Algorithms for Automated Floor Planning
Felix Klesen
v1 v2 v5
v6
e1
e2 e4
e5 e6
v1 v2 v5
v6
e1
e2 e4
e5 e6
w(e1) l(e2)
v1 v2 v5
v6
e1
e2 e4
e5 e6
w(e1) l(e2)
g
(
r, e) =
(1 α
≥
s(r)w(e)2
≥
1α
0 otherwise
v1 v2 v5
v6
e1
e2 e4
e5 e6
w(e1) l(e2)
v1 e1
e2 g
(
r, e) =
(1 α
≥
s(r)w(e)2
≥
1α
0 otherwise
v1 v2 v5
v6
e1
e2 e4
e5 e6
w(e1) l(e2)
v1 e1
e2
≥
d g(
r, e) =
(1 α
≥
s(r)w(e)2
≥
1α
0 otherwise
v1 v2 v5
v6
e1
e2 e4
e5 e6
w(e1) l(e2)
v1 e1
e2
≥
d g(
r, e) =
(1 α
≥
s(r)w(e)2
≥
1α
0 otherwise q
(
r, e, v) =
(1 s
(
r) ≥
s(
v) +
w(
e) ·
d 0 otherwiseFloor-planning is NP-hard.
S
= {
6, 5, 4, 2, 1}
Floor-planning is NP-hard.
S
= {
6, 5, 4, 2, 1}
|
S|
:=
∑s∈S sFloor-planning is NP-hard.
|S|/2 + 1
|S|/2 + 1
|S|/2 + 1
|S|/2 + 1
|S|/2
|S|/2
S
= {
6, 5, 4, 2, 1}
|
S|
:=
∑s∈S sFloor-planning is NP-hard.
S
= {
6, 5, 4, 2, 1}
|
S|
:=
∑s∈S s5 4
1 2 6
|S|/2 + 1
|S|/2 + 1
|S|/2 + 1
|S|/2 + 1
Floor-planning is NP-hard.
xr,e
∈ {
0, 1} ∀
r∈
R, e∈
Exr,e
∈ {
0, 1} ∀
r∈
R, e∈
Eyr,e,v
∈ {
0, 1} ∀
r∈
R, e∈
E, v∈
N(
e)
xr,e
∈ {
0, 1} ∀
r∈
R, e∈
Eyr,e,v
∈ {
0, 1} ∀
r∈
R, e∈
E, v∈
N(
e)
∑e∈E
xr,e
+
∑v∈N(e) yr,e,v=
1∀
r∈
Rxr,e
∈ {
0, 1} ∀
r∈
R, e∈
Eyr,e,v
∈ {
0, 1} ∀
r∈
R, e∈
E, v∈
N(
e)
∑e∈E
xr,e
+
∑v∈N(e) yr,e,v=
1∀
r∈
Rxr,e
+
∑v∈N(e) yr,e,v≤
g(
r, e) ∀
r∈
R, e∈
Exr,e
∈ {
0, 1} ∀
r∈
R, e∈
Eyr,e,v
∈ {
0, 1} ∀
r∈
R, e∈
E, v∈
N(
e)
∑e∈E
xr,e
+
∑v∈N(e) yr,e,v=
1∀
r∈
Rxr,e
+
∑v∈N(e) yr,e,v≤
g(
r, e) ∀
r∈
R, e∈
E∑r∈R ∑e∈N(v) yr,e,v
≤
1∀
v∈
Vxr,e
∈ {
0, 1} ∀
r∈
R, e∈
Eyr,e,v
∈ {
0, 1} ∀
r∈
R, e∈
E, v∈
N(
e)
∑e∈E
xr,e
+
∑v∈N(e) yr,e,v=
1∀
r∈
Rxr,e
+
∑v∈N(e) yr,e,v≤
g(
r, e) ∀
r∈
R, e∈
E∑r∈R ∑e∈N(v) yr,e,v
≤
1∀
v∈
Vyr,e,v
≤
q(
r, e, v) ∀
r∈
R, e∈
E, v∈
N(
e)
xr,e
∈ {
0, 1} ∀
r∈
R, e∈
Eyr,e,v
∈ {
0, 1} ∀
r∈
R, e∈
E, v∈
N(
e)
∑e∈E
xr,e
+
∑v∈N(e) yr,e,v=
1∀
r∈
Rxr,e
+
∑v∈N(e) yr,e,v≤
g(
r, e) ∀
r∈
R, e∈
E∑r∈R ∑e∈N(v) yr,e,v
≤
1∀
v∈
Vyr,e,v
≤
q(
r, e, v) ∀
r∈
R, e∈
E, v∈
N(
e)
∑r∈R
xr,e
·
s(
r) +
∑v∈N(e) yr,e,v·
s(
r) −
s(
v)
≤
s(
e) ∀
e∈
E
xr,e
+
∑v∈N(e) yr,e,v·
γ(
r, c) ≤
ze,c∀
r∈
R, e∈
E, c∈
C
xr,e
+
∑v∈N(e) yr,e,v·
γ(
r, c) ≤
ze,c∀
r∈
R, e∈
E, c∈
C minimize ∑c∈C ∑e∈E ze,c
xr,e
+
∑v∈N(e) yr,e,v·
γ(
r, c) ≤
ze,c∀
r∈
R, e∈
E, c∈
C minimize ∑c∈C ∑e∈E ze,czv,c
∈ {
0, 1} ∀
v∈
V, c∈
Czv,c
∈ {
0, 1} ∀
v∈
V, c∈
Cxr,e
·
γ(
r, c) ≤
ze,c∀
r∈
R, e∈
E, c∈
Czv,c
∈ {
0, 1} ∀
v∈
V, c∈
Cxr,e
·
γ(
r, c) ≤
ze,c∀
r∈
R, e∈
E, c∈
Cyr,e,v
·
γ(
r, c) ≤
zv,c∀
r∈
R, v∈
V, e∈
N(
v)
, c∈
Czv,c
∈ {
0, 1} ∀
v∈
V, c∈
Cxr,e
·
γ(
r, c) ≤
ze,c∀
r∈
R, e∈
E, c∈
Cyr,e,v
·
γ(
r, c) ≤
zv,c∀
r∈
R, v∈
V, e∈
N(
v)
, c∈
C minimize ∑c∈C ∑e1∈E
∑e2∈E ze1,c
·
ze2,c·
δ(
e1, e2) +
∑v∈V ze1,c·
zv,c·
δ(
e1, v)
zv,c
∈ {
0, 1} ∀
v∈
V, c∈
Cxr,e
·
γ(
r, c) ≤
ze,c∀
r∈
R, e∈
E, c∈
Cyr,e,v
·
γ(
r, c) ≤
zv,c∀
r∈
R, v∈
V, e∈
N(
v)
, c∈
C minimize ∑c∈C ∑e1∈E
∑e2∈E ze1,c
·
ze2,c·
δ(
e1, e2) +
∑v∈V ze1,c·
zv,c·
δ(
e1, v)
zv,c
∈ {
0, 1} ∀
v∈
V, c∈
Cxr,e
·
γ(
r, c) ≤
ze,c∀
r∈
R, e∈
E, c∈
Cyr,e,v
·
γ(
r, c) ≤
zv,c∀
r∈
R, v∈
V, e∈
N(
v)
, c∈
C minimize ∑c∈C ∑e1∈E
∑e2∈E ze1,c
·
ze2,c·
δ(
e1, e2) +
∑v∈V ze1,c·
zv,c·
δ(
e1, v)
ue1,e2,c
∈ {
0, 1} ∀
e1∈
E, e2∈
E, c∈
Czv,c
∈ {
0, 1} ∀
v∈
V, c∈
Cxr,e
·
γ(
r, c) ≤
ze,c∀
r∈
R, e∈
E, c∈
Cyr,e,v
·
γ(
r, c) ≤
zv,c∀
r∈
R, v∈
V, e∈
N(
v)
, c∈
C minimize ∑c∈C ∑e1∈E
∑e2∈E ze1,c
·
ze2,c·
δ(
e1, e2) +
∑v∈V ze1,c·
zv,c·
δ(
e1, v)
ue1,e2,c
∈ {
0, 1} ∀
e1∈
E, e2∈
E, c∈
C ue1,e2,c+
1≥
ze1,c+
ze2,c∀
e1∈
E, e2∈
E, c∈
Cminimize cost
(
a, γ) =
∑c∈C ∑f∈F xc,fminimize cost
(
a, γ) =
∑c∈C ∑f∈F xc,fxc,f
=
1 Wr∈R
a
(
r) =
f∧
γ(
r) =
c0 otherwise
minimize cost
(
a, γ) =
∑c∈C ∑f∈F xc,fxc,f
=
1 Wr∈R
a
(
r) =
f∧
γ(
r) =
c0 otherwise
Room-assignment is NP-hard to approximate.
minimize cost
(
a, γ) =
∑c∈C ∑f∈F xc,fminimize cost
(
a, γ) =
∑c∈C ∑f∈F xc,fxc,f
=
(1 a
(
c, f) ≥
1 0 otherwiseminimize cost
(
a, γ) =
∑c∈C ∑f∈F xc,fxc,f
=
(1 a
(
c, f) ≥
1 0 otherwiseSolving area-distribution is NP-hard.
For any instance of area-distribution there is an optimal solution admitting a nice sequence.
For any instance of area-distribution there is an optimal solution admitting a nice sequence.
For any instance of area-distribution we can compute a nice sequence in O
(|
C| + |
F|)
time.For any instance of area-distribution there is an optimal solution admitting a nice sequence.
For any instance of area-distribution we can compute a nice sequence in O
(|
C| + |
F|)
time.For any instance of area-distribution the solution represented by any nice sequence is a 2-approximation.
For any instance of area-distribution there is an optimal solution admitting a nice sequence.
For any instance of area-distribution we can compute a nice sequence in O
(|
C| + |
F|)
time.For any instance of area-distribution the solution represented by any nice sequence is a 2-approximation.
minimize cost
(
a, δ) =
∑c∈C 1+
maxf1,f2∈F xc,f1·
xc,f2·
δ(
f1, f2)
minimize cost
(
a, δ) =
∑c∈C 1+
maxf1,f2∈F xc,f1·
xc,f2·
δ(
f1, f2)
xc,f=
(1 a
(
c, f) ≥
1 0 otherwiseminimize cost
(
a, δ) =
∑c∈C 1+
maxf1,f2∈F xc,f1·
xc,f2·
δ(
f1, f2)
xc,f=
(1 a
(
c, f) ≥
1 0 otherwiseAny optimal solution for an instance of area-distribution in single building can be represented by a nice sequence.
0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6
0 1 2 3 4 5 6 7 8
