2,079,373 research outputs found
Welfare Maximization with Limited Interaction
We continue the study of welfare maximization in unit-demand (matching)
markets, in a distributed information model where agent's valuations are
unknown to the central planner, and therefore communication is required to
determine an efficient allocation. Dobzinski, Nisan and Oren (STOC'14) showed
that if the market size is , then rounds of interaction (with
logarithmic bandwidth) suffice to obtain an -approximation to the
optimal social welfare. In particular, this implies that such markets converge
to a stable state (constant approximation) in time logarithmic in the market
size.
We obtain the first multi-round lower bound for this setup. We show that even
if the allowable per-round bandwidth of each agent is , the
approximation ratio of any -round (randomized) protocol is no better than
, implying an lower bound on the
rate of convergence of the market to equilibrium.
Our construction and technique may be of interest to round-communication
tradeoffs in the more general setting of combinatorial auctions, for which the
only known lower bound is for simultaneous () protocols [DNO14]
Bisector energy and few distinct distances
We introduce the bisector energy of an -point set in ,
defined as the number of quadruples from such that and
determine the same perpendicular bisector as and . If no line or circle
contains points of , then we prove that the bisector energy is
. We also prove the
lower bound , which matches our upper bound when is
large. We use our upper bound on the bisector energy to obtain two rather
different results:
(i) If determines distinct distances, then for any
, either there exists a line or circle that contains
points of , or there exist
distinct lines that contain points of . This result
provides new information on a conjecture of Erd\H{o}s regarding the structure
of point sets with few distinct distances.
(ii) If no line or circle contains points of , then the number of
distinct perpendicular bisectors determined by is
. This appears to
be the first higher-dimensional example in a framework for studying the
expansion properties of polynomials and rational functions over ,
initiated by Elekes and R\'onyai.Comment: 18 pages, 2 figure
Upper and lower limits on the number of bound states in a central potential
In a recent paper new upper and lower limits were given, in the context of
the Schr\"{o}dinger or Klein-Gordon equations, for the number of S-wave
bound states possessed by a monotonically nondecreasing central potential
vanishing at infinity. In this paper these results are extended to the number
of bound states for the -th partial wave, and results are also
obtained for potentials that are not monotonic and even somewhere positive. New
results are also obtained for the case treated previously, including the
remarkably neat \textit{lower} limit with (valid in the Schr\"{o}dinger case, for a class of potentials
that includes the monotonically nondecreasing ones), entailing the following
\textit{lower} limit for the total number of bound states possessed by a
monotonically nondecreasing central potential vanishing at infinity: N\geq
\{\{(\sigma+1)/2\}\} {(\sigma+3)/2\} \}/2 (here the double braces denote of
course the integer part).Comment: 44 pages, 5 figure
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