Concept# Generalization

Summary

A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation.
Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them.
However, the parts cannot be generalized into a whole—until a common relation is established among all parts. This does not mean that the parts are u

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In the present thesis we study the geometry of the moduli spaces of Bradlow-Higgs triples on a smooth projective curve C. There is a family of stability conditions for triples that depends on a positive real parameter Ï. The moduli spaces of Ï-semistable triples of rank r and degree d vary with Ï. The phenomenon arising Ï from this is known as wall-crossing. In the first half of the thesis we will examine how the moduli spaces and their universal additive invariants change as Ï varies, for the case r = 2. In particular we will study the case of Ï very close to 0, for which the moduli space relates to the moduli space of stable Higgs bundles, and Ï very large, for which the moduli space is a relative Hilbert scheme of points for the family of spectral curves. Some of these results will be generalized to Bradlow-Higgs triples with poles. In the second half we will prove a formula relating the cohomology of the moduli spaces for small and odd degree and the perverse filtration on the cohomology of the moduli space of stable Higgs bundles. We will also partially generalize this result to the case of rank greater than 2.

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We study the maximum budgeted allocation problem, i.e., the problem of selling a set of m indivisible goods to n players, each with a separate budget, such that we maximize the collected revenue. Since the natural assignment LP is known to have an integrality gap of , which matches the best known approximation algorithms, our main focus is to improve our understanding of the stronger configuration LP relaxation. In this direction, we prove that the integrality gap of the configuration LP is strictly better than , and provide corresponding polynomial time roundings, in the following restrictions of the problem: (i) the restricted budgeted allocation problem, in which all the players have the same budget and every item has the same value for any player it can be sold to, and (ii) the graph MBA problem, in which an item can be assigned to at most 2 players. Finally, we improve the best known upper bound on the integrality gap for the general case from to and also prove hardness of approximation results for both cases.

Christos Kalaitzis, Aleksander Madry, Lukas Polacek, Ola Nils Anders Svensson

We study the Maximum Budgeted Allocation problem, i.e., the problem of selling a set of m indivisible goods to n players, each with a separate budget, such that we maximize the collected revenue. Since the natural assignment LP is known to have an integrality gap of, which matches the best known approximation algorithms, our main focus is to improve our understanding of the stronger configuration LP relaxation. In this direction, we prove that the integrality gap of the configuration LP is strictly better than, and provide corresponding polynomial time roundings, in the following restrictions of the problem: (i) the Restricted Budgeted Allocation problem, in which all the players have the same budget and every item has the same value for any player it can be sold to, and (ii) the graph MBA problem, in which an item can be assigned to at most 2 players. Finally, we improve the best known upper bound on the integrality gap for the general case from 5/6 to 2√2 2 ≈ 0.828 and also prove hardness of approximation results for both cases. © 2014 Springer International Publishing Switzerland.

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