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{\bssdoz CME 305: Discrete Mathematics and Algorithms}
{\\ \bssten Instructor: Reza Zadeh (rezab@stanford.edu) }
{\\ \bssten HW\#4 -- Due at the beginning of class Thursday 03/12/15}
\vspace{5pt}
\begin{enumerate}
\item Let $G=(V,E)$ be a $c$-edge connected graph. In other words, assume that the size of minimum cut in $G$ is at least $c$. Construct a graph $G'(V, E')$ by sampling each edge of $G$ with probability $p$ independently at random and reweighing each edge with weight $1/p$. Suppose $c > \log n$, and $\epsilon$ is such that $ \frac{\log(n)}{c \epsilon^2} \leq 1$. Show that as long as $p \geq \frac{\log(n)}{c \epsilon^2}$, with high probability the size of every cut in $G'$ is within $(1\pm \epsilon)$ of the cut in the original graph $G$.
\item Consider a random bipartite graph on two vertex sets $L,R$ of size
$n$ each, in which every vertex in $L$ independently picks $d$
uniformly independent random neighbors in $R$. Given $\epsilon >
1/d$, prove that with high probability, every set $S$ in $L$ of
size $\alpha n /d$ has at least $|S|(1-\epsilon)d$ neighbors in
$R$, where $\alpha$ depends only on $\epsilon$ and $d$.
\item Consider scheduling $n$ jobs to $m$ identical machines to minimize the time taken by the machine
with the heaviest load (i.e. to minimize the makespan). One algorithm is
to order the jobs by decreasing processing times $t_1 \geq t_2 \geq \ldots \geq t_n$, then greedily assign jobs
to the machine whose load is the smallest so far (starting with the heaviest job).
\begin{enumerate}
\item Show that this algorithm is a $4/3$ approximation to the optimal makespan.
\item Show that the $4/3$ approximation is tight for this algorithm.
\end{enumerate}
\item Let $V$ be a finite set. A function $f\colon 2^V \to R$ is submodular iff for any $A,B \subseteq V$, we have
$$ f(A \cap B) + f(A \cup B) \leq f(A)+f(B)$$
Now consider a graph with nodes $V$. For any set of vertices $S\subseteq V$ let $f(S)$ denote the number of edges $e=(u,v)$ such that $u\in S$ and $v\in V-S$. Prove that $f$ is submodular.
\item A square integer matrix $A$ is {\bf unimodular} if and only if
its determinant is $-1$ or $1$. A matrix (not necessarily square)
$M$ is {\bf totally unimodular} iff every square submatrix has
determinant $1$, $-1$, or $0$, i.e. every non-singular square submatrix is
unimodular.
Show that for a linear program with totally unimodular
constraint matrix $M$ and integral right-hand side $c$, all extreme
points must be integral.
\item We are given $n$ jobs that each take one unit of processing time. All jobs
are available at time $0$, and job $j$ has a profit of $c_j$ and a deadline $d_j$.
The profit for job $j$ will only be earned if the job completes by time $d_j$.
The problem is to find an ordering
of the jobs that maximizes the total profit. First, prove that if a subset of the jobs can be
completed on time, then they can also be completed on time if they are scheduled in the
order of their deadlines. Now, let $E=\{1, \ldots, n \}$ and let
$$I = \{J \subseteq E\text{ : $J$ can be completed on time } \}$$
Prove that $M=(E,I)$ is a matroid and describe how to find an optimal
ordering for the jobs.
\item Given a list of personnel ($n$ persons) and of list of $k$
vacation periods, each period spanning several contiguous vacation
days. Let $D_j$ be the set of days included in the $j$th vacation
period. You need to produce a schedule satisfying:
\begin{itemize}
\item For a given parameter $c$, each tech support person should be
assigned to work at most $c$ vacation days total.
\item For each vacation period $j$, each person should be assigned to work
at most one of the days during the period.
\item Each vacation day should be assigned a single tech support person.
\item For each person, only certain vacation periods are viable.
\end{itemize}
Describe a polynomial time algorithm to generate an assignment or output
that no assignment exists. Prove correctness.
\item Online social networks carry a huge potential for online advertising.
After a recent controversy, a popular
social networking platform does not allow advertisers to target the
users individually. However, it is allowed to run ads on user
communities.
Formally, let $X$ be the set of all users on a social network, and
$S_1, S_2, \ldots, S_m$ be subsets of $X$, where each $S_i$
represents a user community. Notice that a user can belong to
several communities. Suppose the advertiser can afford placing ads
on at most $k$ communities. The goal is to show the ads to as many
users as possible, i.e.\ to find $S_{i_1}, S_{i_2}, \ldots, S_{i_k}$
such that $|\cup_{j=1}^{k} S_{i_j}|$ is maximized.
Unfortunately, this problem is NP-hard and therefore we are
interested in designing efficient approximation algorithms to solve
it. Consider the following greedy approach: pick the $k$ communities
one at a time, and in each iteration pick the community that
contains the largest number of users that have not been covered yet.
In other words, choose the community that maximizes the current
coverage. Show that this greedy approach yields at least $1 - (1 -
1/k)^k > 1 - 1/e$ fraction of the optimal solution.
Hint: Let $x_i$ denote the number of new elements covered by the
algorithm in the $i$-th set that it picks. Also, let
$y_i=\sum_{j=1}^i x_j,$ and $z_i = OPT - y_i$. Show $x_{i+1} \geq
z_i/k$ and prove by induction that $z_i \leq (1 - 1/k)^{i} OPT$.
\end{enumerate}
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