Add complexity part for algorithms

Still missing definition for Limit Monotonicity and an example. Add a
sentence about Discount Monotonicity with other Budgeting Algorithms.
Write a nice conclusion slide. Mention that he focus is on discrete and
bounded PB and that Approval-Based Voting is a mechanism for this type
of problem. Add a short history of PB with a reference to Porto Alegre
in Brazil to the Introduction.

talk/talk.tex

@@ -8,7 +8,6 @@
 \usepackage{graphicx}
 \usepackage{tikz}
 \usepackage{dsfont}
-\usepackage{comment}
 
 \usetikzlibrary{arrows}
 
@@ -81,13 +80,13 @@
                     $c(p):P\rightarrow\mathbb{R}$
                 \item Projects are either divisible or indivisible (discrete)
             \end{itemize}
-        \item Select a set $P'\subseteq P$ as \emph{winning projects} not
+        \item Select a set $A\subseteq P$ as \emph{winning projects} not
             exceeding total budget $B$
             \begin{itemize}
                 \setlength{\itemsep}{.7\baselineskip}
-                \item Discrete case: $\sum_{p\in P'}c(p)\leq B$
+                \item Discrete case: $\sum_{p\in A}c(p)\leq B$
                 \item Divisible case: $\mu(p): P\rightarrow [0,1]$ with
-                    $\sum_{p\in P'}c(\mu(p))\leq B$
+                    $\sum_{p\in A}c(\mu(p))\leq B$
             \end{itemize}
     \end{itemize}
 \end{frame}
@@ -98,9 +97,9 @@
         \setlength{\itemsep}{1\baselineskip}
         \item Voters $V=\{v_1,\dots,v_n\}$
             \begin{itemize}
-                \setlength{\itemsep}{.5\baselineskip}
+                \setlength{\itemsep}{.4\baselineskip}
                 \item Express preferences over individual projects in $P$ or
-                    over subsets in $\mathcal{P}(P) := \{P'\,|\,P'\subseteq P\}$
+                    over subsets in $\mathcal{P}(P) := \{P_v\,|\,P_v\subseteq P\}$
                 \item Preference elicitation is dependent on the input method
                     (approval-based, ranked orders)
             \end{itemize}
@@ -231,7 +230,81 @@
 \end{frame}
 
 \begin{frame}
-    \frametitle{}
+    \frametitle{Complexity of budgeting algorithms}
+    \begin{itemize}
+        \setlength{\itemsep}{1\baselineskip}
+        \item Computing winners using greedy rules ($\mathcal{R}^g_{sat}$) can
+            be done in polynomial time:
+            \begin{itemize}
+                \setlength{\itemsep}{.4\baselineskip}
+                \item these rules are defined through efficient iterative
+                    processes
+                \item however: making a series of locally optimal choices does
+                    not always lead to a globally optimal choice
+                \item $\mathcal{R}^g_{|A_v|}$ is similar to $k$-Approval and
+                    knapsack voting
+            \end{itemize}
+        \item Max rules ($\mathcal{R}^m_{sat}$) are generally NP-hard
+            \begin{itemize}
+                \setlength{\itemsep}{.4\baselineskip}
+            \item $\mathcal{R}^m_{|A_v|}$ can be solved in polynomial time
+                because one dimension is fixed
+            \item $\mathcal{R}^m_{sat_{0/1}}$: finding a bundle with at least a
+                given total satisfaction is NP-hard
+            \item satisfaction functions can be modeled as integer linear
+                programs
+            \end{itemize}
+    \end{itemize}
+\end{frame}
+
+\begin{frame}
+    \frametitle{Complexity of budgeting algorithms ctd.}
+    {\large Dealing with \emph{intractability}:}
+    \vspace{.3cm}
+    \begin{itemize}
+        \setlength{\itemsep}{1\baselineskip}
+        \item Provide an approximation algorithm, sacrificing exactness
+            \begin{itemize}
+                \setlength{\itemsep}{0.4\baselineskip}
+                \item No algorithm with approx. ratio better than $1-1/\epsilon$
+                    exists for $\mathcal{R}^m_{sat_{0/1}}$
+            \end{itemize}
+        \item Fixed-parameter tractability: fix one parameter to solve problem
+            in reasonable amount of time
+            \begin{itemize}
+                \setlength{\itemsep}{0.4\baselineskip}
+            \item Fix parameter $m$ (the number of items)
+            \item Fix parameter $n$ (the number of voters)
+            \end{itemize}
+    \end{itemize}
+\end{frame}
+
+\begin{frame}
+    \frametitle{Comparing budgeting algorithms}
+        By defining desirable axioms, different budgeting algorithms can
+        be compared:
+        \begin{block}{Discount Monotonicity}
+            Suppose a budgeting algorithm $\mathcal{R}$ outputs a winning set of
+            projects $A$. The cost of project $p\in A$ is lowered (discounted)
+            compared to the previous cost. $\mathcal{R}$ should output another
+            winning set $A'$ where project $p$ is not implemented to a lesser
+            degree.
+        \end{block}
+\end{frame}
+
+\begin{frame}
+    \frametitle{Axiom Examples}
+    \begin{block}{A budgeting scenario}
+        Items $P = \{p_2,p_3,p_4,p_5,p_6\}$ and their associated cost $p_i$
+        where $p_i$ costs $i$. Budget limit $B=10$ and 5 voters vote
+        $v_1=\{p_2,p_5,p_6\}$, $v_2=\{p_2,p_3,p_4,p_5\}$, $v_3=\{p_3,p_4,p_5\}$,
+        $v_4=\{p_4,p_5\}$ and $v_5=\{p_6\}$.
+    \end{block}
+    \begin{exampleblock}{Discount Monotonicity Example}
+        Under $\mathcal{R}^m_{|A_v|}$ the winning bundle is $\{p_2,p_3,p_5\}$.
+        After discounting $p_2$ to $p_1$, we still get $\{p_1,p_3,p_5\}$. The
+        total cost is one unit less but the total satisfaction remains the same.
+    \end{exampleblock}
 \end{frame}
 
 \section{Future Directions}