muthu changes

1 files changed, 5 insertions, 2 deletions

diff --git a/main.tex b/main.tex
index 6fdf39c..384800b 100644
--- a/main.tex
+++ b/main.tex
@@ -367,7 +367,7 @@ Hence, the total payment is bounded by the telescopic sum:
 \end{proof}
 
 Finally, we prove the approximation ratio of the mechanism. We use the
-following lemma which we prove later, which establishes that $OPT'$, the optimal value \eqref{relax} of the fractional relaxation $L$ under the budget constraints
+following lemma  which establishes that $OPT'$, the optimal value \eqref{relax} of the fractional relaxation $L$ under the budget constraints
  is not too far from $OPT$.  
 \begin{lemma}\label{lemma:relaxation}
     %\begin{displaymath}
@@ -375,9 +375,12 @@ following lemma which we prove later, which establishes that $OPT'$, the optimal
         + 2\max_{i\in\mathcal{N}}V(i)$
     %\end{displaymath}
 \end{lemma}
+The proof of Lemma~\ref{lemma:relaxation} is our main technical contribution, and can be found in Section \ref{sec:relaxation}.
+
+
+\paragraph{Finishing Proof of }
 Note that the lower bound $OPT' \geq  OPT
 $ also holds trivially, as $L$ is a fractional relaxation of $V$ over $\mathcal{N}$.
-The proof of Lemma~\ref{lemma:relaxation} is our main technical contribution, and can be found in Section \ref{sec:relaxation}.
 Using this lemma, we can complete the proof of Theorem~\ref{thm:main} by showing that, %\begin{lemma}\label{lemma:approx}
     %C_{\textrm{max}} = \max\left(1+C,\frac{3e}{e-1}\left( 1 + \frac{8e}{C
     %(e-1) -10e  +2}\right)\right)