CS 573: Graduate Algorithms, Fall 2008
Homework 1
Due at 11:59:59pm, Wednesday, September 17, 2008
For this and all future homeworks, groups of up to three students may submit a single,
common solution. Please neatly print (or typeset) the full name, NetID, and alias (if any) of
every group member on the first page of your submission.
1.
Two graphs are said to be
isomorphic
if one can be transformed into the other just by relabeling the
vertices. For example, the graphs shown below are isomorphic; the left graph can be transformed
into the right graph by the relabeling (1, 2, 3, 4, 5, 6, 7)7→ (c,g,b,e,a,f,d).
1 2
345
6 7
c
g
b
e
f
a
d
Two isomorphic graphs.
Consider the following related decision problems:
•
GRA PHIS OM OR PH IS M: Given two graphs
G
and
H
, determine whether
G
and
H
are isomorphic.
•
EVENGRAP HIS OM OR PH IS M: Given two graphs
G
and
H
, such that every vertex in
G
and
H
has
even degree, determine whether Gand Hare isomorphic.
•
SUBGRAPHISOM OR PH IS M: Given two graphs
G
and
H
, determine whether
G
is isomorphic to a
subgraph of H.
(a) Describe a polynomial-time reduction from EVENGRA PH ISO MO RP HI SM to GR AP HIS OM OR PH IS M.
(b) Describe a polynomial-time reduction from GRA PH ISO MO RP HI SM to EVENGR AP HIS OM OR PH IS M.
(c) Describe a polynomial-time reduction from GRA PH ISO MO RP HI SM to SU B GR AP HISO MO RP HI SM .
(d) Prove that SUB G RA PHIS OM OR PH IS M is NP-complete.
(e) What can you conclude about the NP-hardness of GRAP HISO MO RP HI SM ? Justify your answer.
[Hint: These are all easy!]
2. (a)
Atonian path in a graph
G
is a path that goes through at least half of the vertices of
G
. Show
that determining whether a graph has a tonian path is NP-complete.
(b)
Atonian cycle in a graph
G
is a cycle that goes through at least half of the vertices of
G
. Show
that determining whether a graph has a tonian cycle is NP-complete. [Hint: Use part (a).]
3. The following variant of 3SAT is called either EXA CT 3SAT or 1IN 3SAT, depending on who you ask.
Given a boolean formula in conjunctive normal form with 3 literals per clause, is there
an assignment that makes exactly one literal in each clause TRU E?
Prove that this problem is NP-complete.
