Measurement data, snapshots of a system, and traffic or activity logs are
typically collected repeatedly. {\em Difference queries}, which identify and
measure change, are central to anomaly detection, monitoring, and planning.
When the data is sampled, as is often necessary to meet resource constraints,
queries need to be processed over the sampled data. Surprisingly, however, we
are not aware of pre-existing satisfactory estimators even for Euclidean
distances.
Random sampling is an essential tool in the processing and transmission of
data. It is used to summarize data too large to store or manipulate and meet
resource constraints on bandwidth or battery power. Estimators that are applied
to the sample facilitate fast approximate processing of queries posed over the
original data and the value of the sample hinges on the quality of these
estimators.
We study mechanisms for an allocation of goods among agents, where agents
have no incentive to lie about their true values (incentive compatible) and for
which no agent will seek to exchange outcomes with another (envy-free).
Mechanisms satisfying each requirement separately have been studied
extensively, but there are few results on mechanisms achieving both.
We study auctions with additive valuations where agents have a limit on the
number of items they may receive. We refer to this setting as {\em capacitated
allocation games.} We seek truthful and envy free mechanisms that maximize the
social welfare. {\sl I.e.}, where agents have no incentive to lie and no agent
seeks to exchange outcomes with another. In 1983, Leonard showed that VCG with
Clarke Pivot payments (which is known to be truthful, individually rational,
and have no positive transfers), is also an envy free mechanism for the special
case of $n$ items and $n$ unit capacity agents.
We study envy-free mechanisms for scheduling tasks on unrelated machines
(agents) that approximately minimize the makespan. For indivisible tasks, we
put forward an envy-free poly-time mechanism that approximates the minimal
makespan to within a factor of $O(\log m)$, where $m$ is the number of
machines. We also show a lower bound of $\Omega(\log m / \log\log m)$. This
improves the recent result of Hartline {\sl et al.} \cite{Ahuva:2008} who give
an upper bound of $(m+1)/2$, and a lower bound of $2-1/m$.