1
The Set Cover Problem
In the U.S. navy, the SEALS are each specially trained in
a wide variety of skills so that small teams can handle a
multitude of missions.
If there are
k
different skills needed for a mission, and
n
SEAL members that can be assigned to the team, find the
smallest team that will cover all of the required skills.
Andersen
knows
hand-to-hand
,
first aid
, and
camouflage
Butler
knows
hand-to-hand
and
snares
Cunningham
knows
hand-to-hand
Douglas
knows
hand-to-hand
,
sniping
,
diplomacy
, and
snares
Eckers
knows
first-aid
,
sniping
, and
diplomacy
Minimum Set Cover
Problem:
Given a set
S
of subsets {
S
1
,
S
2
, …,
S
m
} out of
a universal set
U
= {
u
1
,
u
2
, …,
u
n
} and an integer
k
, is it
possible to choose only
k
subsets of
S
such that the union
of these subsets is
U
.
Theorem:
Minimum Set Cover is NP-complete.
Proof:
MSC is in NP - given a subset of sets, we can
count them, and show that all elements of
U
are included.
What problem should we choose to reduce this time?
Hamiltonian Cycle
Problem
: Given a graph G, does it contain a cycle that
includes all of the vertices in G?
Theorem:
Hamiltonian Cycle is NP-complete.
Proof:
Hamiltonian cycle is in NP - given an ordering
on the vertices, we can show that and edge connecting
each consecutive pair, and then the final vertex
connecting back to the first
We now have some graph problems to work with, but
how can they really help us with this problem?