A representation of an arbitrary system of strict linear inequalities in R^n as a system of points is proposed. The representation is obtained by using a so-called polarity. Based on this representation an algorithm for constructing a committee solution of an inconsistent plane system of linear inequalities is given. A solution of two problems on minimal committee of a plane system is proposed. The obtained solutions to these problems can be found by means of the proposed algorithm.
Deep Dive into Minimal Committee Problem for Inconsistent Systems of Linear Inequalities on the Plane.
A representation of an arbitrary system of strict linear inequalities in R^n as a system of points is proposed. The representation is obtained by using a so-called polarity. Based on this representation an algorithm for constructing a committee solution of an inconsistent plane system of linear inequalities is given. A solution of two problems on minimal committee of a plane system is proposed. The obtained solutions to these problems can be found by means of the proposed algorithm.
arXiv:0802.1514v3 [cs.DM] 15 Feb 2008
Minimal
Committee
Pr
oblem
f
or
In onsistent
Systems
of
Linear
Inequalities
on
the
Plane
K.
S.
K
ob
ylkin
Institute
of
Mathemati s
and
Me
hani s,
Ural
Bran
h,
Russian
A
adem
y
of
S ien es,
ul.
S.
K
o
v
alevsk
oi
16,
Ek
aterin
burg,
620219
Russia
e-mail:
kobylkinks gmail
. o
m
Abstra t.
A
represen
tation
of
an
arbitrary
system
of
stri t
linear
inequalities
in
I
Rn
as
a
system
of
p
oin
ts
is
prop
osed.
The
represen
tation
is
obtained
b
y
using
a
so- alled
p
olarit
y
.
Based
on
this
represen
tation
an
algorithm
for
onstru ting
a
ommittee
solution
of
an
in onsisten
t
plane
system
of
linear
inequalities
is
giv
en.
A
solution
of
t
w
o
problems
on
minimal
ommittee
of
a
plane
system
is
prop
osed.
The
obtained
solutions
to
these
problems
an
b
e
found
b
y
means
of
the
prop
osed
algorithm.
DOI:
10.1134/S1054661
80
60
40
20
1
1
INTR
ODUCTION
A
problem
of
t
w
o
nite
sets
separation
with
sev
eral
(p
ossibly
small
n
um
b
er
of
)
h
yp
erplanes
is
onsidered.
There
are
dieren
t
w
a
ys
in
whi
h
t
w
o
subsets
in
I
Rn
an
b
e
separated.
W
e
fo
us
on
the
one
that
in
v
olv
es
ma
jorit
y
v
oting
prin iple.
Let A
and B
b
e
arbitrary
subsets
of
I
Rn,
where n
is
an
arbitrary
p
ositiv
e
in
teger.
De nition
1.
A
ommitte
e
of
line
ar
fun tions
[1℄,
whi
h
separates
t
w
o
subsets
of
p
oin
ts A
and B,
is
a
nite
olle tion
(with
p
ossible
rep
etitions)
of
linear
fun tions
su
h
that
at
an
y
p
oin
t
of A
(resp
e tiv
ely
of B
)
more
than
half
of
fun tions
of
this
olle tion
are
p
ositiv
e
(resp
e tiv
ely
negativ
e)
oun
ting
m
ultipli it
y
.
It
generalizes
separating
h
yp
erplane
notion
in
the
ase
where
the
subsets
are
inseparable,
i.e.
if
they
are
linearly
separable
b
y
a
h
yp
erplane (w, x) = α
the
set {f(·) = (w, ·) −α}
of
the
one
fun tion
is
a
ommittee.
Separating
ommittee
notion
has
lear
geometri al
in
terpretation
(Fig.
1).
T
w
o
linearly
inseparable
sets A
and B
are
sho
wn
b
y
three
p
oin
ts
and
three
rosses
resp
e tiv
ely
.
There
is
a
ommittee
of
three
fun tions
separating
them
whi
h
is
sho
wn
in
the
gure
b
y
three
straigh
t
lines.
F
or
ea
h
line
w
e
ha
v
e
its
p
ositiv
e
and
negativ
e
sides
de ned
b
y
orresp
onding
half-planes.
Ev
ery
p
oin
t
of
A
(resp
e tiv
ely
,
of B)
is
on
tained
in
in
terse tion
of
t
w
o
p
ositiv
e
(resp
e tiv
ely
,
negativ
e)
half-planes
among
the
three.
Finding
a
ommittee
of q0
fun tions
that
separates A
and B
giv
en
p
ositiv
e
in
teger q0
is
equiv
alen
t
to
learning
of
a
parti ular
t
yp
e
of
t
w
o
la
y
er
p
er eptrons.
Consider
the
p
er eptron
(Fig.
2)
whose
neurons
are
threshold
units
with
a
single
neuron
in
output
la
y
er
whi
h
sums
up
the
outputs
of q0
neurons
of
one
hidden
1
Ðèñ.
1:
la
y
er
where
training
set
is
giv
en
b
y
the
subsets A
and B.
Here
the i th
hidden
neuron
omputes
the
threshold
fun tion zi(·) = sgn((wi, ·) −αi), i = 1, . . . , q0,
and
the
output
neuron
realizes
the
threshold
fun tion g(z) = sgn(
q0
P
i=1
zi),
where
sgn x =
1,
x < 0,
0,
x = 0,
1,
x > 0.
Input
la
y
er
onsists
of n
no
des x1, . . . , xn ∈
I
R,
and
ea
h
of
them
is
onne ted
to
ev
ery
hidden
neuron.
Then {fi(·) = (wi, ·)−αi}q0
i=1
is
a
separating
ommittee
i
orresp
onding
p
er eptron’s
output
is +1
at
an
y
p
oin
t
of A,
and −1
at
an
y
p
oin
t
of B.
Th
us,
the
problem
is
redu ed
to
training
of
the
p
er eptron
that
implies
adjusting
its
hidden
neurons
w
eigh
ts {wi}q
i=1
and
thresholds {αi}q
i=1.
Ðèñ.
2:
De nition
2.
A
ommitte
e
[1℄
of
a
system
of
stri t
linear
inequalities
(cj, x) > bj, j = 1, . . . , m, cj, x ∈
I
Rn, bj ∈
I
R
(1)
is
a
nite
olle tion
(with
p
ossible
rep
etitions)
of
v
e tors
of
I
Rn
su
h
that
ea
h
inequalit
y
of
the
system
is
satis ed
b
y
more
than
half
of
mem
b
ers
of
2
the
olle tion.
A
ommittee
with
the
minimal
n
um
b
er
of
elemen
ts
(taking
in
to
a oun
t
their
m
ultipli it
y)
for
a
giv
en
system (1)
is
alled
a
minimal
ommitte
e.
Ev
ery
ommittee
[1℄
that
separates
subsets A
and B
an
b
e
easily
transformed
to
a
ommittee
of
a
system
(c, z) > 0, c ∈A′ ∪(−B′), z ∈
I
Rn+1,
(2)
where A′ = {[a, 1] : a ∈A}
and B′ = {[b, 1] : b ∈B}.
The
statemen
t
extends
to
the
ase
of
nonhomogeneous
system (1)
as
follo
ws.
Supp
ose
that bj ̸= 0,
j = 1, . . . , m.
A
olle tion {xi}q
i=1 ⊂
I
Rn
is
a
ommittee
of (1)
i
the
olle tion
{fi(·) = (xi, ·) −1}q
i=1
is
a
ommittee
separating
the
subsets A =
n
cj
bj : bj > 0
o
and B =
n
cj
bj : bj < 0
o
∪{0}.
Th
us,
nding
a
ommittee
of
the
system (1)
is
redu ed
to
onstru ting
a
olle tion
of
linear
fun tions
with
p
ositiv
e
onstan
t
terms
su
h
that
at
an
y
p
oin
t
of A
(resp
e tiv
ely
of B
)
more
than
half
of
fun tions
of
this
olle tion
are
p
ositiv
e
(resp
e tiv
ely
negativ
e).
W
e
fo
us
on
the
problem
Problem.
Find
a
minimal
ommittee
of
the
system (1).
In
the
ase
of
a
system
of
stri t
homogeneous
linear
inequalities
on
the
plane,
this
problem
is
ompletely
solv
ed
[1℄.
The
ase
of
nonhomogeneous
plane
systems,
as
w
ell
as
the
ase
of
homogeneous
systems
for n > 2,
remains
unsolv
ed.
Moreo
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