CS180 Homework 8


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1. Minimum cut of a graph is the minimum over all pairs of vertices of the minimum cut for the
pair. Let m be the number of edges and n be the number of vertices, give an O(m2n) algorithm
that finds the minimum cut of a directed strongly connected unweighted graph. (Hint: do just n
max-flows and argue that the max-flow algorithm we have seen in class that works for integer
capacities will cost at most O(m2
) time).
2. We have learned that the minimum number of edges that need to be removed to disconnect a
pair s and t of vertices equals the maximum number of edge disjoint paths between s and t.
Can we make a similar statement about vertices? Specifically, is the following statement true or
false: the minimum number of vertices that need to be removed to disconnect s and t equals the
maximum number of vertex disjoint paths between s and t. Either prove the correctness or give
a counter-example disproving the statement.
3. Take a standard deck of cards, and deal them out into 13 piles of 4 cards each. Show that it is
always possible to select exactly 1 card from each pile, such that the 13 selected cards contain
exactly one card of each rank (Ace, 2, 3, . . . , Queen, King).
4. UCLA hires you to write an algorithm to schedule their final exams. Each quarter, UCLA offers
n different classes. There are r different rooms on campus and t different time slots in which
exams can be offered. You are given two arrays E[1..n] and S[1..r], where E[i] is the number of
students enrolled in the i
th class, and S[ j] is the number of seats in the j
th room. At most one
final exam can be held in each room during each time slot. Class i can hold its final exam in room
j only if E[i] < S[ j]. Describe and analyze an efficient algorithm to assign a room and a time slot
to each class (or report correctly that no such assignment is possible). Formulate the problem as
max-flow problem and argue the correctness of your construction.
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