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    \makebox[\textwidth][l]{\textbf{MATH 6404: Applied [Combinatorics and] Graph Theory} \hfill CU Denver} \\
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    \makebox[\textwidth][l]{Spring 2026 \hfill Instructor: Carlos Mart\'inez}
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\scribebox{16}{Maximum Flow}{March 16, 2026}{Baihan Liu}


\onehalfspacing
In today's lecture, we will start working toward the ``max flow - min cut theorem", which is a min-max relation with many applications and corollaries in combinatorics.
\section{Flows}



\begin{definition}[Flow]
Let $G=(V,E)$ be a directed multigraph, with source $s \in V$ and target $t \in V$. A flow is an assignment
\(
f_e \in \mathbb{R}_{\ge 0} \text{ to each } e \in E  \text{ }  (f:E \rightarrow \mathbb{R}_{\ge 0})
\)
such that
\[
\sum_{e \in \delta^{-}(u)} f_e
=
\sum_{e \in \delta^{+}(u)} f_e,
\qquad \forall\, u \in V,\; u \neq s,t.
\]
\end{definition}

That is, ``mass conservation" holds at every vertex other than the source and sink.

The value of a flow $f$ is denoted by $|f|$ and given by
\[
|f|
=
\sum_{e \in \delta^{+}(s)} f_e - \sum_{e \in \delta^{-}(s)} f_e.
\]

\subsection{Linear Algebra Formulation of Flows}
You can also think of flows through linear algebra (as a vector). 


Let $B \in \mathbb{R}^{V \times E}$ be the incidence matrix of $G$, where
\[
B_{we}
=
\begin{cases}
1, & \text{if } e=(u,v) \text{ and } v=w,\\
-1, & \text{if } e=(u,v) \text{ and } u=w,\\
0, & \text{otherwise}.
\end{cases}
\]

For example, consider the following directed graph:

\begin{figure}[h!]
\centering
\begin{tikzpicture}[
    >=Stealth,
    scale=1.2,
    every node/.style={font=\small},
    vertex/.style={
        draw,
        circle,
        minimum size=9mm,
        inner sep=0pt,
        line width=1.2pt,
        fill=skyblue!20
    },
    edge/.style={
        ->,
        line width=1.05pt
    }
]
    % vertices
    \node[vertex] (v1) at (0,1.8) {$1$};
    \node[vertex] (v2) at (3,1.8) {$2$};
    \node[vertex] (v3) at (0,0) {$3$};
    \node[vertex] (v4) at (3,0) {$4$};

    % edges
    \draw[edge] (v1) -- node[above=2pt] {$b$} (v2);
    \draw[edge] (v1) -- node[left=2pt] {$a$} (v3);
    \draw[edge] (v2) -- node[pos=0.48, sloped, below=2pt] {$c$} (v3);

    \draw[edge] (v4) to[bend right=18] node[right=2pt] {$d$} (v2);
    \draw[edge] (v2) to[bend right=18] node[left=2pt] {$e$} (v4);
\end{tikzpicture}
\caption{A directed graph with edge set \(E=\{a,b,c,d,e\}\).}
\end{figure}


Its incidence matrix is
\[
B=
\begin{array}{c}
\\[-1.5ex]
1\\
2\\
3\\
4
\end{array}
\left[
\begin{array}{ccccc}
a & b & c & d & e\\
-1 & -1 & 0 & 0 & 0\\
0 & 1 & -1 & 1 & -1\\
1 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & -1 & 1
\end{array}
\right].
\]

For $v \in V$, let $\mathbbm{1}_v \in \{0,1\}^{V}$ be its indicator vector. That is,
\[
(\mathbbm{1}_v)_w
=
\begin{cases}
1, & \text{if } w=v,\\
0, & \text{otherwise}.
\end{cases}
\]

Then, $f \in \mathbb{R}_{\ge 0}^{E}$ is a flow if and only if
\[
Bf=\lambda(\mathbbm{1}_t-\mathbbm{1}_s)
\qquad \text{for some } \lambda \in \mathbb{R}.
\]

In this case,
\[
|f|=\lambda.
\]

This is equivalent to ``mass conservation". The vector \(Bf\) represents the net inflow at each vertex. In particular,
\[
(Bf)_u = 0 \qquad \forall\, u\in V\setminus\{s,t\},
\]
and
\[
(Bf)_s = -\lambda, \qquad (Bf)_t = \lambda.
\]


\subsection{Flow Decomposition}

Flows can be decomposed as a weighted sum of \(s\)-\(t\) paths, \(t\)-\(s\) paths and directed cycles.

For \(S \subseteq E\), let \(\mathbbm{1}_S \in \{0,1\}^E\) be the indicator vector of \(S\), i.e.,
\[
(\mathbbm{1}_S)_e =
\begin{cases}
1, & \text{if } e \in S,\\
0, & \text{otherwise.}
\end{cases}
\]

\begin{lemma}\label{first lemma}
A vector \(f \in \mathbb{R}_{\ge 0}^E\) is an \(s\)-\(t\) flow if and only if
\[
f = \sum_{i=1}^k \lambda_i \mathbbm{1}_{S_i},
\]
where \(S_1,S_2,\dots,S_k \subseteq E\) is a collection of \(s\)-\(t\) paths, \(t\)-\(s\) paths, and directed cycles.
\end{lemma}

\begin{proof}
If \(S\) is an \(s\)-\(t\) path, a \(t\)-\(s\) path, or a directed cycle, then
\[
B\mathbbm{1}_S = \lambda(\mathbbm{1}_t-\mathbbm{1}_s),
\]
where
\[
\lambda =
\begin{cases}
0, & \text{if } S \text{ is a directed cycle},\\
1, & \text{if } S \text{ is an } s\text{-}t \text{ path},\\
-1, & \text{if } S \text{ is a } t\text{-}s \text{ path}.
\end{cases}
\]
Hence, the weighted sum
\[
\sum_{i=1}^k \lambda_i \mathbbm{1}_{S_i}
\]
preserves the identity and, therefore, defines a flow $f$.

Conversely, if \(f\) is a flow, and let
\[
E_+(f)=\{e\in E : f_e>0\}.
\]
We can show that \(E_+(f)\) contains at least one directed cycle, \(s\)-\(t\) path, or \(t\)-\(s\) path, by mass conservation. 


For any such \(S \subseteq E_+(f)\), let
\[
\omega=\min_{e\in S} f_e,
\]
and 
\[
g = f-\omega \cdot \mathbbm{1}_S.
\]
Then \(g\) is again a flow, and moreover
\[
E_+(g)\subsetneq E_+(f).
\] So the statement holds inductively. 
Indeed, since \(\omega=\min_{e\in S} f_e\), there exists at least one edge \(e_0\in S\) such that
\[
g_{e_0}=f_{e_0}-\omega=0,
\]
so at least one positive edge is removed from the support  \(E_+(f)\). Therefore the statement follows by induction on \(|E_+(f)|\).
\end{proof}

\medskip

\section{Maximum Flows}

\subsection{Capacitated Directed Multigraphs and Valid Flows}

Now, consider \(G=(V,E,c)\) be a capacitated directed multigraph, where
\[
c:E\to\mathbb{R}_{\ge 0}
\]
is a capacity function.

A \emph{valid flow} with source \(s\) and sink \(t\) is a flow \(f\) such that
\[
f_e \le c(e) \qquad \forall e\in E.
\]
That is, the capacity constraints are satisfied on every edge.

A \emph{maximum flow} is a valid flow of maximum value.
\medskip


\subsection{Residual Graphs and Augmenting Paths}
\noindent\textbf{Question:} How do we find a maximum flow?

\smallskip

\noindent\textbf{Idea:} Similar to maximum matching, we find augmenting paths.

\medskip

Let \(G=(V,E,c)\) be a capacitated directed multigraph, and let \(f\) be a valid flow.

Let 
\(
G_f=(V_f,E_f,c_f),
\)
be the \emph{residual graph} 
where


\begin{itemize}
    \item \(V_f = V\).
    
    \item \(E_f = E \cup \overline{E}\), where
    \(
    \overline{E}
    =
    \left\{
    \bar e=(v,u)\;:\; e=(u,v)\in E
    \right\}.
    \)
    
    \item The residual capacity function \(c_f\) is given by
    \[
    c_f(e)=c(e)-f_e
    \qquad \text{for } e\in E,
    \]
    and
    \[
    c_f(\bar e)=f_e
    \qquad \text{for } \bar e\in \overline{E}.
    \]
\end{itemize}

An \emph{augmenting path} is a valid \(s\)-\(t\) path in \(G_f\). 
Valid means that no arcs with zero capacity are used. 
The flow value can be increased by augmenting along an $s$-$t$ path in the residual graph.
\begin{example}
\end{example}

\begin{figure}[H]
\centering

\begin{subfigure}[t]{0.48\textwidth}
\centering
\[
\begin{tikzpicture}[
    >=Stealth,
    scale=1.1,
    every node/.style={font=\small},
    vertex/.style={
        draw,
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        fill=skyblue!20
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    edge/.style={
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]
    \node[vertex] (s) at (0,2) {$s$};
    \node[vertex] (u) at (3.4,2) {};
    \node[vertex] (v) at (0,0) {};
    \node[vertex] (t) at (3.4,0) {$t$};

    \node at (1.75,2.9) {\textcolor{blue!70!black}{flow}/\textcolor{vermillion}{capacity}};

    \draw[edge] (s) --
        node[above=2.5pt] {$\textcolor{blue!70!black}{1}/\textcolor{vermillion}{2}$}
        (u);
    \draw[edge] (s) -- node[left=2pt] {$\textcolor{blue!70!black}{1}/\textcolor{vermillion}{1}$} (v);
    \draw[edge] (v) -- node[below=2.5pt] {$\textcolor{blue!70!black}{0}/\textcolor{vermillion}{1}$} (t);
    \draw[edge] (u) -- node[right=2pt] {$\textcolor{blue!70!black}{2}/\textcolor{vermillion}{2}$} (t);
    \draw[edge] (v) -- node[pos=0.58, below right=1pt] {$\textcolor{blue!70!black}{1}/\textcolor{vermillion}{1}$} (u);
\end{tikzpicture}
\]
\caption{A valid flow in \(G\).}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\textwidth}
\centering
\[
\begin{tikzpicture}[
    >=Stealth,
    scale=1.1,
    every node/.style={font=\small},
    vertex/.style={
        draw,
        circle,
        minimum size=8mm,
        inner sep=0pt,
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        fill=skyblue!20
    },
    edge/.style={
        ->,
        line width=1.05pt
    },
    rlabel/.style={text=vermillion}
]

    \node at (1.75,2.9) {\textcolor{vermillion}{capacity}};
    \node[vertex] (s) at (0,2) {$s$};
    \node[vertex] (u) at (3.5,2) {};
    \node[vertex] (v) at (0,0) {};
    \node[vertex] (t) at (3.5,0) {$t$};

    %\node[reddishpurple] at (1.75,2.8) {$G_f$};

    \draw[edge] (s) to[bend left=8] node[rlabel, above] {$1$} (u);
    \draw[edge] (u) to[bend left=8] node[rlabel, below] {$1$} (s);

    \draw[edge] (s) to[bend left=8] node[rlabel, right] {$0$} (v);
    \draw[edge] (v) to[bend left=8] node[rlabel, left] {$1$} (s);

    \draw[edge] (v) to[bend left=8] node[rlabel, above] {$1$} (t);
    \draw[edge] (t) to[bend left=8] node[rlabel, below] {$0$} (v);

    \draw[edge] (u) to[bend left=8] node[rlabel, right] {$0$} (t);
    \draw[edge] (t) to[bend left=8] node[rlabel, left] {$2$} (u);

    \draw[edge] (v) to[bend left=8] node[rlabel, pos=0.4, above] {$0$} (u);
    \draw[edge] (u) to[bend left=8] node[rlabel, pos=0.4, below] {$1$} (v);
\end{tikzpicture}
\]
\caption{Residual graph \(G_f\).}
\end{subfigure}

\vspace{0.8em}

\begin{subfigure}[t]{0.48\textwidth}
\centering
\[
\begin{tikzpicture}[
    >=Stealth,
    scale=1.1,
    every node/.style={font=\small},
    vertex/.style={
        draw,
        circle,
        minimum size=8mm,
        inner sep=0pt,
        line width=1.2pt,
        fill=skyblue!20
    },
    edge/.style={
        ->,
        line width=1.05pt
    },
    wavepath/.style={
        draw=bluishgreen,
        line width=1.5pt,
        decorate,
        decoration={snake,amplitude=0.8mm,segment length=3.8mm}
    },
    rlabel/.style={text=vermillion}
]
    \node[vertex] (s) at (0,2) {$s$};
    \node[vertex] (u) at (3.5,2) {};
    \node[vertex] (v) at (0,0) {};
    \node[vertex] (t) at (3.5,0) {$t$};

    %\node[reddishpurple] at (1.75,2.8) {$G_f$};

    \draw[edge] (s) to[bend left=8] node[above,rlabel] {$1$} (u);
    \draw[edge] (u) to[bend left=8] node[below,rlabel] {$1$} (s);

    \draw[edge] (s) to[bend left=8] node[right,rlabel] {$0$} (v);
    \draw[edge] (v) to[bend left=8] node[left,rlabel] {$1$} (s);

    \draw[edge] (v) to[bend left=8] node[above,rlabel] {$1$} (t);
    \draw[edge] (t) to[bend left=8] node[below,rlabel] {$0$} (v);

    \draw[edge] (u) to[bend left=8] node[right,rlabel] {$0$} (t);
    \draw[edge] (t) to[bend left=8] node[left,rlabel] {$2$} (u);

    \draw[edge] (v) to[bend left=8] node[pos=0.4, above,rlabel] {$0$} (u);
    \draw[edge] (u) to[bend left=8] node[pos=0.4, below,rlabel] {$1$} (v);

    \draw[wavepath] ($(s)!0.12!(u)$) to[bend left=8] ($(s)!0.88!(u)$);
    \draw[wavepath] ($(u)!0.12!(v)$) to[bend left=8] ($(u)!0.88!(v)$);
    \draw[wavepath] ($(v)!0.12!(t)$) to[bend left=8] ($(v)!0.88!(t)$);
\end{tikzpicture}
\]
\caption{An augmenting path in \(G_f\).}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\textwidth}
\centering
\[
\begin{tikzpicture}[
    >=Stealth,
    scale=1.1,
    every node/.style={font=\small},
    vertex/.style={
        draw,
        circle,
        minimum size=8mm,
        inner sep=0pt,
        line width=1.2pt,
        fill=skyblue!20
    },
    edge/.style={
        ->,
        line width=1.1pt
    }
]
    \node[vertex] (s) at (0,2) {$s$};
    \node[vertex] (u) at (3.5,2) {};
    \node[vertex] (v) at (0,0) {};
    \node[vertex] (t) at (3.5,0) {$t$};

    \draw[edge] (s) -- node[above] {$\textcolor{blue!70!black}{2}/\textcolor{vermillion}{2}$} (u);
    \draw[edge] (s) -- node[left] {$\textcolor{blue!70!black}{1}/\textcolor{vermillion}{1}$} (v);
    \draw[edge] (v) -- node[below] {$\textcolor{blue!70!black}{1}/\textcolor{vermillion}{1}$} (t);
    \draw[edge] (u) -- node[right] {$\textcolor{blue!70!black}{2}/\textcolor{vermillion}{2}$} (t);
    \draw[edge] (v) -- node[pos=0.45, below right] {$\textcolor{blue!70!black}{0}/\textcolor{vermillion}{1}$} (u);
\end{tikzpicture}
\]
\caption{Updated flow in \(G\).}
\end{subfigure}

\caption{Illustration of augmenting a flow: starting from a valid flow in \(G\), constructing the residual graph \(G_f\), finding an augmenting path, and updating the flow accordingly.}
\label{fig:augmenting-flow-process}
\end{figure}






\medskip

Given a valid flow \(h\) in \(G_f\), we can define a transformation
\[
\pi(h) \in \mathbb{R}^E,
\]
where for each edge \(e \in E\),
\[
(\pi(h))_e = h_e - h_{\bar e}.
\]
Let \(\bar h\in\mathbb{R}^E\) be defined by
\(
(\bar h)_e := h_{\bar e}, \text{ } e\in E.
\)
Then, \(
\pi(h)=h-\bar h.
\)

\medskip

\noindent
Interpretation:
\begin{itemize}
\item Increase flow in \(e\) by \(h_e\) units.
\item Decrease flow in \(e\) by \(h_{\bar e}\) units.
\end{itemize}


Here \(h_e\) denotes the flow sent through the forward residual arc \(e\), and \(h_{\bar e}\) denotes the flow sent through the backward residual arc \(\bar e\). Thus, \(\pi(h)\) represents the net change of flow on each original edge \(e\in E\).


\medskip

Thus, the different choices of \(h\) (flows in the residual graph) and the corresponding \(\pi(h)\) encode \emph{all} possible ways of modifying the original flow \(f\), while remaining a valid flow.

\begin{lemma}
    If $f$ is a valid flow in $G$ and $h$ is a valid flow in $G_f$, then $f + \pi(h)$ is a valid flow in $G$. Moreover, every valid flow $\tilde{f}$ in $G$ can be written as
    \[\tilde{f} = f+\pi(h)\]

    for some valid flow $h$ in $G_f$. 
\end{lemma}

\begin{proof}
Note that $
B_{G_f}h=\lambda(\mathbbm{1}_t-\mathbbm{1}_s) \text{ for some } \lambda \in \mathbb{R}.$ 


But also, $B_{G_f}h = B_Gh - B_G\overline{h} = B_G (h-\overline{h}) = B_G \pi(h) = \lambda(\mathbbm{1}_t - \mathbbm{1}_s)$. 


Thus, $\pi (h)$ satisfies mass conservation, which implies that $f + \pi(h)$ satisfies mass conservation. In fact, $f +\pi(h)$ is a valid flow in $G$ by our design of $c_f$.


Conversely, if $\tilde{f}$ is a valid flow in $G$, let
\[h_e=\max\{\tilde f_e-f_e,0\}, \qquad e\in E,
\]
\[
h_{\bar e}=\max\{f_e-\tilde f_e,0\}, \qquad e\in E.
\]


One can verify that $h$ defined in this way is a valid flow in $G_f$ and that $\tilde{f} = f + \pi(h)$. 
\end{proof}

\begin{lemma}
    If $f$ is a valid flow in $G$, then it is a maximum flow if and only if $G_f$ does not contain a valid augmenting path.
\end{lemma}

\begin{proof}
    If $f$ is not a maximum flow, let $f^*$ be a maximum flow. Let $h$ be designed as in previous lemma so that $f^* = f + \pi(h)$, where $|h| >0$ since $f$ is not maximum. By lemma \ref{first lemma}, $h$ can be decomposed as s-t paths, t-s paths, and cycles in $G_f$. At least one part of the decomposition is an s-t path since $|h| >0$.
    Thus, $G_f$ contains a valid augmenting path. 

    Conversely, if $P$ is an augmenting path, let $\omega = \min_{e \in P} c_f(e)$ and $h = \omega \cdot \mathbbm{1}_P$, $f+\pi(h)$ is a valid flow with value $|f+\pi(h)| = |f| + \omega > |f|$.
    
\end{proof}





\end{document}