\relax \citation{BL99} \@writefile{toc}{\contentsline {title}{Improving the Competitive Ratios of the Seat Reservation Problem}{1}} \@writefile{toc}{\authcount {2}} \@writefile{toc}{\contentsline {author}{Shuichi Miyazaki\unskip {} \and Kazuya Okamoto\unskip {}}{1}} \@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}} \newlabel{sec:Introduction}{{1}{1}} \citation{BL99} \citation{BM08} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces An example configuration of assignment.}}{2}} \newlabel{fig:WF}{{1}{2}} \citation{BL99} \citation{BL99} \citation{BB03} \citation{BL99} \citation{BB03} \citation{BL01} \citation{BF03} \@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Upper and lower bounds on the competitive ratios.}}{3}} \newlabel{table:bounds}{{1}{3}} \@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces New results (results obtained in this paper are highlighted in boldface).}}{3}} \newlabel{table:ouresults}{{2}{3}} \citation{BM08} \citation{BK04} \citation{KL03} \@writefile{toc}{\contentsline {section}{\numberline {2}The Unit Price Problem}{4}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.1}An Upper Bound}{4}} \newlabel{theorem:unitupper}{{1}{4}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces An example of the unit price problem.}}{5}} \newlabel{fig:UNITEX}{{2}{5}} \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Assignment configuration for $\sigma _{1}$ by algorithm $A$.}}{5}} \newlabel{fig:unit_upper_alg(1)}{{3}{5}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.2}An Upper Bound for Worst-Fit}{6}} \newlabel{theorem:wfitupper}{{2}{7}} \@writefile{toc}{\contentsline {section}{\numberline {3}The Proportional Price Problem}{7}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.1}An Upper Bound}{7}} \newlabel{theorem:propupper}{{3}{7}} \@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Assignment configuration for $\sigma _{1}$ and $\sigma _{2}$ by algorithm $A$.}}{8}} \newlabel{fig:prop_upper_alg(1)}{{4}{8}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Lower Bounds for First-Fit and Best-Fit}{9}} \newlabel{theorem:fbfitlower}{{4}{9}} \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Assignment configurations of FF and $OPT$ for $\sigma $.}}{10}} \newlabel{fig:prop_lower}{{5}{10}} \@writefile{toc}{\contentsline {section}{\numberline {4}Concluding Remarks}{11}} \bibcite{BB03}{1} \bibcite{BF03}{2} \bibcite{BK04}{3} \bibcite{BL99}{4} \bibcite{BL01}{5} \bibcite{BM08}{6} \bibcite{KL03}{7}