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\title
[34th International Joint Conference on Artificial Intelligence (IJCAI25), Guangzhou, China] % Conference
{ % Poster title
\#2001 \; Large-Scale Trade-Off Curve Computation for Incentive Allocation with Cardinality and Matroid Constraints
}

\author{\texorpdfstring{\underline{Yu Cong}}{Yu Cong}, Chao Xu, Yi Zhou}
\institute[UESTC]{University of Electronic Science and Technology of China}

\date{\today}



\begin{document}
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\section{Problem}
We consider the incentive allocation problem with additional constraints.

\textbf{Input}: A set of coupons $E=\bigcupdot_i E_i$, where each coupon $e\in E$ has a value and a cost $v_e,c_e\in \mathbb{Z}_+$. Budget $B\in \mathbb{Z}_+$. Constraints $\mathcal F_i$ on each subset $E_i$.

\textcolor{Gray}{
\textbf{Output}: A subset $X\subset E$ of coupons that maximizes the total value $\sum_{e\in X}v_e$ while satisfying the budget constraint $\sum_{e\in X}c_e\leq B$ and additional constraints $X\cap E_i\in \mathcal F_i$.
}

This problem is NP-hard. Consider its LP relaxation.
\begin{equation*}\label{LP}
\begin{aligned}
\tau(B)=\max_x&\; & v\cdot x& & & \\
s.t.&\;  & c \cdot x &\leq B & &\\
& & x_{E_i}&\in \conv(\mathcal{F}_i)  & &\;\forall i\in [n]\\
& & x&\in [0,1]^m & &
\end{aligned}
\end{equation*}
\textbf{Output}: The entire curve $\tau(B)$ for $B\in [0,\infty)$.

We consider 3 cases of additional constraints $x_{E_i}\in \mathcal{F}_i$ :
\begin{enumerate}
\item Multiple-choice. $\sum\limits_{e\in E_i}x_e\leq 1$;
\item Cardinality. $\sum\limits_{e\in E_i}x_e\leq p$;
\item Matroid. $x_{E_i}\in \text{independence polytope of a matroid}$.
\end{enumerate}

\section{Existing works \& Comparison}

\begin{table}[!htb]
\centering
\small
    \begin{tabular}{cccc}
         Constraint Type & Result & Fixed budget & Trade-off curve \\
         \bottomrule
         \hline
         \multirow{3}{*}{Multiple Choice}& \cite{Dyer84,ZEMEL1984123}& $O(m)$ & -  \\
         &\cite{10.1109/ITSC55140.2022.9922143} & - & $O(m\log m)$ \\
         & \textcolor{OrangeRed}{this paper} & - & $O(m\log m)$ \\
         \hline
         \multirow{4}{*}{Cardinality}& \cite{DavidPisinger} & $O(m\log VC)$  & -\\
           & \cite{DavidPisinger} & $O(mp+nB)$  & -  \\
          & \cite{minimaxoptimization} & $O(m\log m)$  & -  \\
          & \textcolor{OrangeRed}{this paper} & - & $O((k+m)\log m)$ \\
         \hline
         \multirow{3}{*}{Matroid}& \cite{CAMERINI1984157} & $O(m^2 + T \log m)$ & -\\
        & \cite{minimaxoptimization} & $O(T \log m)$ & - \\
        & \textcolor{OrangeRed}{this paper} & - & $O(Tk+k\log m)$\\
        \bottomrule
    \end{tabular}
\caption{Comparison of algorithms for incentive allocation: $m$ is the total number of incentives, $M$ is the maximum number of incentives over each agent, $p$ is the max rank of the matroid constraint over each agent, or the limit in the cardinality constraint. $V$ and $C$ is the maximum value and cost of the incentives, respectively. $B$ is the budget. $k=O(mp^{1/3})$ is the number of breakpoints of the trade-off curve. $T$ is the time complexity of matroid optimum base algorithm.}
\label{runtimetable}
\end{table}

\section{Methods}
The idea is to take advantage of the independence among the constraints $\mathcal{F}_i$ and reduce the optimization problem to one in computational geometry.

\textcolor{DarkOrchid}{\textit{Signature Function.}} Let $f_i(\lambda) = \max\{(v_{E_i}-\lambda c_{E_i}) x | x\in \conv(\mathcal F_i) \}$ be the signature function of agent $i$. The signature function is piecewise-linar and convex.

\textcolor{DarkOrchid}{\textit{Lagrangian Dual.}} The Lagrangian dual of LP\autoref{LP} is therefore
\begin{equation*}
\label{eq:Lagrangiandual}
\begin{aligned}
\min_{\lambda} \left( B\lambda+\sum_i f_i(\lambda)\right).
\end{aligned}
\end{equation*}

\begin{theorem}[4]\large
$\tau(B)$ is piecewise-linear and concave.
\end{theorem}

Computing $\tau(B)$ is straightforward if $f_i(\lambda)$ is known.

\subsection{Finding $f_i(\lambda)$}
\textcolor{DarkOrchid}{\textit{Cardinality constraint.}}
For fixed $\lambda$, computing $f_i(\lambda) = \max\{(v_{E_i}-\lambda c_{E_i})x \mid \mathbf{1}\cdot x \leq p\}$ is the same as finding the $p$ largest coupons with respect to the weights $v_e - \lambda c_e$. If $\lambda$ is not fixed, this is computing the \emph{$k$-level} of univariate linear functions.
\begin{figure}[htb]
    \begin{minipage}[c]{0.6\linewidth} % Minipage for the image
        \centering
        \includegraphics[width=\linewidth]{image/klevel_black.pdf} % Replace with your image
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    \begin{minipage}[c]{0.39\linewidth} % Minipage for the caption
        \caption{The bold line forms a $2$-level in the line arrangement.}
        \label{fig:klevel}
    \end{minipage}
\end{figure}

\textcolor{DarkOrchid}{\textit{Matroid constraint.}}
For fixed $\lambda$ under matroid constraints, computing the signature function is equivalent to finding the optimum-weight base in a matroid.
However, the matroid generalization of $k$-level problem is significantly harder. We use Eisner-Severance method to compute the curve.
\section{Computational results}

\begin{table}[!ht]
\small
    \centering
    \begin{tabular}{ccccccccc}
        \toprule
         \multirow{2}*{$m$} & \multicolumn{2}{c}{$p=20$} & \multicolumn{2}{c}{$p=40$} &  \multicolumn{2}{c}{$p=2000$} & \multicolumn{2}{c}{$p=m/5$}\\
         \cmidrule(lr){2-3}  \cmidrule(lr){4-5} \cmidrule(lr){6-7} \cmidrule(lr){8-9}
         & scan & opt & scan & opt & scan & opt & scan & opt\\
         \midrule
        $1\times 10^3$            & 0.000 & 0.000 & 0.000 & 0.001 & - & - & 0.003& 0.002 \\
        $5\times 10^3$    & 0.003 & 0.005 & 0.006 & 0.005 & 0.137 & 0.027 & 0.091& 0.02\\
        $1\times 10^4$            & 0.008 & 0.010 & 0.014 & 0.012 & 0.384 & 0.048 & 0.384 & 0.048\\
        $5\times 10^4$    & 0.043 & 0.089 & 0.080 & 0.087 & 2.634 & 0.187 & 9.531& 0.326\\
        $1\times 10^5$            & 0.094 & 0.216 & 0.173 & 0.223 & 5.795 & 0.397 & 38.275& 1.222\\
        $5\times 10^5$    & 0.528 & 2.911 & 0.937 & 2.952 & 33.760 & 3.398 & TLE & 10.500 \\
        $1\times 10^6$            & 1.147 & 7.291 & 1.989 & 7.140 & 72.485 & 7.604 & TLE & 23.203\\
        $1\times 10^7$            & 12.994 & 100.512 & 23.863 & 101.675 & TLE & 101.775 & TLE & 133.974\\
        
    %     \bottomrule
    % \end{tabular}
    % \begin{tabular}{ccccc}
    %     % \toprule
    %      \multirow{2}*{$m$} & \multicolumn{2}{c}{$p=2000$} & \multicolumn{2}{c}{$p=m/5$}\\
    %      \cmidrule(lr){2-3}  \cmidrule(lr){4-5} 
    %      & scan & opt & scan & opt \\
    %      \midrule
    %     $1\times 10^3$      & - & - & 0.003& 0.002 \\
    %     $5\times 10^3$      & 0.137 & 0.027 & 0.091& 0.02\\
    %     $1\times 10^4$      & 0.384 & 0.048 & 0.384 & 0.048\\
    %     $5\times 10^4$      & 2.634 & 0.187 & 9.531& 0.326\\
    %     $1\times 10^5$      & 5.795 & 0.397 & 38.275& 1.222\\
    %     $5\times 10^5$      & 33.760 & 3.398 & TLE & 10.500 \\
    %     $1\times 10^6$      & 72.485 & 7.604 & TLE & 23.203\\
    %     $1\times 10^7$      & TLE & 101.775 & TLE & 133.974\\
        
        \bottomrule
    \end{tabular}
    \caption{The time (in seconds) to compute the breakpoints on the signature function under cardinality constraint using the optimum $p$-level algorithm (opt) and the scan line algorithm (scan).}
    \label{tab:klevel}
\end{table}

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