Chapter 6, The Probabilistic Method Video Solutions, Probability and Computing

Problem 1

Consider an instance of SAT with $m$ clauses, where every clause has exactly $k$ literals.
(a) Give a Las Vegas algorithm that finds an assignment satisfying at least $m\left(\mathrm{I}-2^{-\mathrm{k}}\right)$ clauses, and analyze its expected running time.
(b) Give a derandomization of the randomized algorithm using the method of conditional expectations.

Chris Trentman

Numerade Educator

03:41

Problem 2

(a) Prove that, for every integer $n$, there exists a coloring of the edges of the complete graph $K_{n}$ by two colors so that the total number of monochromatic copies of $K_{4}$ is at most $\left(\begin{array}{l}n \\ 4\end{array}\right) 2^{-5}$.
(b) Give a randomized algorithm for finding a coloring with at most $\left(\begin{array}{l}n \\ 4\end{array}\right) 2^{-5}$ monochromatic copies of $K_{4}$ that runs in expected time polynomial in $n$.
(c) Show how to construct such a coloring deterministically in polynomial time using the method of conditional expectations.

Nick Johnson

Numerade Educator

03:13

Problem 3

Given an $n$-vertex undirected graph $G=(V, E)$, consider the following method of generating an independent set. Given a permutation $\sigma$ of the vertices, define a subset $S(\sigma)$ of the vertices as follows: for each vertex $i, i \in S(\sigma)$ if and only if no neighbor $j$ of $i$ precedes $i$ in the permutation $\sigma$.
(a) Show that each $S(\sigma)$ is an independent set in $G$.
(b) Suggest a natural randomized algorithm to produce $\sigma$ for which you can show that the expected cardinality of $S(\sigma)$ is
$$
\sum_{i=1}^{n} \frac{1}{d_{i}+1}
$$
where $d_{i}$ denotes the degree of vertex $i$.
(c) Prove that $G$ has an independent set of size at least $\sum_{i=1}^{n} 1 /\left(d_{i}+1\right)$.

Amany Waheeb

Numerade Educator

04:51

Problem 4

Consider the following two-player game. The game begins with $k$ tokens placed at the number 0 on the integer number line spanning $[0, n] .$ Each round, one player, called the chooser, selects two disjoint and nonempty sets of tokens $A$ and $B$. (The sets $A$ and $B$ need not cover all the remaining tokens; they only need to be disjoint.) The second player, called the remover, takes all the tokens from one of the sets off the board. The tokens from the other set all move up one space on the number line from their current position. The chooser wins if any token ever reaches $n$. The remover wins if the chooser finishes with one token that has not reached $n$.
(a) Give a winning strategy for the chooser when $k \geq 2^{n}$.
(b) Use the probabilistic method to show that there must exist a winning strategy for the remover when $k<2^{n}$.
(c) Explain how to use the method of conditional expectations to derandomize the winning strategy for the remover when $k<2^{n}$.

Hunza Gilgit

Numerade Educator

05:24

Problem 5

We have shown using the probabilistic method that, if a graph $G$ has $n$ nodes and $m$ edges, then there exists a partition of the $n$ nodes into sets $A$ and $B$ such that at least $m / 2$ edges cross the partition. Improve this result slightly: show that there exists a partition such that at least $m n /(2 n-1)$ edges cross the partition.

Chris Trentman

Numerade Educator

17:27

Problem 6

We can generalize the problem of finding a large cut to finding a large $k$-cut. A $k$-cut is a partition of the vertices into $k$ disjoint sets, and the value of a cut is the weight of all edges crossing from one of the $k$ sets to another. In Section $6.2 .1$ we considered 2-cuts when all edges had the same weight 1, showing via the probabilistic method that any graph $G$ with $m$ edges has a cut with value at least $m / 2$. Generalize this argument to show that any graph $G$ with $m$ edges has a $k$-cut with value at least $(k-1) m / k$. Show how to use derandomization (following the argument of Section $6.3$ ) to give a deterministic algorithm for finding such a cut.

Chris Trentman

Numerade Educator

16:44

Problem 7

A hypergraph $H$ is a pair of sets $(V, E)$, where $V$ is the set of vertices and $E$ is the set of hyperedges. Every hyperedge in $E$ is a subset of $V$. In particular, an $r$-uniform hypergraph is one where the size of each edge is $r$. For example, a 2 -uniform hypergraph is just a standard graph. A dominating set in a hypergraph $H$ is a set of vertices $S \subset V$ such that $e \cap S \neq \emptyset$ for every edge $e \in E$. That is, $S$ hits every edge of the hypergraph.

Let $H=(V, E)$ be an $r$-uniform hypergraph with $n$ vertices and $m$ edges. Show that there is a dominating set of size at most $n p+(1-p)^{r} m$ for every real number $0 \leq$ $p \leq 1$. Also, show that there is a dominating set of size at most $(m+n \ln r) / r$.

Chris Trentman

Numerade Educator

03:41

Problem 8

Prove that, for every integer $n$, there exists a way to 2-color the edges of $K_{x}$ so that there is no monochromatic clique of size $k$ when
$$
x=n-\left(\begin{array}{l}
n \\
k
\end{array}\right) 2^{1-\left(\begin{array}{l}
k \\
2
\end{array}\right)}
$$
(Hint: Start by 2-coloring the edges of $K_{n}$, then fix things up.)

Nick Johnson

Numerade Educator

07:17

Problem 9

A toumament is a graph on $n$ vertices with exactly one directed edge between each pair of vertices. If vertices represent players, then each edge can be thought of as the result of a match between the two players: the edge points to the winner. A ranking is an ordering of the $n$ players from best to worst (ties are not allowed). Given the outcome of a tournament, one might wish to determine a ranking of the players. A ranking is said to disagree with a directed edge from $y$ to $x$ if $y$ is ahead of $x$ in the ranking (since $x$ beat $y$ in the tournament).
(a) Prove that, for every tournament, there exists a ranking that disagrees with at most $50 \%$ of the edges.
(b) Prove that, for sufficiently large $n$, there exists a tournament such that every ranking disagrees with at least $49 \%$ of the edges in the tournament.

Bobby Barnes

University of North Texas

05:02

Problem 10

A family of subsets $\mathcal{F}$ of $\{1,2, \ldots, n\}$ is called an antichain if there is no pair of sets $A$ and $B$ in $\mathcal{F}$ satisfying $A \subset B$.
(a) Give an example of $\mathcal{F}$ where $|\mathcal{F}|=\left(\begin{array}{c}n \\ \lfloor n / 2\rfloor\end{array}\right)$.
(b) Let $f_{k}$ be the number of sets in $\mathcal{F}$ with size $k$. Show that
$$
\sum_{k=0}^{n} \frac{f_{k}}{\left(\begin{array}{l}
n \\
k
\end{array}\right)} \leq 1
$$
(Hint: Choose a random permutation of the numbers from 1 to $n$, and let $X_{k}=1$ if the first $k$ numbers in your permutation yield a set in $\mathcal{F}$. If $X=\sum_{k=0}^{n} X_{k}$, what can you say about $X ?)$
(c) Argue that $|\mathcal{F}| \leq\left({ }_{\lfloor n / 2\rfloor}{n}\right)$ for any antichain $\mathcal{F}$.

James Chok

Numerade Educator

01:02

Problem 11

In Section 6.5.1, we bounded the variance of the number of 4-cliques in a random graph in order to demonstrate the second moment method. Show how to calculate the variance directly by using the equality from Exercise 3.9: for $X=\sum_{1=1}^{n} X_{t}$ the sum of Bernoulli random variables,
$$
\mathbf{E}\left[X^{2}\right]=\sum_{i=1}^{n} \operatorname{Pr}\left(X_{i}=1\right) \mathbf{E}\left[X \mid X_{i}=1\right] .
$$

Raj Bala

Numerade Educator

03:06

Problem 12

Consider the problem of whether graphs in $G_{n, p}$ have cliques of constant size $k$. Suggest an appropriate threshold function for this property. Generalize the argument used for cliques of size 4 , using either the second moment method or the conditional expectation inequality, to prove that your threshold function is correct for cliques of size 5 .

Khoobchandra Agrawal

Numerade Educator

04:48

Problem 13

Consider a graph in $G_{n, p}$, with $p=c \ln n / n$. Use the second moment method or the conditional expectation inequality to prove that if $c<1$ then, for any constant $\varepsilon>0$ and for $n$ sufficiently large, the graph has isolated vertices with probability at least $1-\varepsilon$.

Abhirup Pal

Numerade Educator

03:56

Problem 14

Consider a graph in $G_{n, p}$, with $p=1 / n$. Let $X$ be the number of triangles in the graph, where a triangle is a clique with three edges. Show that
$$
\begin{gathered}
\operatorname{Pr}(X \geq 1) \leq 1 / 6 \\
\lim _{n \rightarrow \infty} \operatorname{Pr}(X \geq 1) \geq 1 / 7
\end{gathered}
$$
and that
(Hint: Use the conditional expectation inequality.)

Sagar Singh

Numerade Educator

13:29

Problem 15

Consider the set-balancing problem of Section 4.4. We claim that there is an $n \times n$ matrix $\mathbf{A}$ for which $\|\mathbf{A} \bar{b}\|_{\infty}$ is $\Omega(\sqrt{n})$ for any choice of $\bar{b}$. For convenience here we assume that $n$ is even.
(a) We have shown in Eqn. (5.5) that
$$
n ! \leq \mathrm{e} \sqrt{n}\left(\frac{n}{\mathrm{e}}\right)^{n}
$$
Using similar ideas, show that
$$
n ! \geq a \sqrt{n}\left(\frac{n}{\mathrm{e}}\right)^{n}
$$
for some positive constant $a$.
(b) Let $b_{1}, b_{2}, \ldots, b_{m / 2}$ all equal 1 , and let $b_{m / 2+1}, b_{m / 2+2}, \ldots, b_{m}$ all equal $-1$. Let $Y_{1}, Y_{2}, \ldots, Y_{m}$ each be chosen independently and uniformly at random from $\{0,1\}$. Show that there exists a positive constant $c$ such that, for sufficiently large $m$,
$$
\operatorname{Pr}\left(\left|\sum_{i=1}^{m} b_{i} Y_{t}\right|>c \sqrt{m}\right)>\frac{1}{2} \text {. }
$$
(Hint: Condition on the number of $Y_{i}$ that are equal to 1 .)
(c) Let $b_{1}, b_{2}, \ldots, b_{m}$ each be equal to either 1 or $-1$. Let $Y_{1}, Y_{2}, \ldots, Y_{m}$ each be chosen independently and uniformly at random from $\{0,1\}$. Show that there exists a positive constant $c$ such that, for sufficiently large $m$,
$$
\operatorname{Pr}\left(\left|\sum_{i=1}^{m} b_{i} Y_{i}\right|>c \sqrt{m}\right)>\frac{1}{2} \text {. }
$$
(d) Prove that there exists a matrix $\mathbf{A}$ for which $\|\mathbf{A} \bar{b}\|_{\infty}$ is $\Omega(\sqrt{n})$ for any choice of $\bar{b}$.

Anthony Ramos

Numerade Educator

03:41

Problem 16

Use the Lovasz local lemma to show that, if
$$
4\left(\begin{array}{l}
k \\
2
\end{array}\right)\left(\begin{array}{c}
n \\
k-2
\end{array}\right) 2^{1-\left(\begin{array}{l}
k \\
2
\end{array}\right)} \leq 1
$$
then it is possible to color the edges of $K_{n}$ with two colors so that it has no monochromatic $K_{k}$ subgraph.

Nick Johnson

Numerade Educator

03:21

Problem 17

Use the general form of the Lovasz local lemma to prove that the symmetric version of Theorem $6.11$ can be improved by replacing the condition $4 d p \leq 1$ by the weaker condition e $p(d+1) \leq 1$.

Jacquelyn Trost

Numerade Educator

02:31

Problem 18

Let $G=(V, E)$ be an undirected graph and suppose each $v \in V$ is associated with a set $S(v)$ of $8 r$ colors, where $r \geq 1$. Suppose, in addition, that for each $v \in V$ and $c \in S(v)$ there are at most $r$ neighbors $u$ of $v$ such that $c$ lies in $S(u) .$ Prove that there is a proper coloring of $G$ assigning to each vertex $v$ a color from its class $S(v)$ such that, for any edge $(u, v) \in E$, the colors assigned to $u$ and $v$ are different. You may want to let $A_{u, v, c}$ be the event that $u$ and $v$ are both colored with color $c$ and then consider the family of such events.

Nick Johnson

Numerade Educator