473: CS Fundamental Algorithms, Spring 2012
Homework 8 (due Tuesday, 23:55:00, April 3, 2012)
Collaboration Policy & submission guidelines: See homework 1.
Each student individually have to also do quiz 8 online.
Version: 1.3
1. (35 pts.) Recomputing correct ow.
Given a ow network G with integer capacities you have computed a maximum ow f between
s and t. However you have made a mistake in noting the capacity of an edge e.
(15 pts) Suppose you under estimated the capacity of e by k > 0 units. Show that you
can compute the correct maximum ow in O(km) time using the current ow f .
(20 pts) Do the same as above if you over estimated the capacity of e by k > 0 units.
Hint: First assume that f is acyclic. How do you reduce ow on e?
2. (35 pts.) Special minimum cuts.
Let G = (V, E ) be a ow network with integer edge capacities. We have seen algorithms that
compute a minimum s-t cut. For both problems below assume that you only have black box
access to an algorithm that given G and nodes s, t outputs a minimum cut between s and t.
(15 pts) Given G and s, t describe an algorithm that computes a minimum cut with a
minimum number of edges.
(20 pts) Given G and s, t describe an algorithm that decides whether G has a unique
minimum s-t cut.
3. (30 pts.) Committee Assignment.
The Computer Science Department at UIUC has n professors. They handle department
duties by taking part in various committees. There are m committees and the j th committee
requires kj professors. The head of the department asked each professor to volunteer for set
of a committees. Let Si {1, 2, . . . , m} be the set of committees that professor i has volunteered
for. A committee assignment consists of sets S1 , S2 , . . . , Sn where Si {1, 2, . . . , m} is the
set of committees that professor i will participate in. A valid committee assignment has
to satisfy two constraints: (i) for each professor i, Si Si , that is each professor is only
given committees that he/she has volunteered for, and (ii) each committee j has kj professors
assigned to it, or in other words j occurs in at least kj of the sets S1 , S2 , . . . , Sn .
(a) Describe a polynomial time algorithm that the head of the department can employ
to check if there is a valid committee assignment given the lists S1 , S2 , . . . , Sn . The
algorithm should output a valid assignment if there is one.
(b) The head of the department notices that often there is no valid committee assignment
because professors naturally are inclined to volunteer for as few committees as possible.
To overcome this, the denition of a valid assignment is relaxed as follows. Let be
some integer. An assignment S1 , S2 , . . . , Sn is now said to be valid if (i) |Si Si |
and (ii) each committee j has kj professors assigned to it. The new condition (i) means
that a professor i may be assigned up to committees not on the list Si that he/she
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volunteered for. Describe an algorithm to check if there is a valid committee assignment
with the relaxed denition.
(b) is a more general problem than (a) so you can do (b) for full points or (a) only for 15 pts.
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