From 19c544e04ca206d50dee86964552810dc0c179e4 Mon Sep 17 00:00:00 2001
From: Yu Cong <sxlxcsxlxc@gmail.com>
Date: Fri, 29 Aug 2025 20:55:07 +0800
Subject: [PATCH] z

---
 slides.tex | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/slides.tex b/slides.tex
index c4183b9..cf1cd65 100644
--- a/slides.tex
+++ b/slides.tex
@@ -56,15 +56,15 @@ Three problems with this formulation:
 \begin{enumerate}
 \item Finding the exact optimum is NP-hard. So we consider solving it approximately.
 \item Companies may run multiple campaigns at the same time. So a trade-off curve between budget and profit will be useful.
-\item The multiple-choice constraint \textcolor{Red}{$|K\cap K_i|\leq 1$} is too weak for real-world applications.
+\item The multiple-choice constraint \textcolor{Red}{$|K\cap K_i|\leq 1$} is too strong for real-world applications.
 \end{enumerate}
 
 \end{frame}
 
 \begin{frame}{Linear programming relaxation}
-\textcolor{gray}{
+% \textcolor{gray}{
 \textbf{Input}: $n$ sets of coupons $K_1,\dots,K_n$. Each coupon $e\in K_i$ has a non-negative cost $c_e\in \Z_+$ and value $v_e\in \Z_+$. \st{A positive budget $b\in \Z_+$.}
-}
+% }
 \pause
 \begin{equation*}
 \begin{aligned}
@@ -74,7 +74,7 @@ s.t.&   &   c\cdot x&\leq b     &   &\\
 \end{aligned}
 \end{equation*}
 
-\textbf{Output}: A compact representation of $\tau(b)$.
+\textbf{Output}: function $\tau(b)$ for $b\in (0,+\infty)$.
 \pause
 
 We focus on 2 kinds of constraints of \textcolor{Plum}{$x_{K_i}\in P_{K_i}$}.