. We provide an algorithm to calculate the columns of a binet matrix using the underlying graphical structure. Using this method, we prove some results about binet matrices and demonstrate that several interesting classes of matrices are binet. We show that binet matrices are 2-regular, therefore they provide half-integral vertices for a polyhedron with a binet constraint matrix and integral right hand side vector. We also prove that optimization on such a polyhedron can be carried out very efficiently, as there exists an extension of the network simplex method for binet matrices. Furthermore, the integer optimization with binet matrices is equivalent to solving a matching problem. We also describe the connection of k-regular and binet matrices to other parts of combinatorial optimization, notably to matroid theory and regular vectorspaces.

This thesis is concerned with examining issues of theoretical validity and bias in contingent valuation (CV) and discrete choice experiment (DCE) methods in low income countries. This thesis contributes to the small body of literature on the application of CV and DCEs in low income countries and in populations which have little or no formal education. Theoretical validity is examined by testing whether willingness to pay corresponds to theoretical expectations focusing on gender and willingness to pay, sensitivity to scope, starting point bias, and strategic bias in CV. The theoretical validity of the DCE method in populations with no formal education is also explored. It is found that whilst iterative methods to elicit willingness to pay often mimic local market conditions in low income countries they are prone to starting point bias and strategic bias. An association between gender and willingness to pay was also found. Issues of gender, starting point bias and strategic behaviour can be tested for and controlled for in the estimation of willingness to pay and do not present an insurmountable problem. Willingness to pay was also found to be insensitive to the size of the benefit in CV. Using the DCE method, it was found that with the use of visual aids, DCEs can be used among respondents with no formal education. It is concluded that CV and DCEs are feasible and valid in populations with low levels of education when surveys are conducted using trained enumerators and administered using face-to-face interviews. This suggests that both techniques are capable of being used in wide variety of settings. The exception to this is a lack of evidence on sensitivity to scope. Further research is therefore required into sensitivity to scope. Further research is also required to examine the association between gender and willingness to pay.