Topics in Tropical Linear Algebra and Applied Probability

Topics in Tropical Linear Algebra and Applied Probability by Ngoc Mai Tran Doctor of Philosophy in Statistics University of California, BERKELEY Professor Bernd Sturmfels, Chair Tropical linear algebra is the study of classical linear algebra problems with arithmetic done over the tropical semiring, namely with addition replaced by max, and multiplication replaced by addition. It allows one to reformulate nonlinear problems into piecewise-linear ones. This approach has successfully been employed to solve and characterize solutions to many problems in combinatorial optimization, control theory and game theory [5]. Tropical spectral theory, the study of tropical eigenvalues and eigenspaces, often plays a central role in these applications. We derive the basics of this theory in Chapter 1. In Chapter 2 we give a combinatorial description of the cones of linearity of the tropical eigenvector map. In Chapter 3 we extend this work to cones of linearity of the tropical eigenspace and polytrope map. Our results contribute to a better understanding of the polyhedral foundations of tropical linear algebra. Chapter 4 illustrates the above results in the context of pairwise ranking. Here one assigns to each pair of candidates a comparison score, and the algorithm produces a cardinal (numerically quantified) ranking of candidates. This setup is natural in sport competitions, business and decision making. The difficulty lies in the existence of inconsistencies of the form A > B > C > A, since pairwise comparisons are performed independently. TropicalRank is an algorithm pioneered by Elsner and van den Driessche. Solution sets of this ranking method are precisely the polytropes studied in Chapter 3. For generic input pairwise comparison matrices, this set contains one unique point that is the tropical eigenvector, which is then interpreted as the comparison score. In particular, the results in Chapter 3 provide a complete classification of all possible solution sets to the optimization problem that TropicalRank solves. This answers open questions from several papers [22, 32] in the area. In Chapter 4 we also show that TropicalRank belongs to the same parametrized family of ranking methods as two other commonly used algorithms, PerronRank and HodgeRank. Furthermore, we show that HodgeRank and PerronRank asymptotically give the same score under certain random ranking models. Despite their mathematical connections, we can construct instances in which these three methods produce arbitrarily different rank order.