The possibility of writing this book occurred to us in the late fall of 1991 when we were both participating in the program on Applied Linear Algebra at the Institute for Mathematics and its Applications (IMA) in Minnesota. A few years earlier we had been attracted to the subject of sign-solvability because of the beautiful interplay it afforded among linear algebra, combinatorics, and theoretical computer science (combinatorial algorithms). The subject, begun in 1947 by the economist P. Samuelson, was developed from various perspectives in the linear algebra, combinatorics, and economics literature. We thought that it would be A worthwhile project to organize the subject and to give A unified and self-contained presentation. Because there were no previous books or even survey papers on the subject, the tasks of deciding what was fundamental and how the material should be ordered for exposition had to be thought out very carefully. Our organization of the material has resulted in new connections among various results in the literature. In addition, many new results and many new and simpler proofs of previously established results are given throughout the book. We began the book in earnest in early 1992 and completed approximately three quarters of it while in residence at the IMA. After we returned to our home institutions, with the other duties that that entails, it was difficult to find the time for completing the book.

One of the features of this book is that we have explicitly described algorithms that are implicit in many of the proofs and have commented on their complexity. Throughout we have given credit for results that have previously occurred in the literature.