diff --git a/beamerthemePoster.sty b/beamerthemePoster.sty index a647a10..b5ed7a0 100644 --- a/beamerthemePoster.sty +++ b/beamerthemePoster.sty @@ -6,12 +6,12 @@ \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} diff --git a/poster.tex b/poster.tex index e0ecad5..4045367 100644 --- a/poster.tex +++ b/poster.tex @@ -8,9 +8,16 @@ 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}