From 6914bc40cecb0cf3bacbe8bf44f8fb1bfb5d690b Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Thu, 22 Nov 2012 17:14:24 +0100
Subject: Typos. There was an error in the proof of lemma 4. Fixing this error
 changes our approximation ratio to 12.98 instead of 19.68...

---
 related.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

(limited to 'related.tex')

diff --git a/related.tex b/related.tex
index 24ba359..16b2309 100644
--- a/related.tex
+++ b/related.tex
@@ -5,7 +5,7 @@ Budget feasible mechanism design was originally proposed by Singer \cite{singer-
  Singer shows that there exists a randomized, 112-approximation mechanism for submodular maximization that is \emph{universally truthful} (\emph{i.e.}, it is a randomized mechanism sampled from a distribution over truthful mechanisms). Chen \emph{et al.}~\cite{chen}  improve this result by providing a 7.91-approximate mechanism, and show a corresponding lower bound of $2$ among universally truthful mechanisms for submodular maximization.
 
 \sloppy
-In contrast to the above results, no truthful, constant approximation mechanism that runs in polynomial time is presently known for submodular maximization.  However, assuming access to an oracle providing the optimum in the full-information setup, Chen \emph{et al.},~provide a truthful, $8.34$-approximate mechanism; in cases for which the full information problem is NP-hard, as the one we consider here, this mechanism is not poly-time, unless P=NP.  Chen \emph{et al.}~also prove a $1+\sqrt{2}$ lower bound for truthful mechanisms, improving upon an earlier bound of 2 by Singer \cite{singer-mechanisms}. 
+In contrast to the above results, no deterministic, truthful, constant approximation mechanism that runs in polynomial time is presently known for submodular maximization.  However, assuming access to an oracle providing the optimum in the full-information setup, Chen \emph{et al.},~provide a truthful, $8.34$-approximate mechanism; in cases for which the full information problem is NP-hard, as the one we consider here, this mechanism is not poly-time, unless P=NP.  Chen \emph{et al.}~also prove a $1+\sqrt{2}$ lower bound for truthful mechanisms, improving upon an earlier bound of 2 by Singer \cite{singer-mechanisms}. 
 
 \fussy
 Improved bounds, as well as deterministic polynomial mechanisms, are known for specific submodular objectives. For symmetric submodular functions, a truthful mechanism with approximation ratio 2 is known, and this ratio is tight \cite{singer-mechanisms}. Singer also provides a 7.32-approximate truthful mechanism for the budget feasible version of \textsc{Matching}, and a corresponding lower bound of 2 \cite{singer-mechanisms}. Improving an earlier result by Singer,  Chen \emph{et al.}~\cite{chen}, give a truthful, $2+\sqrt{2}$-approximate mechanism for \textsc{Knapsack}, and a lower bound of $1+\sqrt{2}$. Finally, a truthful, 31-approximate  mechanism is also known for the budgeted version of \textsc{Coverage} \cite{singer-mechanisms,singer-influence}.
-- 
cgit v1.2.3-70-g09d2

