From ff48f359b3b0bd0bd1b7252457bfd9fce71ed545 Mon Sep 17 00:00:00 2001 From: Stratis Ioannidis Date: Mon, 5 Nov 2012 10:25:58 -0800 Subject: removed erroneous statement --- problem.tex | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) (limited to 'problem.tex') diff --git a/problem.tex b/problem.tex index 527c34a..f313be0 100644 --- a/problem.tex +++ b/problem.tex @@ -146,7 +146,8 @@ Ideally, motivated by the $D$-optimality criterion, we would like to design a me We present our results with this version of the objective function because it is simple and captures the versions we will need later (including an arbitrary matrix in place of $I_d$ or a zero matrix which will correspond to ~\eqref{dcrit} ). -Note that maximizing \eqref{modified} is equivalent to maximizing \eqref{dcrit} in the full-information case. In particular, $\det(\T{X_S}X_S)> \det(\T{X_{S'}}X_{S'})$ iff $\det(I_d+\T{X_S}X_S)>\det(I_d+\T{X_{S'}}X_{S'})$. In addition, \eqref{edp}---and the equivalent problem with objective \eqref{dcrit}---are NP-hard; to see this, note that \textsc{Knapsack} reduces to EDP with dimension $d=1$ by mapping the weight of each item $w_i$ to an experiment with $x_i=w_i$. Nevertheless, \eqref{modified} is submodular, monotone and satisfies $V(\emptyset) = 0$. +%Note that maximizing \eqref{modified} is equivalent to maximizing \eqref{dcrit} in the full-information case. In particular, $\det(\T{X_S}X_S)> \det(\T{X_{S'}}X_{S'})$ iff $\det(I_d+\T{X_S}X_S)>\det(I_d+\T{X_{S'}}X_{S'})$. In addition, +\EDP{} \emph{and} the corresponding problem with objective \eqref{dcrit} are NP-hard; to see this, note that \textsc{Knapsack} reduces to EDP with dimension $d=1$ by mapping the weight of each item $w_i$ to an experiment with $x_i=w_i$. Moreover, \eqref{modified} is submodular, monotone and satisfies $V(\emptyset) = 0$. %, allowing us to use the extensive machinery for the optimization of submodular functions, as well as recent results in the % context of budget feasible auctions \cite{chen,singer-mechanisms}. @@ -162,7 +163,8 @@ For any $M>1$, there is no $M$-approximate, truthful, budget feasible, individua \begin{proof} \input{proof_of_lower_bound1} \end{proof} -This negative result motivates us to study a problem with a modified objective:}\end{comment} +This negative result motivates us to study a problem with a modified objective:} +\end{comment} \begin{comment} -- cgit v1.2.3-70-g09d2