Classification and enumeration of special classes of posets and polytopes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 91-93).This thesis concerns combinatorial and enumerative aspects of different classes of posets and polytopes. The first part concerns the finite Eulerian posets which are binomial, Sheffer or triangular. These important classes of posets are related to the theory of generating functions and to geometry. Ehrenborg and Readdy [ER2] gave a complete classification of the factorial functions of infinite Eulerian binomial posets and infinite Eulerian Sheffer posets, where infinite posets are those posets which contain an infinite chain. We answer questions asked by R. Ehrenborg and M. Readdy [ER2]. We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets; We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases; In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets. This work is also motivated by the work of R. Stanley about recognizing the boolean lattice by looking at smaller intervals. In the second topic concerns lattice path matroid polytopes. The theory of matroid polytopes has gained prominence due to its applications in algebraic geometry, combinatorial optimization, Coxeter group theory, and, most recently, tropical geometry. In general matroid polytopes are not well understood. Lattice path matroid polytopes (LPMP) belong to two famous classes of polytopes, sorted closed matroid polytopes [LP] and polypositroids [Pos]. We study several properties of LPMPs and build a new connection between the theories of matroid polytopes and lattice paths. I investigate many properties of LPMPs, including their face structure, decomposition, and triangulations, as well as formulas for calculating their Ehrhart polynomial and volume.by Hoda Bidkhori.Ph.D

Scalable Robust Kidney Exchange

In barter exchanges, participants directly trade their endowed goods in a constrained economic setting without money. Transactions in barter exchanges are often facilitated via a central clearinghouse that must match participants even in the face of uncertainty---over participants, existence and quality of potential trades, and so on. Leveraging robust combinatorial optimization techniques, we address uncertainty in kidney exchange, a real-world barter market where patients swap (in)compatible paired donors. We provide two scalable robust methods to handle two distinct types of uncertainty in kidney exchange---over the quality and the existence of a potential match. The latter case directly addresses a weakness in all stochastic-optimization-based methods to the kidney exchange clearing problem, which all necessarily require explicit estimates of the probability of a transaction existing---a still-unsolved problem in this nascent market. We also propose a novel, scalable kidney exchange formulation that eliminates the need for an exponential-time constraint generation process in competing formulations, maintains provable optimality, and serves as a subsolver for our robust approach. For each type of uncertainty we demonstrate the benefits of robustness on real data from a large, fielded kidney exchange in the United States. We conclude by drawing parallels between robustness and notions of fairness in the kidney exchange setting.Comment: Presented at AAAI1

Analyzing process flexibility: A distribution-free approach with partial expectations

We develop a distribution-free model to evaluate the performance of process flexibility structures when only the mean and partial expectation of the demand are known. We characterize the worst-case demand distribution under general concave objective functions, and apply it to derive tight lower bounds for the performance of chaining structures under the balanced systems (systems with the same number of plants and products). We also derive a simple lower bound for chaining-like structures under unbalanced systems with different plant capacities. Keywords: Process flexibility; Distributionally-robust analysis; Chaining; Production system desig