165,812 research outputs found
Optimal Scoring Rules for Multi-dimensional Effort
This paper develops a framework for the design of scoring rules to optimally
incentivize an agent to exert a multi-dimensional effort. This framework is a
generalization to strategic agents of the classical knapsack problem (cf.
Briest, Krysta, and V\"ocking, 2005, Singer, 2010) and it is foundational to
applying algorithmic mechanism design to the classroom. The paper identifies
two simple families of scoring rules that guarantee constant approximations to
the optimal scoring rule. The truncated separate scoring rule is the sum of
single dimensional scoring rules that is truncated to the bounded range of
feasible scores. The threshold scoring rule gives the maximum score if reports
exceed a threshold and zero otherwise. Approximate optimality of one or the
other of these rules is similar to the bundling or selling separately result of
Babaioff, Immorlica, Lucier, and Weinberg (2014). Finally, we show that the
approximate optimality of the best of those two simple scoring rules is robust
when the agent's choice of effort is made sequentially
Cluster Dynamical Mean Field Theories
Cluster Dynamical Mean Field Theories are analyzed in terms of their
semiclassical limit and their causality properties, and a translation invariant
formulation of the cellular dynamical mean field theory, PCDMFT, is presented.
The semiclassical limit of the cluster methods is analyzed by applying them to
the Falikov-Kimball model in the limit of infinite Hubbard interaction U where
they map to different classical cluster schemes for the Ising model.
Furthermore the Cutkosky-t'Hooft-Veltman cutting equations are generalized and
derived for non translation invariant systems using the Schwinger-Keldysh
formalism. This provides a general setting to discuss causality properties of
cluster methods. To illustrate the method, we prove that PCDMFT is causal while
the nested cluster schemes (NCS) in general and the pair scheme in particular
are not. Constraints on further extension of these schemes are discussed.Comment: 26 page
Virtual Knot Cobordism
This paper defines a theory of cobordism for virtual knots and studies this
theory for standard and rotational virtual knots and links. Non-trivial
examples of virtual slice knots are given. Determinations of the four-ball
genus of positive virtual knots are given using the results of a companion
paper by the author and Heather Dye and Aaron Kaestner. Problems related to
band-passing are explained, and a theory of isotopy of virtual surfaces is
formulated in terms of a generalization of the Yoshikawa moves.Comment: 32 pages, 43 figures, LaTeX documen
Quantum Information Complexity and Amortized Communication
We define a new notion of information cost for quantum protocols, and a
corresponding notion of quantum information complexity for bipartite quantum
channels, and then investigate the properties of such quantities. These are the
fully quantum generalizations of the analogous quantities for bipartite
classical functions that have found many applications recently, in particular
for proving communication complexity lower bounds. Our definition is strongly
tied to the quantum state redistribution task.
Previous attempts have been made to define such a quantity for quantum
protocols, with particular applications in mind; our notion differs from these
in many respects. First, it directly provides a lower bound on the quantum
communication cost, independent of the number of rounds of the underlying
protocol. Secondly, we provide an operational interpretation for quantum
information complexity: we show that it is exactly equal to the amortized
quantum communication complexity of a bipartite channel on a given state. This
generalizes a result of Braverman and Rao to quantum protocols, and even
strengthens the classical result in a bounded round scenario. Also, this
provides an analogue of the Schumacher source compression theorem for
interactive quantum protocols, and answers a question raised by Braverman.
We also discuss some potential applications to quantum communication
complexity lower bounds by specializing our definition for classical functions
and inputs. Building on work of Jain, Radhakrishnan and Sen, we provide new
evidence suggesting that the bounded round quantum communication complexity of
the disjointness function is \Omega (n/M + M), for M-message protocols. This
would match the best known upper bound.Comment: v1, 38 pages, 1 figur
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