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Is there an algorithm package that allows one to achieve the style of the algorithms in the book Combinatorial Optimization by Korte and Vygen? I am interested specifically in the numbering of the steps by encircled numbers.

3

This is view of the original algorithm:

Edmond's branching algorithm

Aside from duplicating the algorithm (with heading and content), since your were particularly interested in the numbering, the following does something similar using PGF/TikZ and algorithmicx. Moreover, I've used the following two answers to obtain the desired result:

\documentclass{article}
\usepackage{algpseudocode}
\usepackage{tikz}
% https://tex.stackexchange.com/questions/7032/good-way-to-make-textcircled-numbers/7045#7045
\newcommand*\circled[1]{\tikz[baseline=(char.base)]{%
            \node[shape=circle,draw,inner sep=1pt] (char) {#1};}}
\begin{document}

% https://tex.stackexchange.com/questions/25019/how-to-place-a-program-and-a-text-side-by-side/25026#25026
\algrenewcommand{\alglinenumber}[1]{\scriptsize\circled{#1}}% circled line numbers
\begin{algorithmic}[1]% Taken from the algorithmicx package documentation
  \Procedure{Euclid}{$a,b$}
  \State $r\gets a\bmod b$
  \While{$r\not=0$}
    \State $a\gets b$
    \State $b\gets r$
    \State $r\gets a\bmod b$
  \EndWhile\label{euclidendwhile}
  \State \textbf{return} $b$
  \EndProcedure
\end{algorithmic}

\end{document}

Euclid's algorithm

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