From 5db33d6a133669cb876f1b4da3c1c1c6fedd0d19 Mon Sep 17 00:00:00 2001
From: Stratis Ioannidis <stratis@stratis-Latitude-E6320.(none)>
Date: Wed, 3 Jul 2013 20:05:28 -0700
Subject: approx

---
 appendix.tex | 7 ++-----
 1 file changed, 2 insertions(+), 5 deletions(-)

(limited to 'appendix.tex')

diff --git a/appendix.tex b/appendix.tex
index 67ae7cc..3685c2f 100644
--- a/appendix.tex
+++ b/appendix.tex
@@ -1,4 +1,3 @@
-\section{Proof of Proposition~\ref{prop:relaxation}}
 
 \subsection{Proof of Lemma~\ref{lemma:relaxation-ratio}}\label{proofofrelaxation-ratio}
 %\begin{proof}
@@ -217,9 +216,7 @@ the proof of the lemma. \qed
     attained at one of its limit, at which either the $i$-th or $j$-th component of
     $\lambda_\varepsilon$ becomes integral.
 \end{proof}
-
-\subsection*{End of the proof of Proposition~\ref{prop:relaxation}}
-
+\subsection{Proof of Proposition~\ref{prop:relaxation}}\label{proofofproprelaxation}
 Let us consider a feasible point $\lambda^*\in[0,1]^{n}$ such that
 $L(\lambda^*) = L^*_c$. By applying Lemma~\ref{lemma:relaxation-ratio} and
 Lemma~\ref{lemma:rounding} we get a feasible point $\bar{\lambda}$ with at most
@@ -242,7 +239,7 @@ one fractional component such that
     \begin{equation}\label{eq:e2}
         F(\bar{\lambda}) \leq  OPT + \max_{i\in\mathcal{N}} V(i).
     \end{equation}
-Together, \eqref{eq:e1} and \eqref{eq:e2} imply the proposition.\qedhere
+Together, \eqref{eq:e1} and \eqref{eq:e2} imply the proposition.\qed
 
 \section{Proof of Proposition~\ref{prop:monotonicity}}
 
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