From 24fdab84d29f2d2fb0bd78a4c7ad7131100afe94 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Mon, 16 Dec 2013 17:23:26 -0500
Subject: More space saving

---
 problem.tex | 5 ++---
 1 file changed, 2 insertions(+), 3 deletions(-)

(limited to 'problem.tex')

diff --git a/problem.tex b/problem.tex
index 46e72c5..6033e86 100755
--- a/problem.tex
+++ b/problem.tex
@@ -31,8 +31,7 @@ $\sigma^2$ is the noise variance).  Then, \E\ estimates $\beta$ through
 \emph{maximum a posteriori estimation}: \emph{i.e.}, finding the parameter
 which maximizes the posterior distribution of $\beta$ given the observations
 $y_S$. Under the linearity assumption and the Gaussian prior on $\beta$,
-maximum a posteriori estimation leads to the following maximization
-\cite{hastie}: 
+maximum a posteriori estimation leads to \cite{hastie}: 
 \begin{align}
     \begin{split}
         \hat{\beta} = \textstyle\argmax_{\beta\in\reals^d} \prob(\beta\mid y_S)
@@ -159,7 +158,7 @@ We relax the notion of truthfulness to \emph{$\delta$-truthfulness}, requiring t
 In addition, we would like the allocation $S(c)$ to be of maximal value; however, $\delta$-truthfulness, as well as the hardness of \EDP{}, preclude achieving this goal. Hence, we seek mechanisms with that  are \emph{$(\alpha,\beta)$-approximate}, \emph{i.e.}, there exist $\alpha\geq 1$ and $\beta>0$ s.t.~$OPT \leq \alpha V(S(c))+\beta$, and are \emph{computationally efficient}, in that $S$ and $p$ can be computed in polynomial time.
 
 We note that  the constant approximation algorithm \eqref{eq:max-algorithm} breaks
-truthfulness. Though this is not true for all submodular functions (see, \emph{e.g.}, \cite{singer-mechanisms}), it is true for the objective of \EDP{}: we show this in \cite{arxiv}. This motivates our study of more complex mechanisms.
+truthfulness. Though this is not true for all submodular functions (see, \emph{e.g.}, \cite{singer-mechanisms}), it is true for the objective of \EDP{}: we show this in \cite{arxiv}, motivating our study of more complex mechanisms.
 
 \begin{comment}
 As noted in \cite{singer-mechanisms, chen}, budget feasible reverse auctions are  \emph{single parameter} auctions: each agent has only one
-- 
cgit v1.2.3-70-g09d2

