332 research outputs found
On the Complexity of Computing an Equilibrium in Combinatorial Auctions
We study combinatorial auctions where each item is sold separately but
simultaneously via a second price auction. We ask whether it is possible to
efficiently compute in this game a pure Nash equilibrium with social welfare
close to the optimal one.
We show that when the valuations of the bidders are submodular, in many
interesting settings (e.g., constant number of bidders, budget additive
bidders) computing an equilibrium with good welfare is essentially as easy as
computing, completely ignoring incentives issues, an allocation with good
welfare. On the other hand, for subadditive valuations, we show that computing
an equilibrium requires exponential communication. Finally, for XOS (a.k.a.
fractionally subadditive) valuations, we show that if there exists an efficient
algorithm that finds an equilibrium, it must use techniques that are very
different from our current ones
Compactness for Holomorphic Supercurves
We study the compactness problem for moduli spaces of holomorphic supercurves
which, being motivated by supergeometry, are perturbed such as to allow for
transversality. We give an explicit construction of limiting objects for
sequences of holomorphic supercurves and prove that, in important cases, every
such sequence has a convergent subsequence provided that a suitable extension
of the classical energy is uniformly bounded. This is a version of Gromov
compactness. Finally, we introduce a topology on the moduli spaces enlarged by
the limiting objects which makes these spaces compact and metrisable.Comment: 38 page
Volume distortion in homotopy groups
Given a finite metric CW complex and an element ,
what are the properties of a geometrically optimal representative of ?
We study the optimal volume of as a function of . Asymptotically,
this function, whose inverse, for reasons of tradition, we call the volume
distortion, turns out to be an invariant with respect to the rational homotopy
of . We provide a number of examples and techniques for studying this
invariant, with a special focus on spaces with few rational homotopy groups.
Our main theorem characterizes those in which all non-torsion homotopy
classes are undistorted, that is, their distortion functions are linear.Comment: 49 pages, 4 figures. Accepted for publication in Geometric and
Functional Analysis (GAFA
A formula for the minimal coordination number of a parallel bundle
An exact formula for the minimal coordination numbers of the parallel packed
bundle of rods is presented based on an optimal thickening scenario. Hexagonal
and square lattices are considered.Comment: 12 pages, 4 figures, to appear in J. Chem. Phy
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