%--A0 beamer slide------------------------------------------------------------- \documentclass[final]{beamer} \usepackage[orientation=portrait,size=a0, scale=1.5 % font scale factor ]{beamerposter} \geometry{ hmargin=2.5cm, % little modification of margins } \usepackage{lipsum} \usepackage{MnSymbol} \usepackage{multirow} \usepackage{booktabs} \usepackage{soul} \usepackage{graphicx} \usepackage{natbib} \usepackage{bibentry} % \usepackage{hyperref}[colorlinks=true,urlcolor=Blue,citecolor=Green,linkcolor=BrickRed,unicode] \DeclareMathOperator*{\conv}{conv} \linespread{1.15} % %==The poster style============================================================ \usetheme{Poster} %==Title, date and authors of the poster======================================= \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} % larger font \large \begin{frame}[t] \begin{multicols}{2} \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 value and 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 $\sum_{e\in X}c_e\leq B$ 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 to 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 \end{minipage} \hfill % Optional: Adds horizontal space between minipages \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} \bibliographystyle{plainnat} \nobibliography{ijcai25} \end{multicols} \end{frame} \end{document}