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Yu Cong
2025-08-29 20:55:07 +08:00
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@@ -56,15 +56,15 @@ Three problems with this formulation:
\begin{enumerate} \begin{enumerate}
\item Finding the exact optimum is NP-hard. So we consider solving it approximately. \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 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{enumerate}
\end{frame} \end{frame}
\begin{frame}{Linear programming relaxation} \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 $b\in \Z_+$.} \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 \pause
\begin{equation*} \begin{equation*}
\begin{aligned} \begin{aligned}
@@ -74,7 +74,7 @@ s.t.& & c\cdot x&\leq b & &\\
\end{aligned} \end{aligned}
\end{equation*} \end{equation*}
\textbf{Output}: A compact representation of $\tau(b)$. \textbf{Output}: function $\tau(b)$ for $b\in (0,+\infty)$.
\pause \pause
We focus on 2 kinds of constraints of \textcolor{Plum}{$x_{K_i}\in P_{K_i}$}. We focus on 2 kinds of constraints of \textcolor{Plum}{$x_{K_i}\in P_{K_i}$}.