CSci 5403, Spring 2010
Brian Berzins
Derived with: Stefan NelsonLindall, Hannah Jaber
Homework 1.3 Solution
(a) Show that
quadeq
is
NP
complete.
First: We show that
quadeq
∈
NP
by demonstrating that there is a
polynomial time verifier for it. One such verifier takes the quadratic sys
tem and a variable assignment
~x
∈ {
0
,
1
}
n
and checks if each quadratic
equation in the system evaluates to true. This requires
m
·
n
2
(polyno
mial) time. Therefore
quadeq
∈
NP
.
Second: We show that
quadeq
is
NP
hard by a reduction from
3sat
.
Let
φ
be a
3sat
boolean formula. Construct the reduction as follows:
for each clause let the three literals be
a
,
b
, and
c
. Add the following
equation to the system of quadratics:
az
1
+
bz
2
+
cz
3
= 1 mod 2
where
z
1
,
z
2
and
z
3
are variables that do not appear in any previous
equation in our system. The variables
a
,
b
, and
c
represent the appro
priate literals in the clause and remain consistent across all clauses in
φ
. If a negated literal
x
appears, we substitute (
x
+1) into this equation
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 Spring '08
 Sturtivant,C
 Equations, Quadratic equation, Elementary algebra, clause

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