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Quindi includiamo l'arco $e' = (v, u)$ in $G_f$ , con una capacità di $f(e)$. Notare che $e'$ ha le stesse estremità di $e$, ma la sua direzione è \textbf{invertita}; chiameremo gli archi inclusi in questo modo \textbf{backward edges}. +\end{itemize} +\end{myblockquote} + +Si noti che ogni arco $e$ in $G$ può dare origine a uno o due archi +in $G_f$ : Se $0 < f (e) < c_e$ risulta che sia un arco in avanti +che uno all'indietro siano inclusi in $G_f$ . Quindi $G_f$ ha al +massimo il doppio degli archi rispetto a $G$. A volte ci riferiremo +alla capacità di un arco nel grafo residuo come \textbf{residual +capacity}, per aiutare a distinguerla dalla capacità dell'arco +corrispondente nella rete di flusso originale $G$. + + +\subsection{Augmenting Paths in a Residual Graph} + +Ora vogliamo rendere preciso il modo in cui viene spinto il flusso da +$s$ a $t$ in $G_f$ . Sia $P$ un cammino $s-t$ in $G_f$ (un +path nel grafo residuale viene chiamato \textbf{augmenting path}), cioè +$P$ non visita nessun nodo più di una volta. Definiamo +\texttt{bottleneck(P,\ f)} come la \textbf{minima capacità residua tra +tutti gli archi di $P$}, rispetto al flusso $f$. Definiamo ora la +seguente operazione \texttt{augment(f\ ,\ P)}, che produce un nuovo +flusso $f'$ in $G$.\\ + +\texttt{augment(f, P)} + +\begin{lstlisting}[language=Python, mathescape=true] + Let b = bottleneck(P, f) + +For each edge (u, v) $\in$ P + If e = (u, v) is a forward edge then + increase f (e) in G by b + Else ((u, v) is a backward edge, and let e = (v, u)) + decrease f (e) in G by b + Endif +Endfor + +Return(f) + +\end{lstlisting} + +Proprio per poter eseguire questa operazione abbiamo definito il grafo +residuale; per riflettere l'importanza dell'\textbf{augment} (aumento), +ci si riferisce a qualsiasi cammino $s-t$ nel grafo residuale come +\textbf{\emph{augmenting path}}. Il risultato di +\texttt{augment(f\ ,\ P)} è un nuovo flusso $f'$ in $G$, ottenuto +aumentando e diminuendo i valori di flusso sugli archi di $P$.\\ + +Questa operazione di \textbf{augmentation} cattura il tipo di spinta +avanti e indietro (forward and backward) del flusso che abbiamo discusso +in precedenza. Consideriamo ora il seguente algoritmo per calcolare un +flusso $s-t$ in $G$.\\ + +\texttt{Max-Flow(G)} + +\begin{lstlisting}[language=Python, mathescape=true] +Start with f (e) = 0 for each edge e \in E. + +While there is an s-t path in the residual graph G_f + Find an s \rightsquigarrow t path P in the residual network G_f + f' = augment(f , P) + Update the residual graph Gf using f' +Endwhile + +Return f + +\end{lstlisting} + +Lo chiameremo Algoritmo di \textbf{Ford-Fulkerson}, dal nome dei due +ricercatori che lo svilupparono nel 1956. + +\begin{figure}[H] + \centering + \includegraphics[width = 12 cm]{Network_flow/imgs/flow3.png} + \caption{Esempio di un esecuzione dell'algoritmo di Ford Fulkerson} +\end{figure} + +L'algoritmo Ford-Fulkerson è davvero molto semplice. Per quanto riguarda +il modo in cui vengono trovati i path nel grafo residuale, non è stato +specificato nell'algoritmo, ma si ha libera scelta sull'utilizzo di +algoritmi di esplorazione dei grafi, un esempio è l'utilizzo della DFS +(il costo complessivo dell'algoritmo di Ford-Fulkerson dipenderà anche +da questa scelta). Ciò che non è affatto chiaro è se il suo ciclo +\texttt{While} centrale termini e se il flusso restituito sia un flusso +massimo. Le risposte a entrambe queste domande si rivelano abbastanza +sottili. + +\subsection{Analyzing the Algorithm: Termination and Running Time} + +Per prima cosa consideriamo alcune proprietà che l'algoritmo mantiene +per induzione sul numero di iterazioni del ciclo \texttt{While}, +basandoci sulla nostra ipotesi che tutte le \emph{capacità} siano numeri +interi. + +\begin{myblockquote} +Ad ogni stadio intermedio dell'algoritmo di Ford-Fulkerson, i valori di +flusso $f(e)$ e le capacità residue in $G_f$ sono interi. +\end{myblockquote} + +Possiamo usare questa proprietà per dimostrare che l'algoritmo di +Ford-Fulkerson termina. Per prima cosa mostriamo che il valore del +flusso aumenta strettamente quando applichiamo \emph{augmentation} + +\paragraph{DEF 7.3} +\begin{myblockquote} + Sia $f$ un flusso in $G$, e sia $P$ un semplice cammino $s-t$ in + $G_f$ . Allora $v(f') = v(f)$ + \texttt{bottleneck(P,\ f)}; e poiché + \texttt{bottleneck(P,\ f)} \textgreater{} 0, abbiamo $v(f') > v(f)$. +\end{myblockquote} + +Abbiamo bisogno di un'altra osservazione per dimostrare la terminazione. +Dobbiamo essere in grado di limitare il massimo valore di flusso +possibile. Ecco un \textbf{upper bound}: + +\begin{myblockquote} + Se tutti gli archi al di + fuori di $s$ potessero essere completamente saturati dal flusso, il + valore del flusso sarebbe $\sum_{e \text{ out of }s} c_e$. Sia $C$ + questa somma. Quindi abbiamo $v(f) \le C$ per tutti i flussi $f$ + $s-t$ +\end{myblockquote} + +\textbf{N.B.} + +$C$ può essere un'enorme sovrastima del valore massimo +di un flusso in $G$, ma è utile per noi come limite finito. +\\ +\\ +Usando l'affermazione (7.3), ora possiamo dimostrare la terminazione: + +\begin{myblockquote} + Supponiamo, come sopra, che tutte le capacità nella rete + di flusso $G$ siano numeri interi. Quindi l'algoritmo di + Ford-Fulkerson termina al massimo in $C$ iterazioni del ciclo + \texttt{While}. +\end{myblockquote} + + +\subsection{Costo} + +Successivamente consideriamo il tempo di esecuzione dell'algoritmo +Ford-Fulkerson. Sia $n$ il numero di nodi in $G$, ed $m$ il numero +di archi in $G$. Abbiamo supposto che tutti i nodi abbiano almeno un +arco incidente, quindi $m \ge n/2$, e quindi possiamo usare +$O(m + n ) = O(m)$ per semplificare i limiti. + +\begin{myblockquote} +Supponiamo, come sopra, che tutte le capacità nella rete di flusso $G$ +siano numeri interi. Quindi l'algoritmo Ford-Fulkerson può essere +implementato per funzionare in tempo $O(mC)$. +\end{myblockquote} + +Una versione un po' più efficiente dell'algoritmo manterrebbe le linked +lists di archi nel grafo residuo $G_f$ come parte della procedura di +augmentation per il flusso $f$. + + +\section{Maximum Flows and Minimum Cuts in a Network} + +Proseguiamo ora con l'analisi dell'algoritmo Ford-Fulkerson. + + +\subsection{Analyzing the Algorithm: Flows and Cuts} + +Il nostro prossimo obiettivo è dimostrare che il flusso restituito +dall'algoritmo di Ford-Fulkerson ha il massimo valore possibile per il +flusso in $G$.\\ + +Per compiere progressi verso questo obiettivo, torniamo ad un problema +già descritto: \textbf{il modo in cui la struttura della rete di flusso +pone upper bounds al valore massimo di un flusso $s-t$}. Abbiamo già +visto un upper bound: + +\begin{myblockquote} + il valore $v(f)$ di qualsiasi + flusso $s-t$ $f$ è al massimo + $C = \sum_{e \text{ out of } S} c_e$. A volte questo limite è utile, + ma a volte è molto debole. +\end{myblockquote} + +Usiamo ora la nozione di \textbf{taglio} per sviluppare un metodo molto +più generale per porre upper bound al valore del flusso massimo. + +\begin{myblockquote} + Si consideri la possibilità di dividere i nodi del grafo + in due insiemi, $A$ e $B$, in modo che $s \in A$ e $t \in B$. + + Formalmente diciamo che un \textbf{taglio} $s-t$ è una + partizione $(A, B)$ dell'insieme di vertici $V$, tale che + $s \in A$ e $t \in B$. + + La \textbf{capacità di un taglio} $(A, B)$, che indicheremo con $c(A , B)$, è la somma delle + capacità di tutti gli archi che escono da A: + $ + c(A, B) = \sum_{e \text{ out of } A} c_e. + $\\ + I tagli risultano fornire upper bounds molto naturali sui valori dei flussi. +\end{myblockquote} + +Lo precisiamo attraverso una sequenza di teoremi e/o definizioni. + +\paragraph{DEF(7.6)} + +\begin{myblockquote} + Sia $f$ un flusso $s-t$ qualsiasi, e $(A, B)$ un taglio $s-t$, allora: + $ + v(f) = f^{\text{out}}(A) - f^{\text{in}}(A) + $ +\end{myblockquote} + +Questa affermazione è in realtà molto più forte di un semplice upper +bound. Dice che osservando la quantità di flusso che $f$ invia +attraverso un taglio, possiamo misurare esattamente il valore del +flusso: \textbf{è la quantità totale che lascia A, meno la quantità che +``torna indietro'' in A}.\\ + +Se $A = {s}$, allora $f^{out}(A) = f^{out}(s)$ e $f^{in}(A) = 0$ +poiché non ci sono archi che entrano nella sorgente per ipotesi. Quindi +l'affermazione per questo insieme $A = {s}$ è esattamente la +definizione del valore del flusso $v(f)$. Si noti che se $(A, B)$ è +un taglio, allora gli archi entranti in $B$ sono esattamente gli archi +che escono da $A$. Allo stesso modo, gli archi che escono da $B$ +sono esattamente gli archi che entrano in $A$. Quindi abbiamo +$f^{out}(A) = f^{in}(B)$, semplicemente confrontando le definizioni di +queste due espressioni. Quindi possiamo riformulare la (7.6) nel modo +seguente. + +\paragraph{DEF(7.7)} + +\begin{myblockquote} +Sia $f$ un flusso $s-t$ qualsiasi, e $(A, B)$ un taglio $s-t$, allora +$ +v(f) = f^{\text{in}}(B) - f^{\text{out}}(B) +$ +\end{myblockquote} + +Se poniamo $A = V - {t}$ e $B = {t}$ nella (7.7), abbiamo +$v(f) = f^{in}(B) - f^{out}(B) = f^{in}(t) - f^{out}(t)$. In base alla +nostra assunzione il \textbf{sink} $t$ non ha archi uscenti, quindi +abbiamo $f^{out}(t) = 0$. Questo dice che avremmo potuto definire +originariamente il \emph{valore} di un flusso altrettanto bene in +termini del sink $t$: è $f^{in}(t)$, la quantità di flusso che +arriva al \textbf{sink}.\\ + +Una conseguenza molto utile della (7.6) è il seguente upper bound. + + +\paragraph{DEF(7.8)} + +\begin{myblockquote} +Sia $f$ un flusso $s-t$ qualsiasi, e $(A, B)$ un taglio $s-t$, allora +$ + v(f) \le c(A, B) +$ +\end{myblockquote} + +In un certo senso, la (7.8) sembra più debole della (7.6), poiché è solo +una disuguaglianza piuttosto che un'uguaglianza. Tuttavia, ci sarà +estremamente utile, poiché il suo lato destro è indipendente da un +flusso particolare $f$. Quello che dice la (7.8) è che \textbf{il +valore di ogni flusso è limitato superiormente dalla capacità di ogni +taglio}. In altre parole, se eseguiamo un qualsiasi taglio $s-t$ in +$G$ di un certo valore $c^{*}$, sappiamo immediatamente dalla (7.8) +che non può esserci un flusso $s-t$ in $G$ di valore maggiore di +$c^{*}$. Al contrario, se valutiamo un qualsiasi flusso $s-t$ in $G$ +di un certo valore $v^{*}$, sappiamo immediatamente dalla (7.8) che non +può esserci un taglio $s-t$ in $G$ di valore minore di $v^{*}$. + +\section{Analyzing the Algorithm: Max-Flow Equals Min-Cut} + +Sia $\bar{f}$ il flusso restituito dall'algoritmo di +\textbf{Ford-Fulkerson}. Vogliamo dimostrare che $\bar{f}$ ha il +massimo valore possibile di qualsiasi flusso in $G$, e lo facciamo con +il metodo discusso sopra: Lo facciamo con un taglio $s-t$ +$(A^{*} , B^{*})$ per il quale $v(\bar{f}) = c(A^{*} , B^{*})$. Questo +stabilisce immediatamente che $\bar{f}$ ha il valore massimo di +qualsiasi flusso, e che $(A^{*} , B^{*})$ ha la capacità minima di +qualsiasi taglio $s-t$.\\ + +L'algoritmo di Ford-Fulkerson \textbf{termina quando il flusso $f$ non +ha un cammino $s-t$ nel grafo residuale $G_f$}. Questa risulta +essere l'unica proprietà necessaria per dimostrare la sua massimalità.\\ + +\paragraph{DEF(7.9)} + +\begin{myblockquote} +Se $f$ è un flusso $s-t$ tale che non esiste un cammino $s-t$ nel +grafo residuale $G_f$ , allora esiste un taglio $s-t$ +$(A^{*} , B^{*})$ in $G$ per cui $v(f) = c(A^{*} , B^{*})$. Di +conseguenza, $f$ ha il valore massimo di qualsiasi flusso in $G$, e +$(A^{*} , B^{*})$ ha la capacità minima di qualsiasi taglio $s-t$ in +$G$. +\end{myblockquote} + + +\paragraph{Dimostrazione} + +Dobbiamo identificare un taglio che dimostri la precedente proprietà. A +tal fine, indichiamo con $A^{*}$ l'insieme di tutti i nodi $v$ in +$G$ per i quali esiste un cammino $s-v$ in $G_f$ . Sia $B^{*}$ +l'insieme di tutti gli altri nodi: $B^{*} = V - A^{*}$.\\ + +Per prima cosa stabiliamo che $(A^{*} , B^{*})$ è effettivamente un taglio +$s-t$. È chiaramente una partizione di $V$. La sorgente $s$ +appartiene ad $A^{*}$ poiché c'è sempre un cammino da $s$ a $s$. +Inoltre, $t \notin A^{*}$ assumendo che non ci sia un cammino $s-t$ +nel grafo residuale; quindi $t \in B^*$ come desiderato.\\ + +Supponiamo ora che $e = (u, v)$ sia un arco in $G$ per il quale +$u \in A^{*}$ e $v \in B^{*}$, come mostrato nella Figura seguente. +Affermiamo che $f(e) = c_e$ . Infatti, in caso contrario, $e$ +sarebbe un arco \emph{forward} nel grafo residuale $G_f$ , e poiché +$u \in A^{*}$ esiste un cammino $s-u$ in $G_f$ ; aggiungendo $e$ a +questo cammino, otterremmo un cammino $s-v$ in $G_f$ , +contraddicendo la nostra ipotesi che $v \in B^{*}$.\\ + +Supponiamo ora che $e' = (u' , v')$ sia un arco in $G$ per cui +$u' \in B^{*}$ e $v' \in A^{*}$. Affermiamo che $f(e') = 0$. In caso +contrario, $e'$ darebbe luogo a un arco \emph{backward} +$e'' = (v' , u')$ nel grafo residuale $G_f$ , e poiché +$v' \in A^{*}$, allora è un cammino $s-v'$ in $G_f$ ; aggiungendo +$e''$ a questo cammino, otterremmo un cammino $s-u'$ in $G_f$ , +contraddicendo la nostra ipotesi che $u' \in B^{*}$.\\ + +Quindi tutti gli archi uscenti da $A^{*}$ sono completamente saturati di +flusso, mentre tutti gli archi entranti in $A^{*}$ sono completamente +inutilizzati. Possiamo ora usare la (7.6) per raggiungere la conclusione +desiderata: +$$ +v(f) = f^{out}(A^{*}) - f^{in}(A^{*}) = \sum_{e \text{ out of }A^{*}} f(e) - \sum_{e \text{ into }A^{*}}f(e) = \sum_{e \text{ out of }A^{*}} c_e - 0 = c(A^{*}, B^{*}) +$$ + +\begin{figure}[H] + \centering + \includegraphics[width = 10 cm]{Network_flow/imgs/flow4.png} + \caption{La dimostrazione della 7.9} +\end{figure} + +\paragraph{DEF(7.10)} + +\begin{myblockquote} +Il flusso $\bar{f}$ restituito dall'algoritmo di Ford-Fulkerson è un +flusso massimo. +\end{myblockquote} + +\paragraph{DEF(7.11)} + +\begin{myblockquote} +Dato un flusso f di valore massimo, possiamo calcolare un taglio $s-t$ +di capacità minima in tempo $O(m)$. +\end{myblockquote} + + +\paragraph{DEF(7.12)} + +\begin{myblockquote} +In ogni rete di flussi esiste un flusso $f$ e un taglio $(A, B)$ +tale che $v(f) = c(A, B)$. +\end{myblockquote} + +Il punto è che $f$ nella (7.12) deve essere un flusso massimo $s-t$; +poiché se ci fosse un flusso $f'$ di valore maggiore, il valore di +$f$ supererebbe la capacità di $(A, B)$, e ciò contraddirebbe la +(7.8). Allo stesso modo segue che $(A, B)$ nella (7.12) è un taglio +minimo (nessun altro taglio può avere capacità minore) perché se ci +fosse un taglio $(A , B)$ di capacità minore, sarebbe minore del +valore di $f$ , e anche questo contraddirebbe la (7.8). A causa di +queste implicazioni, la (7.12) è spesso chiamata \textbf{teorema del +taglio minimo del flusso massimo} ed è formulata come segue. + +\paragraph{Teorema del Taglio Minimo del Flusso Massimo} + +\begin{myblockquote} +\textbf{In ogni rete di flussi, il valore massimo di un flusso $s-t$ è +uguale alla capacità minima possibile per un taglio $s-t$.} +\end{myblockquote} \ No newline at end of file diff --git a/magistrale/Anno 1/Advanced and Distributed Algorithms/Pinotti/latex/Network_flow/imgs/bipartite1.png b/magistrale/Anno 1/Advanced and Distributed Algorithms/Pinotti/latex/Network_flow/imgs/bipartite1.png new file mode 100644 index 0000000000000000000000000000000000000000..5fb4865c0f0ea77e9c9a66715d106eabb166a67e GIT binary patch literal 43961 zcmd43$cN~e@`N=SD~NP|dAh|=94-FMCV zo^$Sp`wv{cSTN7b?7i3e)iUsvoWz~mq_T-QQDx%LSC7W@s{Z+;s1 z<)-}$DJ69Haz{4^fIpKsh^aX!S{pey>Dn2h7+YCe8Zz1I+Zh^K*_&89Y~N@Sgm0on zzDdN+P}jlK+UlW_sih%`rJ>zJ_J@|b#t)yeK4oWo$i~IT`jn4@OIL-o76s)Yij=6Z 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