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\title{CSP Exercise 02 Solution}

\author{Eugen Sawin}

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\section*{Exercise 2.1}

(a) The strongly connected components are the subgraphs induced by following sets of vertices.

\begin{itemize}

  \item $V_1 = \{v_1,v_2,v_5,v_6\}$

  \item $V_2 = \{v_3,v_4\}$

\end{itemize}

(b) The cliques are the the complete subgraphs induced by the following sets of vertices.

\begin{itemize}

  \item $V_1 = \{v_1,v_2,v_5\}$

  \item $V_{2-8} \in E$

  \item $V_{9-14} \in \{\{v_i\} \mid v_i \in V\}$

\end{itemize}

\section*{Exercise 2.2}

We define the following undirected simple graph $G = \langle W, E \rangle$ with $\{w_i,w_j\} \in E$ iff $w_i$ and $w_j$ dislike each other. Additionally we define two sets $B_1 = B_2 = \{\}$ for the buildings and an auxiliary set $F = \{\}$ for remembering \emph{expanded} nodes.

\begin{enumerate}

  \item For all $w_i \in W$ with $w_i \notin \bigcup E$ add $w_i$ to $B_1$ and $F$, i.e. $B_1 = B_1 \cup \{w_i\}$ and $F = F \cup \{w_i\}$. This way we assign all workers, who are liked by everyone to building $B_1$.

  \item If $F = W$ terminate, we have found a solution.\\

  Otherwise select any $w_i \notin F$ and add it to $B_1$ and $F$, i.e. $B_1 = B_1 \cup \{w_i\}$ and $F = F \cup \{w_i\}$.\\

  Add all its disliked neighbours to $B_2$, i.e. $B_2 = B_2 \cup \{w_j \mid \{w_i,w_j\} \in E\}$.\\

  If $B_1 \cap B_2 \neq \emptyset$ terminate, there was a conflict and therefore no possible solution.

  \item If there a $w_i \notin F$ with $w_i \in B_1 \cup B_2$ select it, otherwise continue with $2.$\\

  Add all its disliked neighbours to the other building, i.e. $w_i \in B_1 \implies B_2 = B_2 \cup \{w_j \mid (w_i,w_j) \in E\}$, $w_i \in B_2 \implies B_1 = B_1 \cup \{w_j \mid (w_i,w_j) \in E\}$ and $F = F \cup \{w_i\}$.\\

  If $B_1 \cap B_2 \neq \emptyset$ terminate, there was a conflict and therefore no possible solution.\\

  Otherwise continue with $3.$

\end{enumerate}

This is essentially a breadth-first search with random-walk used to jump to disconnected nodes. Assumming that set containment can be tested in $\mathcal{O}(1)$ and set intersection in $\mathcal{O}(n)$ we have an asymptotic complexity of $\mathcal{O}(|W| \cdot |E|)$.

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