Multidimensional Allocation: In Apportionment and Bin Packing
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In this thesis, we deal with two problems on multidimensional allocation, specifically in apportionment and in bin packing. The apportionment problem models the allocation of seats in a House of Representatives such that it is proportional to the dimensions being represented. One example is the allocation of the 435 House seats to the 50 U.S. states, which demands being proportional in the one dimension of state population. It is also common to demand proportionality in both state population and political affiliation, where we now have to allocate to two dimensions simultaneously. We begin by investigating what it means for an 1-D apportionment to be "fair", and use this to judge the various methods of apportionment that have been used throughout history. This motivates the study of divisor methods, a certain class of apportionment methods that avoid any paradoxes. We then formally tackle the problem of 1-dimensional and 2-dimensional apportionment with divisor methods through the lens of optimization. The optimization approach generalizes well to higher dimensions, but a proportional apportionment is not always possible in 3 or more dimensions. Our thesis outlines the current method for finding "approximate" apportionments and improves it in certain regimes. As for bin packing, we model the allocation of virtual machines to servers (with limited capacity) in cloud computing, with the goal of designing and analyzing efficient algorithms that optimize the expected cost of the allocation. This builds off previous work that only considered the case where the items being packed had a one-dimensional size. We extend some of those results to items with multi-dimensional size in this thesis.