Up-growing On-line Linear Discrepancy of Triple-optimal Partially Ordered Sets (thesis)

Abstract

Whether we acknowledge it as a poset or not, posets arise in many natural contexts, and many also seem to warrant linear extensions (or rankings) of the poset. In some sense, the linear discrepancy of a linear extension L of a poset P indicates the unfairness of L. We describe triple-optimal posets, a class of posets where there exists at least one linear extension which has linear discrepancy three times the minimum linear discrepancy l. This is the worst case scenario; there is no way to have a worse linear discrepancy than triple the optimal linear discrepancy. Two players, a Builder and an Assigner, play an on-line game to construct a linear extension. The Builder gives the Assigner points from P that the Assigner subsequently irrevocably places in a linear extension LA using an algorithm. The Builder's goal is to maximize the linear discrepancy of LA while the Assigner battles to minimize the linear discrepancy of LA. Restrictions can be placed on the Builder, such as up-growing where the Builder cannot give points less than those points already given. In the context of up-growing, we play this on-line game using triple-optimal posets and develop an algorithm that caps the linear discrepancy of LA at 2l on triple-optimal posets with linear discrepancy l.