CS 473G: Combinatorial Algorithms, Fall 2005
Homework 1
Due Tuesday, September 13, 2005, by midnight (11:59:59pm CDT)
Name:
Net ID:
Alias:
Name:
Net ID:
Alias:
Name:
Net ID:
Alias:
Starting with Homework 1, homeworks may be done in teams of up to three people. Each team
turns in just one solution, and every member of a team gets the same grade.
Neatly print your name(s), NetID(s), and the alias(es) you used for Homework 0 in the boxes
above. Staple this sheet to the top of your answer to problem 1.
There are two steps required to prove NPcompleteness: (1) Prove that the problem is in NP,
by describing a polynomialtime verification algorithm.
(2) Prove that the problem is NPhard,
by describing a polynomialtime reduction from some other NPhard problem. Showing that the
reduction is correct requires proving an ifandonlyif statement; don’t forget to prove both the “if”
part and the “only if” part.
Required Problems
1. Some NPComplete problems
(a) Show that the problem of deciding whether one graph is a subgraph of another is NP
complete.
(b) Given a boolean circuit that embeds in the plane so that no 2 wires cross,
PlanarCir
cuitSat
is the problem of determining if there is a boolean assignment to the inputs
that makes the circuit output true. Prove that
PlanarCircuitSat
is NPComplete.
(c) Given a set
S
with 3
n
numbers,
3partition
is the problem of determining if
S
can be
partitioned into
n
disjoint subsets, each with 3 elements, so that every subset sums to the
same value. Given a set
S
and a collection of three element subsets of
S
, X3M (or
exact
3dimensional matching
) is the problem of determining whether there is a subcollection
of
n
disjoint triples that exactly cover
S
.
Describe a polynomialtime reduction from
3partition
to X3M.
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CS 473G
Homework 1 (due September 13, 2005)
Fall 2005
(d) A
domino
is a 1
×
2 rectangle divided into two squares, each of which is labeled with an
integer.
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 Spring '11
 Smith
 Graph Theory, Computational complexity theory, NPcomplete, Polynomialtime reduction

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