Fix some typos and inconsistent notations

1 files changed, 3 insertions, 3 deletions

diff --git a/main.tex b/main.tex
index e112470..d8635db 100644
--- a/main.tex
+++ b/main.tex
@@ -1,7 +1,7 @@
 \label{sec:main}
 
 In this section we present our mechanism for \SEDP. Prior approaches to budget
-feasible mechanisms for sudmodular maximization build upon the full information
+feasible mechanisms for submodular maximization build upon the full information
 case, which we discuss first.
 
 \subsection{Submodular Maximization in the Full Information Case}
@@ -67,7 +67,7 @@ $\mathcal{N}$ if $L(\id_S) = V(S)$ for all $S\subseteq\mathcal{N}$, where $\id_S
 denotes the indicator vector of $S$. The optimization program
 \eqref{eq:non-strategic} extends naturally to such relaxations:
 \begin{align}
-    OPT' = \argmax_{\lambda\in[0, 1]^{n}}
+    OPT' \defeq \max_{\lambda\in[0, 1]^{n}}
     \left\{L(\lambda) \Big| \sum_{i=1}^{n} \lambda_i c_i
     \leq B\right\}\label{relax}
 \end{align}
@@ -102,7 +102,7 @@ We show this by establishing that $L$ is within a constant factor from the so-ca
     \begin{algorithmic}[1]
     \State $\mathcal{N} \gets \mathcal{N}\setminus\{i\in\mathcal{N} : c_i > B\}$
     \State $i^* \gets \argmax_{j\in\mathcal{N}}V(j)$
-    \State $OPT'_{-i^*} \gets \argmax_{\lambda\in[0,1]^{n}} \{L(\lambda)
+    \State $OPT'_{-i^*} \gets \max_{\lambda\in[0,1]^{n}} \{L(\lambda)
                                     \mid \lambda_{i^*}=0,\sum_{i \in \mathcal{N}\setminus\{i^*\}}c_i\lambda_i\leq B\}$
         \Statex
         \If{$OPT'_{-i^*} < C \cdot V(i^*)$} \label{c}