From d8ff5dc86a5d87561a88c0a8f796ca5cb2928129 Mon Sep 17 00:00:00 2001 From: KC_PC <kevin.conrads@rwth-aachen.de> Date: Fri, 13 Mar 2020 16:41:04 +0100 Subject: [PATCH] Version 1.1.1 Chapter 9 completed --- README.md | 5 +++-- ems.last-change | 2 +- ems.pdf | Bin 415224 -> 439378 bytes ems.tex | 51 +++++++++++++++++++++++++++--------------------- 4 files changed, 33 insertions(+), 25 deletions(-) diff --git a/README.md b/README.md index 4050126..143aec3 100644 --- a/README.md +++ b/README.md @@ -12,9 +12,10 @@ Das Dokument ist noch im Aufbau und wird sich unter Umständen noch ein wenig ve - 5.3.2: Algo EFTF Lesbarkeit erhöht - 6.1.1: Def induzierter Teilgraph: Lesbarkeit erhöht - 7.4.1: Fixed Typo in Spacing Definition +- 13.03.2020 - Version 1.1.1: + - Kapitel 9 hinzugefügt - TBA - Version 1.2: - - 4.1: Typo - - Kapitel 9 - Project Selection hinzugefügt + - Kapitel 9 - 11 hinzugefügt diff --git a/ems.last-change b/ems.last-change index ff5f362..f8921ae 100644 --- a/ems.last-change +++ 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zOhJPKV4jOYw2i5Wp{a$5i>tG<v#Gg}k*m3zxudCzi;J<Tv7?!rp}Cy`VI{E&c6MCF aC5c5P6-B9OTqdB|T_bibRaIAiH!c8YSl##l diff --git a/ems.tex b/ems.tex index 7867b5e..a2cbe13 100644 --- a/ems.tex +++ b/ems.tex @@ -1723,7 +1723,7 @@ Um nun eine Gewinn-maximierende Auswahl an Projekten zu finden, erweitern wir di Angewendet auf unser vorheriges Beispiel $P'$, wobei nun auch die Revenues definiert werden, ergibt sich also folgendes Flussnetzwerk: \begin{bsp}{Project Selection: Flussnetzwerk} - Sei $P' = \{a,b,c,d,e\}$ eine Projektmenge mit $a \prec b,\, a \prec c,\, a \prec d$ und $ d \prec c$. Außerdem sei $w_a = 3, \, w_b = -2, \, w_c = 6, \, w_d = -4, \, w_e = 2$. Das Flussnetzwerk von $P'$ in der min-cut-Formulierung sieht dann wie folgt aus: + Sei $P' = \{a,b,c,d,e\}$ eine Projektmenge mit $a \prec b,\, a \prec c,\, a \prec d$ und $ d \prec c$. Außerdem sei $w_a = -3, \, w_b = 5, \, w_c = 3, \, w_d = -4, \, w_e = 2$. Das Flussnetzwerk von $P'$ in der min-cut-Formulierung sieht dann wie folgt aus: \begin{center} \begin{tikzpicture} @@ -1736,14 +1736,14 @@ Angewendet auf unser vorheriges Beispiel $P'$, wobei nun auch die Revenues defin \node[draw,circle,minimum size=5mm] (s) at (0,-1) {$s$}; \node[draw,circle,minimum size=5mm] (t) at (5.5,-1) {$t$}; - \draw[->] (b) -- (a) node[midway, below] {$\infty$} ; - \draw[->] (c) -- (d) node[midway, above] {$\infty$}; + \draw[->] (b) -- (a) node[midway, above] {$\infty$} ; + \draw[->] (c) -- (d) node[midway, below] {$\infty$}; \draw[->] (d) -- (a) node[midway, left] {$\infty$}; - \draw[->] (s) to[bend left=60] node[midway,above right] {$3$} (a) ; - \draw[->] (s) to[bend right=30] node [below,midway] {$6$} (c) ; + \draw[->] (s) to[bend left=30] node[midway,above right] {$5$} (b) ; + \draw[->] (s) to[bend right=30] node [below,midway] {$3$} (c) ; \draw[->] (s) -- (e) node[midway, above] {$2$} ; - \draw[->] (b) to[bend left=60] node[above right] {$2$} (t) ; + \draw[->] (a) -- node[above right] {$3$} (t) ; \draw[->] (d) -- node[below right] {$4$} (t) ; \end{tikzpicture} @@ -1756,37 +1756,44 @@ Als letzter Schritt bleibt nur noch, aus diesem Flussnetzwerk eine zulässige Pr $S \subseteq P$ ist eine zulässige Projektauswahl mit maximalem Revenue $\sum_{j \in S} w_j \Leftrightarrow S \cup \{s \}$ ist ein s-t Schnitt in der Flussnetzwerk-Konstruktion mit minimaler Kapazität. \end{theo} -Warum funktioniert diese Konstruktion und berechnet eine optimale Projektauswahl? Dazu schaut man sich erneut das Beispiel an (in dem schon ein maximale Fluss bestimmt wurde): +Warum funktioniert diese Konstruktion und berechnet eine optimale Projektauswahl? Dazu schaut man sich erneut das Beispiel an (in dem schon ein maximaler Fluss bestimmt wurde): \begin{bsp}{Project Selection: Flussnetzwerk} - Sei $P' = \{a,b,c,d,e\}$ eine Projektmenge mit $a \prec b,\, a \prec c,\, a \prec d$ und $ d \prec c$. Außerdem sei $w_a = 3, \, w_b = -2, \, w_c = 6, \, w_d = -4, \, w_e = 2$. Das Flussnetzwerk von $P'$ in der min-cut-Formulierung und die zugehörige Berechnung des minimalen Schnittes sehen dann wie folgt aus: + Sei $P' = \{a,b,c,d,e\}$ eine Projektmenge mit $a \prec b,\, a \prec c,\, a \prec d$ und $ d \prec c$. Außerdem sei $w_a = -3, \, w_b = 5, \, w_c = 3, \, w_d = -4, \, w_e = 2$. Das Flussnetzwerk von $P'$ in der min-cut-Formulierung und die zugehörige Berechnung des minimalen Schnittes sehen dann wie folgt aus: \begin{center} \begin{tikzpicture} - \node[draw,circle,minimum size=5mm] (a) at (4,0) {$a$}; - \node[draw,circle,minimum size=5mm] (b) at (2.5,0) {$b$}; + \node[draw,circle,minimum size=5mm,fill={rgb:red,2;green,15;blue,2}] (a) at (4.5,0) {$a$}; + \node[draw,circle,minimum size=5mm,fill={rgb:red,2;green,15;blue,2}] (b) at (2.5,0) {$b$}; \node[draw,circle,minimum size=5mm] (c) at (2.5,-2) {$c$}; - \node[draw,circle,minimum size=5mm] (d) at (4,-2) {$d$}; - \node[draw,circle,minimum size=5mm] (e) at (1.5,-1) {$e$}; + \node[draw,circle,minimum size=5mm] (d) at (4.5,-2) {$d$}; + \node[draw,circle,minimum size=5mm,fill={rgb:red,2;green,15;blue,2}] (e) at (1.5,-1) {$e$}; - \node[draw,circle,minimum size=5mm] (s) at (0,-1) {$s$}; - \node[draw,circle,minimum size=5mm] (t) at (5.5,-1) {$t$}; + \node[draw,circle,minimum size=5mm,fill={rgb:red,10;green,10;blue,-10}] (s) at (0,-1) {$s$}; + \node[draw,circle,minimum size=5mm] (t) at (6,-1) {$t$}; - \draw[->] (b) -- (a) node[midway, below] {$\infty$} ; - \draw[->] (c) -- (d) node[midway, above] {$\infty$}; - \draw[->] (d) -- (a) node[midway, left] {$\infty$}; + \draw[->] (b) -- (a) node[midway, above] {${\color{blue}3}/\infty$} ; + \draw[->] (c) -- (d) node[midway, below] {${\color{blue}3}/\infty$}; + \draw[->] (d) -- (a) node[midway, left] {${\color{blue}0}/\infty$}; - \draw[->] (s) to[bend left=60] node[midway,above right] {$3$} (a) ; - \draw[->] (s) to[bend right=30] node [below,midway] {$6$} (c) ; - \draw[->] (s) -- (e) node[midway, above] {$2$} ; - \draw[->] (b) to[bend left=60] node[above right] {$2$} (t) ; - \draw[->] (d) -- node[below right] {$4$} (t) ; + \draw[->] (s) to[bend left=30] node[midway,above right] {${\color{blue}3}/5$} (b) ; + \draw[->] (s) to[bend right=30] node [below,midway] {${\color{blue}3}/3$} (c) ; + \draw[->] (s) -- (e) node[midway, above] {${\color{blue}0}/2$} ; + \draw[->] (a) -- node[above right] {${\color{blue}3}/3$} (t) ; + \draw[->] (d) -- node[below right] {${\color{blue}3}/4$} (t) ; + + \draw[thick, color=red] plot [smooth,tension=0.75] coordinates{(5.25,0.5) (4.75, -1) (2, -1.5) (0,-2)}; \end{tikzpicture} + + In rot ist dabei der minimale Schnitt und in Grün die daraus folgende optimale Projektauswahl zu sehen. \end{center} \end{bsp} +Man kann sich den Fluss in diesem Netzwerk grundsätzlich als zu investierende Kosten vorstellen: Das Projekt $c$ bringt $5$ Geldeinheiten Profit, weswegen die Kante $(s,c)$ mit Kapazität $5$ existiert. Es kann $c$ nur ausgewählt werden, wenn auch $d$ ausgewählt wird (durch die Kante $(c,d)$ ausgedrückt). Das Projekt $d$ hingegen verursacht Kosten in Höhe von $4$, dargestellt durch die Kante $(d,t)$ mit Kapazität $4$. Es fließen also Flusseinheiten von $s$ zu gewinnbringenden Knoten und von verlustbringenden Knoten zu $t$. Da $c$ uns $3$ Profit bringt, wählen wir das Projekt temporär einmal aus. Um nun aber zu schauen, ob die Abhängigkeiten von $c$ eventuell insgesamt zu Verlust führen würden, versuchen wir also, den maximalen Fluss zu bestimmen, der über die Kante $(s,c)$ laufen kann. Wenn die Kante voll ausgelastet sein sollte (wie oben im Beispiel), heißt das, dass alle drei Einheiten an Profit von $c$ aus zu $t$ fließen können, also das sämtlicher Profit von $c$ durch die Abhängigkeiten von $c$ ``aufgebraucht'' wurde, denn es kann ja nur von Projekten, die Verlust bringen etwas zu $t$ fließen (oben $3$ Einheiten über $(d,t)$, die ursprünglich von $c$ kommen. Die Wahl von $c$ würde also durch die Kosten von $d$ nicht zu Profit führen. +Der Knoten $b$ stellt dabei das gleiche Szenario dar, jedoch sind die Kosten durch die Wahl von $b$ kleiner als der Profit durch $b$, weswegen hier $5-3 = 2$ Profit durch $b$ verursacht werden. Als letztes bleibt das Projekt $e$, das keinerlei Abhängigkeiten hat, weswegen auch $2-0= 2$ Profit durch die Wahl von $e$ verursacht werden. + \section{Matching Markets} ... \textit{folgt zeitnah} ... -- GitLab