Transcript Homework 2 Problem 1 Families 1…..N go out for dinner together.

Homework 2
Problem 1
Families 1…..N go out for dinner together. To increase
their social interaction, no two members of the same
family use the same table. Family j has a(j) members.
There are M tables. Table j can seat b(j) people. Find a
valid seating assignment if one exists.
Problem 2
A commander is located at one node p in a network.
His subordinates constitute a node set S. The enemy
needs to cut off the communication between the
commander and his subordinates (commander
should not be able to communicate to any of his
subordinates). Enemy needs w(e) effort to remove
an edge e in the network. Compute the minimum
effort required to cut off the communication
between the commander and his subordinates.
Problem 3
A network has some infinite capacity edges. Replace
these capacities by finite numbers such that the
maximum flow between any source and destination is
not reduced.
Problem 4
Prove or disprove (give counter examples) for the
following:
1. For any maximum flow allocation, for all pairs (u, v)
either flow in edge (u, v) or flow in (v, u) must be 0
2. There exists one maximum flow for which for all pairs
(u, v) either flow in edge (u, v) or flow in (v, u) is 0
3. If all edges have unique capacities, the network has a
unique minimum cut.
4. If we add a positive number b to the capacity of every
edge, the minimum cut remains the same.
Problem 5
Consider a network with a source and a destination. An
edge is called upward critical if increasing the capacity of
an edge increases the maximum flow. Does all networks
have an upward critical edge?
An edge is called downward critical if decreasing the
capacity of the edge decreases the maximum flow. Does
all networks have a downward critical edge?
Justify your answer in both cases.
Problem 6
Consider a network with a source and a destination.
Links have integer edge capacities. Suppose a
maximum flow allocation is known.
Increase the capacity of an edge by 1 unit. Give an
algorithm for updating the maximum flow. (Your
algorithm should be substantially faster than max flow
computation capacity).
Problem 7
Consider a convex function f(x) from RM to R (x is an M
dimensional vector). Prove that f(x) q is a convex set.
Consider (1,…… N) such that j j = 1, j 0. Prove that
ixi  j j xi (Hint: -ln x is a convex function).
Problem 8
The definition for maxmin fairness can be generalized
as follows. A vector (not necessarily rate allocation
vector) is maxmin fair in a feasible set if none of its
components can be increased without hurting an equal
or lower component.
Consider a feasible set: r  1. Is there a maxmin fair
vector in this feasible set? (Here, any vector consists of
a single component).
What is the answer for the feasible set r  1?
How about the feasible set with x  [0, 0.2] [0.8, 1]
and y = 1-x. Here, you are considering vectors with 2
components (x, y).
Problem 9
There are M faculties and N courses. Every faculty ranks 2
courses in order of preference. A faculty can teach one
course and a course can be taught by one faculty only.
Find a feasible course allocation if one exists (A feasible
course allocation allows a faculty to teach one of the two
courses he prefers).
Find a k-feasible course allocation if one exists. (A faculty
is dissatisfied if he is allotted his second choice course. A
k-feasible allocation is one which dissatisfies at most k
faculties).
Problem 10
Consider a complete bipartite graph with weighted edges (There
is an edge between any pair (u, v) if u and v belong to different
partitions). The number of vertices in both partitions are the
same. There exists a matching which matches all vertices in any
such graph. Such a matching is called perfect matching. A
perfect matching is lexicographically greater than another if the
minimum weight in the matching is strictly less than that in the
other, or if the minimum weights are identical but the second
minimum is greater, etc….Find a lexicographically greatest
matching. (Hint: Use weighted matching. You may assume a
subroutine for computing a maximum weighted matching, and
don’t need to present an algorithm for the same.).
Analyze the complexity of your algorithm.