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Yu Cong
2025-08-21 15:42:13 +08:00
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beamerthemePoster.sty
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\ProvidesPackage{beamerthemePoster}
\RequirePackage{tikz}
\usetikzlibrary{arrows,decorations.pathmorphing,backgrounds,calc}
\RequirePackage[dvipsnames]{xcolor}
% \RequirePackage{tikz}
% \usetikzlibrary{arrows,decorations.pathmorphing,backgrounds,calc}
% \RequirePackage{lmodern}
% \RequirePackage{textcomp}
% \RequirePackage{amsmath,amssymb}
\RequirePackage{amsmath,amssymb,amsthm}
% \usefonttheme{professionalfonts}
\usepackage{ragged2e}
@@ -88,16 +88,40 @@
%set some beamer theme options
\setbeamertemplate{title page}[default][colsep=-4bp,rounded=true]
\setbeamertemplate{sections/subsections in toc}[square]
\setbeamertemplate{items}[circle]
\setbeamertemplate{items}[default]
\setbeamertemplate{blocks}[width=0.0]
\beamertemplatenavigationsymbolsempty
%set bibliography style
\setbeamertemplate{bibliography item}[text]
\setbeamercolor{bibliography item}{fg=color0,bg=color3}
\setbeamercolor{bibliography entry author}{fg=color0,bg=color3}
\setbeamerfont{bibliography item}{size=\small}
\setbeamerfont{bibliography entry author}{size=\small}
% \setbeamertemplate{bibliography item}[author]
% \setbeamercolor{bibliography item}{fg=color0,bg=color3}
% \setbeamercolor{bibliography entry author}{fg=color0,bg=color3}
% \setbeamerfont{bibliography item}{size=\small}
% \setbeamerfont{bibliography entry author}{size=\small}
% theorem env
\setbeamertemplate{theorem begin}{%
{
\vspace{5pt}%
\usebeamerfont*{block title}%
\selectfont%
\usebeamercolor[fg]{block title}%
\textbf{%
\inserttheoremname
% \inserttheoremnumber
\ifx \inserttheoremaddition \empty \else\ \inserttheoremaddition\fi
}
}
}
\setbeamertemplate{theorem end}{\vspace{5pt}}
% proof env
\setbeamertemplate{proof begin}{%
{\vspace{5pt}
\usebeamercolor[fg]{block title}
\textit{\textbf{Proof:}}}
}
\setbeamertemplate{proof end}{
\qedhere
\vspace{5pt}
}
%
%==============================================================================
@@ -105,9 +129,9 @@
%==============================================================================
\setbeamertemplate{headline}{
\leavevmode
\begin{columns}
% \begin{columns}
\begin{column}{\linewidth}
% \begin{column}{\linewidth}
\vskip2cm
\centering
%\usebeamercolor{title in headline}
@@ -118,14 +142,14 @@
%\usebeamercolor{institute in headline}
{\color{fg} \large{\insertinstitute}\\[1ex]}
\vskip2cm
\end{column}
% \end{column}
% \begin{column}{.2\linewidth}
% \begin{center}
% \includegraphics[width=0.55\linewidth]{image/uestc.png}
% \end{center}
% \end{column}
% \vspace{1cm}
\end{columns}
% \end{columns}
%%% additional bar under titles
%\begin{beamercolorbox}[colsep=0.5cm]{colorbar}

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poster.tex
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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]
%
\usepackage[utf8]{inputenc}
\DeclareMathOperator*{\conv}{conv}
\linespread{1.15}
%
@@ -24,7 +31,7 @@
\#2001 \; Large-Scale Trade-Off Curve Computation for Incentive Allocation with Cardinality and Matroid Constraints
}
\author{\underline{Yu Cong}, Chao Xu, Yi Zhou}
\author{\texorpdfstring{\underline{Yu Cong}}{Yu Cong}, Chao Xu, Yi Zhou}
\institute[UESTC]{University of Electronic Science and Technology of China}
\date{\today}
@@ -40,20 +47,141 @@
\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 $E_i$.
\section{Existing works}
\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.
\section{Results}
\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}