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\title[Incentive allocation]{Large-Scale Trade-Off Curve Computation for Incentive Allocation with Cardinality and Matroid Constraints}
\date{August 30, 2025}
\author{\underline{Yu Cong}, Chao Xu, Yi Zhou}
\institute[UESTC]{University of Electronic Science and Technology of China}
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    \scriptsize 34th International Joint Conference on Artificial Intelligence (IJCAI25)
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% introduce the problem. 3 things: trade-off curve, approximation, general constraints

\begin{frame}{Incentive allocation with constraints}
A ride sharing company wants to send riders promotional coupons in the hope of more rides.

%  each agent gets at most 1 coupon.
\begin{figure}
\includegraphics[width=.7\textwidth]{image/chatgpt.png}

\scriptsize Image courtesy: ChatGPT-5
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\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 (multi)set 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 $\sum_{e\in K} c_e\leq b$.

\vspace{1em}
\pause

Three problems with this modeling:
\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.
\end{enumerate}

\end{frame}

\begin{frame}{Linear programming formulation}
\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 $b\in \Z_+$.}
}
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\begin{equation*}
\begin{aligned}
\tau(b)= \max_x&    &   v\cdot x&   &   &\\
s.t.&   &   c\cdot x&\leq b     &   &\\
&   &       \mathcolor{Plum}{x_{K_i}}&\mathcolor{Plum}{\in P_{K_i}} &   &\forall i\in [n]\\
\end{aligned}
\end{equation*}

\textbf{Output}: A compact representation of $\tau(b)$.
\pause

We focus on 2 kinds of constraints of \textcolor{Plum}{$x_{K_i}\in P_{K_i}$}.
\begin{enumerate}
\item Cardinality. \textcolor{Plum}{$x_{K_i}\in P_{K_i}$}→ $\sum_{e\in K_i}x_e\leq p$.
\item Matroid. \textcolor{Plum}{$x_{K_i}\in P_{K_i}$}→ $x_{K_i}$ is in the base polytope of a matroid $M_i$.
\end{enumerate} 
\end{frame}

\begin{frame}{Results}
We compute the curve $\tau(b)$ fast.
\begin{theorem}
Consider an incentive allocation problem with a total of $m$ incentives. 
The trade-off curve is a piecewise linear concave function with $k$ breakpoints.
\begin{itemize}
\item Cardinality constraint.
$k=O(mp^{1/3})$ and $\tau$ can be computed in $O((k+m)\log m)$ time.
\item Matroid constraint. $k=O(mr^{1/3})$ and $\tau$ can be computed in $O(Tk+k\log m)$ time.
\end{itemize} 
\end{theorem}
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\centering
\Large \textit{Let's discuss this in detail at my poster!}

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