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inputs/lectures/lec_01.tex
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inputs/lectures/lec_01.tex
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\section{Introduction}
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\begin{question}
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Which problems can be modelled as flow problems?
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\end{question}
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\begin{example}
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One interesting application are \emph{Traffic Flows}.
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\begin{itemize}
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\item For example, a central authority might coordinate how people evacuate in case
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of a nearing flood.
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This is an example of a \vocab{flow over time}
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\item In a typical real-life situation, however, there is no such central authority
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that dictates peoples' routes.
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Thus, the setting resembles much rather a \emph{game theoretical problem},
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since each participant will individually decide which route to take,
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potentially leading to solutions that are not globally optimal.
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This can lead to unexpected outcomes:
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\end{itemize}
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\end{example}
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\begin{example}[Breass's Paradox]
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Consider the following traffic network, where we want to route a total flow of $1$:
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\missingfigure{tikz}
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Here, an edge of cost $x$ will have cost equal to the amount of flow routed along it.
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The optimum solution (in terms of maximum travel time)
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is of course to route flow $\frac{1}{2}$ along each of the paths.
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It can be shown that this is also the solution that people will actually choose,
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since it is always better to use the currently less-congested path,
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leading to an equal spread.
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However, now considering adding an additional
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\todo{finish remark}.
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\end{example}
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\section{Max Flows over time}
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\begin{definition}[Flow over time]
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Consider a graph $G = (V,E)$ with \vocab{transit times} $\set{ τ_e } _{e\in E}$
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and capacities $\set{ u_e } _{e\in E}$.
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A \vocab{flow over time} $f$ time horizon $T$ is a
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Lebesgue integrable function $f_e\colon [0, T) \to \mathbb{R}_{\geq 0}$
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for $e\in E$ such that $f_{e}(\theta) = 0$ for $\theta \geq T - τ_e$.
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\end{definition}
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\begin{oral}
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The assumption on the Lebesgue integrability is just here for technical reasons,
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we will not deal much with it.
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In practice, most functions that we will encounter are piecewise constant.
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\end{oral}
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\begin{remark}+
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We can interpret this definition as edges signaling pipes,
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and the flow $f_e$ denoting the inflow rate to edge $e$ over a certain time.
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Then, it takes $τ_e$ time for the flow to traverse the pipes.
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Thus, $f_e(\theta - τ_e)$ denotes the excess flow of edge $e$ at time $\theta$.
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\end{remark}
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\setcounter{toplevel}{2}
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\begin{definition}
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Let $f$ be a flow over time with horizon $T$.
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\begin{enumerate}[a)]
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\item The \vocab{capacity constraint} is that $f_e(\theta) \leq u_e$ for all $e,\theta$.
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\item The \vocab{excess for a vertex} $v\in V$ at $\theta$ is defined as
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\[
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\ex_f(v, \theta) \coloneq
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\sum_{e\in δ^-(v)} \int_0^{\zeta - τ_e} f_e(\zeta) d \zeta
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-
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\sum_{e\in δ^+(v)} \int_0^{\zeta} f_e(\zeta) d \zeta
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\]
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\item We say that $f$ has \vocab{weak flow conservation} if
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\begin{IEEEeqnarray*}{rCLl}
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\ex_f(v, \theta) &\geq& 0 & \forall v \in V \setminus \set{ s } \forall \theta \in [0,T)
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\\
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\ex_f(v, T) = 0 & \forall v \in V \setminus \set{ s,t }
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\end{IEEEeqnarray*}
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\item We say that $f$ has \vocab{strict flow conservation} if
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\[
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\ex_f(v, \theta) = 0 \forall v\in V \\ \set{ s , t} \forall \theta \in T
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\]
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\item The \vocab{value of f} is $\abs{f} \coloneqq \ex_f(t,T)$.
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\item A flow is called \vocab{feasible} if it satisfies the capacity constraints
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and has weak flow conservation.
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\end{enumerate}
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\end{definition}
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Consider a graph $G = (V,E)$ with capacities $c\colon E \to \mathbb{R}_{\geq 0}$,
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transit times $τ\colon E\to \mathbb{R}_{\geq 0}$,
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a time horizon $T$ and source/sink vertices $s$, $t$.
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The \textsc{Maximum Flow over Time Problem} asks for a feasible $s$-$t$-flow
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over time with time horizon $T$ and maximum value of $\abs{f}$.
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\begin{definition}[Temporally repeated flow with time horizon $T$]
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Let $x$ be a static\footnotemark $s$-$t$ (feasible) flow and consider a flow decomposition
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$(X_p)_{p \in \mathcal{P} \cup \mathcal{C}}$
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with $\abs{\mathcal{P} \cup \mathcal{C}} \leq \abs{E}$.
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Now, define a flow over time by
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\[
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f_e(\theta) \coloneqq \sum_{P \in P_e(\theta) } x_p \qquad
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\forall e = (v,w) \in E \; \theta \in [0, T)
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,
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\]
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where
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\[
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P_e(\theta) \coloneqq \set{
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p\in \mathcal{P} \suchthat
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p \in \mathcal{P},
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τ(P_{s,v)} \leq \theta,
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τ(P_{v, t}) \leq T - \theta
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}
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.
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\]
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Here, the transit times along a fixed path $P$ are defined canonically as
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\[
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τ(P_{v,w}) \coloneqq \sum_{e\in P_{v,w}} τ_e
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.
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\]
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The resulting flow is called the \vocab{temporally repeated flow with time horizon $T$}
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associated to the static flow $x$.
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\end{definition}
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\footnotetext{Here, \enquote{static} means just \enquote{not temporal}}
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\begin{observation}
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\label{obs:temporally-repeated-flow-construction}
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The Temporally repeated flow can be obtained as follows:
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For each $P \in \mathcal{P}$, we send $x_p$ into $P$
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from $s$ during $[0, T - τ(P))$ and then pass the flow along $P$.
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Note that the temporally repeated flow even fulfills strict flow conservation.
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\end{observation}
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\setcounter{toplevel}{6}
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\begin{lemma}
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Let $x$ be a static, feasible $s$-$t$ flow with
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flow decomposition $(x_P)_{P \in \mathcal{P} \cup \mathcal{C}}$
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and $x_P = 0$ for all $P \in \mathcal{P}$ with $τ(P) > T$
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and for all cycles.
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Denote by $f$ the corresponding temporally repeated flow $f$.
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Then,
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\[
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\abs{f} = T \cdot \abs{x} - \sum_{e\in E} τ_e \cdot x_e
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.
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\]
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\end{lemma}
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\begin{proof}
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Motivated by \autoref{obs:temporally-repeated-flow-construction},
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we deduce:
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\begin{IEEEeqnarray*}{rCl}
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\abs{f}
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&=& \sum_{P\in \mathcal{P}} (T - τ(P)) \cdot x_P \\
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&=& T \cdot \sum_{P\in \mathcal{P}} x_P - \sum_{P\in \mathcal{P}} τ(P) \cdot x_P \\
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&=& T \cdot \abs{x} - \sum_{e\in E} τ_e \underbrace{\sum_{\substack{P\in \mathcal{P} \\ e \in P} } x_p}_{= x_e}
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\end{IEEEeqnarray*}
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\end{proof}
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\begin{corollary}
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Let $x$ be a static, feasible $s$-$t$ flow and $(x_P)_{P\in \mathcal{P} \cup \mathcal{C}}$
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a flow decomposition.
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Then,
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\[
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\abs{f} \geq T \cdot \abs{x} - \sum_{e\in E} τ_e x_e
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.
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\]
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\end{corollary}
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\begin{proof}
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Define a new flow by
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\[
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\tilde{x}_P \coloneqq \begin{cases}
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x_P & \text{if $P\in \mathcal{P}$ and $τ(P) \leq T$} \\
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0 & \text{otherwise}
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\end{cases}
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\]
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for all $P \in \mathcal{P}\cup \mathcal{C}$.
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Then, the associated temporally repeated flows satisfy
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\begin{IEEEeqnarray*}{rCl}
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\abs{f} &=& \abs{\tilde{f}} \\
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&=& T \abs{\tilde{x}} - \sum_{e \in E} τ_e \tilde{x}_e \\
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&\geq & T \abs{x} - \sum_{e\in E} τ_e x_e
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\end{IEEEeqnarray*}
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Note that to check the last inequality,
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it suffices to see that cycles only contribute to the negative sum
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and do not change the flow value of $\tilde{x}$ compared to $x$.
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For paths $P$ with $τ(P) > T$, these contribute to $\abs{x} $
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but not to $\tilde{x}$, however, the effect cancels out by summing
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the $τ_e$ along this path in the negative summand.
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\end{proof}
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\begin{algorithm}[H]
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\SetKwInput{KwInput}{Input}
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\SetKwInput{KwOutput}{Output}
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\SetKw{KwGoTo}{go to}
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\SetKwProg{Fn}{Def}{:}{}
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\DontPrintSemicolon
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\caption{Ford-Furkerson Algorithm for maximum flows over time}
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\KwInput{$G = (V, E)$ with capacities $c$, transit times $τ$, time horizon $T$ and source/sink vertices $s$, $t$.}
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\KwOutput{A feasible temporally repeated flow}
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\;
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Compute a feasible static $s$-$t$-flow $x$ maximizing
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\[
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T \cdot \abs{x} - \sum_{e \in E} τ_e x_e
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\]
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\;
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Compute a flow decomposition $(x_P)_{P\in \mathcal{P} \cup \mathcal{C}}$.
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\;
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Output corresponding temporally repeated flow.
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\end{algorithm}
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\begin{remark}
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To implement step 1, consider the auxiliary graph $G'$ obtained from $G$
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by adding an edge $(t,s)$ with infinite capacity.
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For each edge, set its cost to $τ_e$.
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Now, for each static $s$-$t$ flow $x$ in $G$ gives rise to a circulation
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in $G'$ by sending $\abs{x}$ along $(t,s)$, whose cost will be precisely
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\[
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\sum_{e\in E} x_e \cdot τ_e + \abs{x} \cdot (-T)
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.
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\]
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Thus, solving the minimum cost flow problem in $G'$ yields a desired flow $f$.
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\end{remark}
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