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new poster img & fix typo in slides
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Yu Cong
2025-08-27 15:40:41 +08:00
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slides.tex
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@@ -47,21 +47,21 @@ A ride sharing company wants to send riders promotional coupons in the hope of m
\begin{frame}{Multiple-choice knapsack}
\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 $v_e\in \Z_+$. A positive budget $b\in \Z_+$.
\textbf{Output}: A subset of coupons $K$ that maximizes the total value $\sum_{e\in K} c_e$ while satisfying \textcolor{Red}{$|K\cap K_i|\leq 1$} and $\sum_{e\in K} c_e\leq b$.
\textbf{Output}: A subset of coupons $K$ that maximizes the total value $\sum_{e\in K} v_e$ while satisfying \textcolor{Red}{$|K\cap K_i|\leq 1$} and $\sum_{e\in K} c_e\leq b$.
\vspace{1em}
\pause
Three problems with this modeling:
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 applications.
\item The multiple-choice constraint \textcolor{Red}{$|K\cap K_i|\leq 1$} is too weak for real-world applications.
\end{enumerate}
\end{frame}
\begin{frame}{Linear programming formulation}
\begin{frame}{Linear programming relaxation}
\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 $b\in \Z_+$.}
}