\chapter{—‰È‰Âş‰¼ “‰Â¡‰ü �¥ �ó‰Ú‰�¤ş‰µ‰İû‰‘ \label{algorithms}}
¢¤ �ş‰ß ê‰Ê‰Û “‰Â¡‰ü �¥ �ó‰Ú‰�¤ş‰µ‰İû‰‘ş‰ü  “‰Â�ı Ÿ‰Û õ‰Æ‰ÿ‰Ü‰‚ı •‰ƒ‰À� î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â
•‰ƒ‰È‰€‰ú‰‘¢ ª‰Àù �¨‰´  ¤� “‰Â¤¨‰ü ¡‰��û‰ƒ‰İ î‰Â¢. ş‰Ø‰ü �¥ �ş‰ß �ó‰Ú‰�¤ş‰µ‰İû‰‘, •‰‘¨‰¾ ¢ì‰ƒ‰Õ 
¤� “‰‚ õ‰‘ õ‰ü¢û‰À ø ¢ş‰Ú‰Âı •‰‘¨‰¿‰ü  —‰Ö‰Âş‰±‰ü ¤� ¢¤ ¥õ‰‘ö ‰µ‰Âı •‰ƒ‰À� õ‰ü‰À.
¢¤ �ş‰ß ›‰‘ û‰Â ş‰× �¥ �ş‰ß ¢ø õ‰�¤¢ ¤� “‰‚ ω�¤ ›‰À�ğ‰‘÷‰‚  ¢¤ ş‰× “‰¿‰Ç “‰Â¤¨‰ü ¡‰��û‰ƒ‰İ î‰Â¢.
�ó‰Ú‰�¤ş‰µ‰İû‰‘ı õ‰Î‰Â� ª‰Àù ¢¤ �ş‰ß ê‰Ê‰Û, û‰Ş‰Ú‰ü “‰Â�ı •‰ƒ‰À� î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â
“‰Â ¤øı ¨‰Î‰¼ ş‰× �‰€‰Àø›‰ú‰ü ¢�¢ù ª‰Àù û‰Æ‰µ‰€‰À. õ‰Æ‰ÿ‰Ü‰‚ı �¬‰Ü‰ü �ş‰ß ¢¤ �ş‰ß •‰Âø¦ù,
•‰ƒ‰À� î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¤øı ş‰× ¨‰Î‰¼ ÷‰‘õ‰€‰Ñ‰İ õ‰·‰Ü‰¶“‰€‰Àı ª‰Àù �¨‰´  ş‰× Ÿ‰‘ó‰´
¡‰‘« �¥ �‰€‰Àø›‰ú‰ü �¨‰´. ¢¤ �ş‰ß ê‰Ê‰Û, �ó‰Ú‰�¤ş‰µ‰İû‰‘ ¤� ¢¤ û‰Ş‰ƒ‰ß Ÿ‰‘ó‰´ î‰Ü‰ü õ‰Î‰Â�
¡‰��û‰ƒ‰İ î‰Â¢; øó‰ü ¢¤ ê‰Ê‰Û “‰ã‰À, “‰Â�ı •‰ƒ‰‘¢ù¨‰‘¥ı �ó‰Ú‰�¤ş‰µ‰İ ¢ì‰ƒ‰Õ õ‰Î‰Â� ª‰Àù
¢¤ �ş‰ß ê‰Ê‰Û, “‰‚ û‰Ş‰‘ö õ‰Æ‰ÿ‰Ü‰‚ı �¬‰Ü‰ü, ş‰ã‰€‰ü •‰ƒ‰À� î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â “‰Â ¤øı ş‰× 
¨‰Î‰¼ ÷‰‘õ‰€‰Ñ‰İ õ‰·‰Ü‰¶“‰€‰Àı ª‰Àù “‰‘¥ğ‰È‰´ ¡‰��û‰ƒ‰İ î‰Â¢.

% ¢¤ �ş‰ß›‰‘ ì‰Æ‰Ş‰´û‰‘ş‰ü Ÿ‰Áé ª‰Àù �¨‰´.

\section{�ó‰Ú‰�¤ş‰µ‰İ ¢ì‰ƒ‰Õ \label{alg}}
¢¤ �ş‰ß “‰¿‰Ç “‰‚ —‰�®‰ƒ‰¼ �ó‰Ú‰�¤ş‰µ‰İ �¤��‰‚ ª‰Àù ¢¤ \cite{paper}  “‰ú‰µ‰Âş‰ß �ó‰Ú‰�¤ş‰µ‰İ
ª‰€‰‘¡‰µ‰‚ ª‰Àù “‰Â�ı Ÿ‰Û õ‰Æ‰ÿ‰Ü‰‚ �¨‰´ ¡‰��û‰ƒ‰İ •‰Â¢�¡‰´. “‰Â�ı �ş‰ß õ‰€‰Ñ‰�¤, �ş‰ß �ó‰Ú‰�¤ş‰µ‰İ ¤�
¢¤ �‰€‰À õ‰ÂŸ‰Ü‰‚ “‰Â¤¨‰ü ¡‰��û‰ƒ‰İ î‰Â¢. ÷‰¿‰Æ‰´ “‰Â�ı ¨‰‘¢ù—‰Â ª‰Àö “‰½‰¶, �ó‰Ú‰�¤ş‰µ‰İ ¤� ¢¤ Ÿ‰‘󉵉ü
 �‰€‰Àø›‰ú‰ü õ‰�¤¢ ÷‰Ñ‰Â õ‰½‰À’ “‰‘ª‰À, ¢¤ �‰€‰À õ‰ÂŸ‰Ü‰‚ —‰È‰Âş‰¼ ¡‰��û‰ƒ‰İ î‰Â¢ ø •‰Å �¥ �ö 
�ó‰Ú‰�¤ş‰µ‰İ ¤� “‰‚ Ÿ‰‘ó‰´ î‰Ü‰ü )艃‰Â õ‰½‰À’( —‰ã‰Ş‰ƒ‰İ õ‰ü¢û‰ƒ‰İ.

\subsection{�ó‰Ú‰�¤ş‰µ‰İ �ø󉃉‚ ¢¤ Ÿ‰‘ó‰´ õ‰½‰À’ }
¤øª‰ü  ¢¤ �è‰Ü‰° �ó‰Ú‰�¤ş‰µ‰İû‰‘ı î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â “‰‚ ¤  ¤ê‰µ‰‚ �¨‰´, ¤ø© 
{\em'“‰‘¥î‰Â¢ö"}\footnote{\InE{}Unfolding\EnE{}} �¨‰´. 
�ş‰ß ¤ø© “‰‚ �ş‰ß ¬‰�¤– �¨‰´: ê‰Â­ ‰ƒ‰À  ¢÷‰±‰‘ó‰‚ı 
\InE{}$f_1, e_1, f_2, e_2, \ldots, f_i, e_i, f_{i+1}, \ldots, e_{m-1}, f_m$\EnE{}
¢÷‰±‰‘ó‰‚�ı �¥ ø›‰�ù ø ş‰‘ñû‰‘ı �‰€‰Àø›‰ú‰ü “‰‘ª‰À, “‰‚ ω�¤ı \InE{}$f_1$\EnE{} ø›‰ú‰ü “‰‘ª‰À  ª‰‘õ‰Û
÷‰Ö‰Î‰‚ı õ‰±‰À\hamze � �¨‰´ ø ş‰‘ñ \InE{}$e_i$\EnE{}  ê‰Ê‰Û õ‰È‰µ‰Âí ø›‰�ù \InE{}$f_i$\EnE{} ø \InE{}$f_{i+1}$\EnE{} “‰‘ª‰À. 
¤øş‰‚ı {\em “‰‘¥î‰Â¢ö } �ş‰ß ¢÷‰±‰‘ó‰‚, “‰‚ ¬‰�¤– ¥ş‰Â �¨‰´:

\begin{enumerate}
\item \InE{}$F$\EnE{} ¤� õ‰Æ‰‘øı “‰‘ \InE{}$\{f_1\}$\EnE{} ì‰Â�¤ “‰Àù.
\item “‰Â�ı û‰Â \InE{}$i$\EnE{} �¥ 1 —‰‘ \InE{}$m-1$\EnE{} �ş‰ß ¤ ¤� �÷‰¹‰‘ô “‰Àù:
\begin{description}
\item[2-1(] \InE{}$F$\EnE{} ¤� Ÿ‰�ñ \InE{}$e_i$\EnE{} “‰‚ �÷‰À�¥ù�ı ¢ø¤�ö “‰Àù  \InE{}$F$\EnE{} ø \InE{}$f_{i+1}$\EnE{} ¢¤ ş‰× 
            ¬‰Ô‰½‰‚ ì‰Â�¤  ğ‰ƒ‰Â÷‰À ø ¢¤ ¢ø ωÂé õ‰¿‰µ‰Ü‰Ó \InE{}$e_i$\EnE{}  ÷‰ƒ‰Ã “‰‘ª‰€‰À.
\item[2-2(] \InE{}$F$\EnE{} ¤� õ‰Æ‰‘øı “‰‘ \InE{}$F \cup \{e_i, f_{i+1}\}$\EnE{} ì‰Â�¤ “‰Àù.
\end{description}
\end{enumerate}

•‰Å �¥ •‰‘ş‰‘ö ş‰‘ꉵ‰ß �ó‰Ú‰�¤ş‰µ‰İ ê‰�ë, û‰Ş‰‚ı ø›‰�ù ¢÷‰±‰‘ó‰‚ı õ‰�¤¢÷‰Ñ‰Â ¤øı ş‰× ¬‰Ô‰½‰‚ı õ‰È‰µ‰Âí
ì‰Â�¤ ğ‰Âꉵ‰‚�÷‰À. ì‰Ì‰ƒ‰‚ı ¥ş‰Â  ¢¤ \cite{ref23} “‰‚ �™‰±‰‘– ¤¨‰ƒ‰Àù �¨‰´, �û‰Ş‰ƒ‰´ ¤øş‰‚ı
“‰‘¥î‰Â¢ö ¤� õ‰È‰¿‰É õ‰ü‰À:

\begin{theorem}
\label{convex}
û‰Â î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Âı  �¥ ¢÷‰±‰‘ó‰‚ı ø›‰�ù \InE{}$f_1, f_2, \ldots, f_m$\EnE{} 
䉱‰�¤ ‰À, •‰Å �¥ �÷‰¹‰‘ô ¤øş‰‚ı {\em “‰‘¥î‰Â¢ö} “‰‚ ¬‰�¤– ş‰× •‰‘¤ù¡‰Í õ‰Æ‰µ‰Ö‰ƒ‰İ ¢¤ õ‰ü�ş‰À.
\end{theorem}

% ¢¤ �ş‰ß›‰‘ ì‰Æ‰Ş‰´û‰‘ş‰ü Ÿ‰Áé ª‰Àù �¨‰´.

“‰‘ —‰�›‰‚ “‰‚ ÷‰Ş‰‘¢ğ‰Á�¤ıû‰‘ı ê‰�ë �ó‰Ú‰�¤ş‰µ‰İ ¤� õ‰ü—‰��ö “‰‚ ¬‰�¤– ¥ş‰Â ÷‰�ª‰´: )\InE{}$S$\EnE{} ¤\hamze �§ 
õ‰±‰À\hamze � î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â �¨‰´.(

\begin{algorithm}
\label{alg1}
\begin{enumerate}
\item ¤ş‰È‰‚ı ¢¤¡‰´ )\InE{}root\EnE{}( ¤� õ‰Æ‰‘øı “‰‘ \InE{}$S$\EnE{} ì‰Â�¤ “‰Àù.
\item “‰Â�ı û‰Â ş‰‘ñ \InE{}$e$\EnE{}  õ‰Ö‰‘“‰Û \InE{}$S$\EnE{} ì‰Â�¤ ğ‰Âꉵ‰‚ �¨‰´ 
      )\InE{}$e$\EnE{} ş‰‘ó‰ü �¥ ø›‰ú‰ü �¨‰´  \InE{}$S$\EnE{} ¤øı �ö ì‰Â�¤ ğ‰Âꉵ‰‚ �¨‰´( 
      �ş‰ß ¤ ¤� �÷‰¹‰‘ô “‰Àù:
\begin{description}
\item[2-1( ] \InE{}$(e,S,e)$\EnE{} ¤� “‰‚ 䉀‰��ö ê‰Â¥÷‰À ¤ş‰È‰‚ ø�¤¢ ¢¤¡‰´ î‰ß.
\end{description}
\item “‰Â�ı \InE{}$i$\EnE{} �¥ 1 —‰‘ \InE{}$n$\EnE{} )—‰ã‰À�¢ ø›‰‚û‰‘( �ş‰ß ¤ ¤� �÷‰¹‰‘ô “‰Àù:
\begin{description}
\item[3-1( ] “‰Â�ı û‰Â “‰Âï \InE{}$(e, I, Proj_e^I)$\EnE{} ¢¤ ¨‰Î‰¼ \InE{}$i$\EnE{} �ô ¢¤¡‰´ ¢÷‰±‰‘ó‰‚ 
           �ş‰ß ¤ ¤� �÷‰¹‰‘ô “‰Àù:
\begin{description}
\item[3-1-1( ] \InE{}$I$\EnE{} ¤� Ÿ‰�ñ \InE{}$e$\EnE{} ¢ø¤�ö “‰Àù —‰‘ ÷‰Ö‰Î‰‚ı \InE{}$\overline{I}$\EnE{} “‰‚ ¢¨‰´ �ş‰À. 
      )—‰‘ \InE{}$\overline{I}$\EnE{} “‰‘ õ‰·‰Ü‰¶ \InE{}$ABC$\EnE{}, ø›‰‚ ¨‰‘ş‰‚ı \InE{}$e$\EnE{}, û‰İ¬‰Ô‰½‰‚ ª‰�¢.(
\item[3-1-2( ] “‰Â�ı û‰Â ş‰× �¥ ¢ø ş‰‘ñ \InE{}$e':= AB, CA$\EnE{} �ş‰ß ¤ ¤� �÷‰¹‰‘ô “‰Àù:
\begin{description}
\item[3-1-2-1( ] \InE{}$Proj_{e'}^{\overline{I}}$\EnE{} ¤� õ‰½‰‘¨‰±‰‚ î‰ß.
\item[3-1-2-2( ] �ğ‰Â \InE{}$Proj_{e'}^{\overline{I}}$\EnE{} ÷‰‘—‰ú‰ü “‰‘ª‰À, ğ‰Âùı 
               \InE{}$(e', \overline{I}, Proj_{e'}^{\overline{I}})$\EnE{} ¤� “‰‚ 䉀‰��ö
               ê‰Â¥÷‰À ğ‰Âùı \InE{}$(e, I, Proj_e^I)$\EnE{} ¢¤ ¢¤¡‰´ ø�¤¢ î‰ß.
\end{description}
\end{description}
\end{description}
\end{enumerate}
\end{algorithm}

% ¢¤ �ş‰ß›‰‘ ì‰Æ‰Ş‰´û‰‘ş‰ü Ÿ‰Áé ª‰Àù �¨‰´.

\subsection{�ó‰Ú‰�¤ş‰µ‰Ş‰ü “‰‘ ¥õ‰‘ö �›‰Â�ı �‰€‰À›‰Ş‰Ü‰‚�ı \label{poly}}
¢¤ �ş‰ß “‰¿‰Ç, ¨‰ã‰ü ¡‰��û‰ƒ‰İ î‰Â¢ “‰‘ —‰ç‰ƒ‰ƒ‰Â �ó‰Ú‰�¤ş‰µ‰İ \ref{alg1}, �ó‰Ú‰�¤ş‰µ‰Ş‰ü “‰‘ ¥õ‰‘ö �›‰Â�ı �‰€‰À›‰Ş‰Ü‰‚�ı
“‰Â�ı õ‰Æ‰ÿ‰Ü‰‚ı •‰ƒ‰À� î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ωÂ�Ÿ‰ü ‰ƒ‰İ. “‰Â�ı �ş‰ß õ‰€‰Ñ‰�¤, ì‰Ì‰ƒ‰‚�ı ¤�
�™‰±‰‘– ¡‰��û‰ƒ‰İ î‰Â¢  —‰ã‰À�¢ ğ‰Âùû‰‘ı ¢¤¡‰´ ¢÷‰±‰‘ó‰‚ ¤� õ‰½‰Àø¢ õ‰ü‰À. �ş‰ß ì‰Ì‰ƒ‰‚ ™‰‘“‰´ õ‰ü‰À
 �¥ “‰ƒ‰ß ğ‰Âùû‰‘ş‰ü  ¨‰‘ş‰‚ı �öû‰‘ ş‰× ¥�øş‰‚ ¤� õ‰ü•‰�ª‰‘÷‰À, Ÿ‰À�‰Â ş‰Ø‰ü �¥ �öû‰‘ õ‰ü—‰��÷‰À
¢ø ê‰Â¥÷‰À ¢�ª‰µ‰‚ “‰‘ª‰À, ø “‰Â�ı ¢ş‰Ú‰Â ğ‰Âùû‰‘ş‰ü  �ş‰ß ¥�øş‰‚ ¤� õ‰ü•‰�ª‰‘÷‰€‰À, ê‰ü �¨‰´
ş‰× ê‰Â¥÷‰À ì‰Â�¤ ¢û‰ƒ‰İ. 

% ¢¤ �ş‰ß›‰‘ ì‰Æ‰Ş‰´û‰‘ş‰ü Ÿ‰Áé ª‰Àù �¨‰´.

“‰€‰‘“‰Â�ş‰ß �ó‰Ú‰�¤ş‰µ‰İ ¤� õ‰ü—‰��ö “‰‚ ¬‰�¤– ¥ş‰Â ÷‰�ª‰´:

\begin{algorithm}
\label{n2alg}
\begin{enumerate}
\item ¤ş‰È‰‚ı ¢¤¡‰´ )\InE{}root\EnE{}( ¤� õ‰Æ‰‘øı “‰‘ \InE{}$S$\EnE{} ì‰Â�¤ “‰Àù.
\item “‰Â�ı û‰Â ş‰‘ñ \InE{}$e$\EnE{}  õ‰Ö‰‘“‰Û \InE{}$S$\EnE{} ì‰Â�¤ ğ‰Âꉵ‰‚ �¨‰´ 
      )\InE{}$e$\EnE{} ş‰‘ó‰ü �¥ ø›‰ú‰ü �¨‰´  \InE{}$S$\EnE{} ¤øı �ö ì‰Â�¤ ğ‰Âꉵ‰‚ �¨‰´( 
      �ş‰ß ¤ ¤� �÷‰¹‰‘ô “‰Àù:
      \begin{description}
      \item[2-1( ] \InE{}$(e,S,e)$\EnE{} ¤� “‰‚ 䉀‰��ö ê‰Â¥÷‰À ¤ş‰È‰‚ ø�¤¢ ¢¤¡‰´ î‰ß.
      \end{description}

% ¢¤ �ş‰ß›‰‘ ì‰Æ‰Ş‰´û‰‘ş‰ü Ÿ‰Áé ª‰Àù �¨‰´.

\end{enumerate}
\end{algorithm}

\subsubsection{—‰½‰Ü‰ƒ‰Û ¥õ‰‘ö �›‰Â�ı �ó‰Ú‰�¤ş‰µ‰İ}
û‰Ş‰‘ö ω�¤  쉱‰\nasb … ğ‰Ô‰µ‰ƒ‰İ, �ó‰Ú‰�¤ş‰µ‰İ \ref{n2alg} ¢�¤�ı ¥õ‰‘ö �›‰Â�ş‰ü �¥ \InE{}$O(n^2)$\EnE{}
�¨‰´. ¢¤ �ş‰ß›‰‘ ¨‰ã‰ü ¢�¤ş‰İ �ş‰ß õ‰�®‰�á ¤� �™‰±‰‘– ‰ƒ‰İ.
\begin{theorem}
\label{n2proof}
¥õ‰‘ö �›‰Â�ı �ó‰Ú‰�¤ş‰µ‰İ \ref{n2alg} �¥ \InE{}$O(n^2)$\EnE{} �¨‰´.
\end{theorem}
\begin{proof}
¡‰‘¬‰ƒ‰µ‰ü  ¢¤ ì‰Ì‰ƒ‰‚ı 쉱‰Û “‰‚ �™‰±‰‘– ¤¨‰ƒ‰À ø “‰Â �¨‰‘§ �ö, ¢¤ �ó‰Ú‰�¤ş‰µ‰İ
\ref{n2alg}, “‰Â�ı û‰Â ¥�øş‰‚, Ÿ‰À�‰Â ş‰Ø‰ü �¥ ğ‰Âùû‰‘ı ¢¤¡‰´ ¢�¤�ı ¢ø ê‰Â¥÷‰À 
õ‰üª‰À, —‰Ì‰Ş‰ƒ‰ß õ‰ü‰À  •‰Å �¥ û‰Â “‰‘¤ �›‰Â�ı Ÿ‰Ü‰Ö‰‚ı �¬‰Ü‰ü �ó‰Ú‰�¤ş‰µ‰İ \ref{n2alg}
)Ÿ‰Ü‰Ö‰‚ı ¡‰Í 3( —‰ã‰À�¢ “‰Âïû‰‘ı ¢¤¡‰´ �¥ \InE{}$O(n)$\EnE{} �¨‰´. )—‰�›‰‚ ‰ƒ‰À  —‰ã‰À�¢ î‰Û
¥�øş‰‚û‰‘ �¥ \InE{}$O(n)$\EnE{} �¨‰´.( “‰Â�ı õ‰½‰‘¨‰±‰‚ı ¥õ‰‘ö �›‰Â�ı �ó‰Ú‰�¤ş‰µ‰İ, �“‰µ‰À� ¥õ‰‘ö ¬‰Âé
ª‰Àù “‰Â�ı ¨‰‘¡‰µ‰ß ¢¤¡‰´ )ø ÷‰‚ Ÿ‰Áé ¥ş‰Â¢¤¡‰´û‰‘( ¤� ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰À. “‰‘ —‰�›‰‚ “‰‚ �ş‰ß
 —‰ã‰À�¢ “‰Âïû‰‘ �¥ \InE{}$O(n)$\EnE{} �¨‰´, û‰Â “‰‘¤ �›‰Â�ı Ÿ‰Ü‰Ö‰‚ı ¡‰Í 3 “‰‚ �÷‰À�¥ùı \InE{}$O(n)$\EnE{}
ø쉴 “‰Â�ı —‰�󉃉À ¢¤¡‰´ õ‰üğ‰ƒ‰Â¢. “‰€‰‘“‰Â�ş‰ß ¥õ‰‘÷‰ü  ¬‰Âé �ş‰¹‰‘¢ ¢¤¡‰´ õ‰üª‰�¢ ¢¤
î‰Û �ó‰Ú‰�¤ş‰µ‰İ “‰Â�“‰Â “‰‘  \InE{}$O(n^2)$\EnE{} �¨‰´.

Ÿ‰‘„ ¥õ‰‘ö õ‰Ê‰Âé ª‰Àù “‰Â�ı Ÿ‰Áé ¥ş‰Â¢¤¡‰´û‰‘ ¤� õ‰½‰‘¨‰±‰‚ õ‰ü‰ƒ‰İ. “‰Â�ı õ‰½‰‘¨‰±‰‚ı �ş‰ß
õ‰Ö‰À�¤ �¥ ş‰× —‰Ø‰€‰ƒ‰× ›‰‘ó‰° �¨‰µ‰Ô‰‘¢ù õ‰ü‰ƒ‰İ: ê‰Â­ ‰ƒ‰À  “‰Â�ı û‰Â ş‰× �¥ ğ‰Âùû‰‘ı
¢¤¡‰´, ¢¤ û‰Ş‰‘ö ¥õ‰‘ö �ş‰¹‰‘¢ î‰Â¢öª‰‘ö, “‰‚ �÷‰À�¥ùı ş‰× ø�Ÿ‰À ¥õ‰‘÷‰ü ø쉴 “‰Â�ı
Ÿ‰Áé �öû‰‘ �¥ •‰ƒ‰Ç ÷‰Ú‰‚ õ‰ü¢�¤ş‰İ. �ğ‰Â �ş‰ß ğ‰Âù “‰ã‰À\nasb � Ÿ‰Áé ª‰À, �¥ �ş‰ß ø�Ÿ‰À
¥õ‰‘÷‰ü “‰Â�ı Ÿ‰Áé �ö �¨‰µ‰Ô‰‘¢ù õ‰ü‰ƒ‰İ ø �ğ‰Â Ÿ‰Áé ÷‰È‰À, �ş‰ß ø�Ÿ‰À ¥õ‰‘÷‰ü õ‰Ê‰Âé 
÷‰È‰Àù õ‰üõ‰‘÷‰À.  �ş‰ß �¨‰µ‰À„ñ ÷‰È‰‘ö õ‰ü¢û‰À  ¥õ‰‘ö ¬‰Âé ª‰Àù “‰Â�ı Ÿ‰Áé ¥ş‰Â¢¤¡‰´û‰‘
¢¤ ¢¤¡‰´ ¢÷‰±‰‘ó‰‚ ÷‰ƒ‰Ã �¥ \InE{}$O(n^2)$\EnE{} �¨‰´. “‰€‰‘“‰Â�ş‰ß •‰ƒ‰»‰ƒ‰Àğ‰ü ¥õ‰‘÷‰ü �ó‰Ú‰�¤ş‰µ‰İ
\ref{n2alg} �¥ \InE{}$O(n^2)$\EnE{} �¨‰´.
\end{proof}

\subsection{—‰ã‰Ş‰ƒ‰İ “‰‚ Ÿ‰‘ó‰´ è‰ƒ‰Â õ‰½‰À’ \label{nonconvexsec}}
�ó‰Ú‰�¤ş‰µ‰Ş‰ü  ¢¤ “‰¿‰Ç 쉱‰Û “‰Â�ı •‰ƒ‰À� î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¢¤ Ÿ‰‘ó‰´
õ‰½‰À’ ¢ş‰Àş‰İ, õ‰\nasb … ø�“‰Æ‰µ‰‚ “‰‚ ì‰Ì‰ƒ‰‚ı \ref{convex} “‰�¢. ¢¤ Ÿ‰‘󉵉ü  
�‰€‰Àø›‰ú‰ü õ‰�¤¢ ÷‰Ñ‰Â õ‰‘, ş‰× �‰€‰Àø›‰ú‰ü 艃‰Âõ‰½‰À’ “‰‘ª‰À, �ş‰ß ì‰Ì‰ƒ‰‚ ¢ş‰Ú‰Â ¢¤¨‰´ ÷‰ƒ‰Æ‰´.
¢¤ �ş‰ß Ÿ‰‘ó‰´, î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â “‰ƒ‰ß ¢ø ¤\hamze �§ �¥ �‰€‰Àø›‰ú‰ü õ‰Ş‰Ø‰ß �¨‰´ �¥ “‰ã‰Ì‰ü �¥
¤�‰�§ 䉱‰�¤ ‰À, øó‰ü ¢¤ ꉑ¬‰Ü‰‚ı “‰ƒ‰ß �ş‰ß ¤�‰�§, ¢�¤�ı ¡‰‘¬‰ƒ‰´ £î‰Â ª‰Àù ¢¤ ì‰Ì‰ƒ‰‚ı
\ref{convex} �¨‰´. “‰‚ 䉱‰‘¤– ¢ş‰Ú‰Â ¢¤ Ÿ‰‘ó‰´ è‰ƒ‰Âõ‰½‰À’ ì‰Ì‰ƒ‰‚ı ¥ş‰Â “‰Âì‰Â�¤ �¨‰´.

% ¢¤ �ş‰ß›‰‘ ì‰Æ‰Ş‰´û‰‘ş‰ü Ÿ‰Áé ª‰Àù �¨‰´.

\section{�ó‰Ú‰�¤ş‰µ‰İ —‰Ö‰Âş‰±‰ü \label{approxalgsec}}
¢¤ �ş‰ß “‰¿‰Ç, “‰‚ —‰È‰Âş‰¼ ş‰× �ó‰Ú‰�¤ş‰µ‰İ —‰Ö‰Âş‰±‰ü “‰Â�ı Ÿ‰Û õ‰Æ‰ÿ‰Ü‰‚ı î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â “‰Â 
¤øı ¨‰Î‰¼ ş‰× �‰€‰Àø›‰ú‰ü õ‰½‰À’  —‰�¨‰Í \InE{}J.~Hershberger\EnE{} ø \InE{}S.~Suri\EnE{} ¢¤ \cite{approx} 
�¤��‰‚ ª‰Àù �¨‰´ ¡‰��û‰ƒ‰İ •‰Â¢�¡‰´.

% ¢¤ �ş‰ß›‰‘ ì‰Æ‰Ş‰´û‰‘ş‰ü Ÿ‰Áé ª‰Àù �¨‰´.