Polynomiality for Bin Packing with
a Constant Number of Item Types
Michel X. Goemans & Thomas Rothvoß
MIP 2013, Madison
Bin Packing / Cutting Stock
Input:
item 1 . . . item d
Bin Packing / Cutting Stock
Input:
◮ Item sizes s1 , . . . , sd ∈ [0, 1]
Input:
s1
item 1 . . . item d
Bin Packing / Cutting Stock
Input:
◮ Item sizes s1 , . . . , sd ∈ [0, 1]
◮ Multiplicities a1 , . . . , ad ∈ N
Input:
a1 many
s1
item 1 . . . item d
Bin Packing / Cutting Stock
Input:
◮ Item sizes s1 , . . . , sd ∈ [0, 1]
◮ Multiplicities a1 , . . . , ad ∈ N
Goal: Pack items into minimum number of bins of size 1.
Input:
a1 many
s1
item 1 . . . item d
1
Solution:
bin 1 . . .
Bin Packing / Cutting Stock
Input:
◮ Item sizes s1 , . . . , sd ∈ [0, 1]
◮ Multiplicities a1 , . . . , ad ∈ N
Goal: Pack items into minimum number of bins of size 1.
Input:
a1 many
s1
item 1 . . . item d
1
Solution:
bin 1 . . .
Bin Packing / Cutting Stock
Input:
◮ Item sizes s1 , . . . , sd ∈ [0, 1]
◮ Multiplicities a1 , . . . , ad ∈ N
Goal: Pack items into minimum number of bins of size 1.
Input:
a1 many
s1
item 1 . . . item d
1
Solution:
bin 1 . . .
Bin Packing / Cutting Stock
Input:
◮ Item sizes s1 , . . . , sd ∈ [0, 1]
◮ Multiplicities a1 , . . . , ad ∈ N
Goal: Pack items into minimum number of bins of size 1.
Input:
a1 many
s1
item 1 . . . item d
1
Solution:
bin 1 . . .
Polynomial time algorithms
For general d:
◮ NP-hard to distinguish OP T ≤ 2 or OP T ≥ 3
[Garey & Johnson ’79]
Polynomial time algorithms
For general d:
◮ NP-hard to distinguish OP T ≤ 2 or OP T ≥ 3
[Garey & Johnson ’79]
◮ Asymptotic FPTAS
OP T + O(log2 d) [Karmarkar & Karp ’82]
OP T + O(log d · logP
log d) [R. ’13]
(running time poly( di=1 ai ))
Polynomial time algorithms
For general d:
◮ NP-hard to distinguish OP T ≤ 2 or OP T ≥ 3
[Garey & Johnson ’79]
◮ Asymptotic FPTAS
OP T + O(log2 d) [Karmarkar & Karp ’82]
OP T + O(log d · logP
log d) [R. ’13]
(running time poly( di=1 ai ))
◮ ∈ NP [Eisenbrand & Shmonin ’06]
Polynomial time algorithms
For general d:
◮ NP-hard to distinguish OP T ≤ 2 or OP T ≥ 3
[Garey & Johnson ’79]
◮ Asymptotic FPTAS
OP T + O(log2 d) [Karmarkar & Karp ’82]
OP T + O(log d · logP
log d) [R. ’13]
(running time poly( di=1 ai ))
◮ ∈ NP [Eisenbrand & Shmonin ’06]
For constant d:
Polynomial time algorithms
For general d:
◮ NP-hard to distinguish OP T ≤ 2 or OP T ≥ 3
[Garey & Johnson ’79]
◮ Asymptotic FPTAS
OP T + O(log2 d) [Karmarkar & Karp ’82]
OP T + O(log d · logP
log d) [R. ’13]
(running time poly( di=1 ai ))
◮ ∈ NP [Eisenbrand & Shmonin ’06]
For constant d:
◮ Polytime for d = 2 [McCormick, Smallwood, Spieksma ’97]
Polynomial time algorithms
For general d:
◮ NP-hard to distinguish OP T ≤ 2 or OP T ≥ 3
[Garey & Johnson ’79]
◮ Asymptotic FPTAS
OP T + O(log2 d) [Karmarkar & Karp ’82]
OP T + O(log d · logP
log d) [R. ’13]
(running time poly( di=1 ai ))
◮ ∈ NP [Eisenbrand & Shmonin ’06]
For constant d:
◮ Polytime for d = 2 [McCormick, Smallwood, Spieksma ’97]
O(d)
◮
OP T + 1 in time 22
· poly [Jansen & Solis-Oba ’10]
Polynomial time algorithms
For general d:
◮ NP-hard to distinguish OP T ≤ 2 or OP T ≥ 3
[Garey & Johnson ’79]
◮ Asymptotic FPTAS
OP T + O(log2 d) [Karmarkar & Karp ’82]
OP T + O(log d · logP
log d) [R. ’13]
(running time poly( di=1 ai ))
◮ ∈ NP [Eisenbrand & Shmonin ’06]
For constant d:
◮ Polytime for d = 2 [McCormick, Smallwood, Spieksma ’97]
O(d)
◮
OP T + 1 in time 22
· poly [Jansen & Solis-Oba ’10]
Open problem [ES’06, MSS’97, F’07]
Solvable in poly-time for d = 3?
A geometric view
◮
Define P = {x ∈ Rd≥0 | sT x ≤ 1}
a
Rd
b
b
b
b
b
sT x ≤ 1
b
b
b
b
b
b
b
b
b
b
b
b
P
b
b
b
b
b
b
b
b
A geometric view
◮
Define P = {x ∈ Rd≥0 | sT x ≤ 1}
a
Rd
b
b
b
b
b
b
sT x ≤ 1
b
b
b
b
b
b
b
b
b
b
b
1
b
2
b
b
b
P
b
b
b
b
A geometric view
◮
Define P = {x ∈ Rd≥0 | sT x ≤ 1}
a
Rd
b
b
b
b
b
b
sT x ≤ 1
b
b
b
b
b
b
b
b
b
b
b
1
b
2
b
b
b
P
b
b
b
b
A geometric view
◮
Define P = {x ∈ Rd≥0 | sT x ≤ 1}
a
Rd
b
b
b
b
b
b
sT x ≤ 1
b
b
b
b
b
b
b
b
b
b
b
1
b
2
b
b
b
P
b
b
b
b
A geometric view
◮
Define P = {x ∈ Rd≥0 | sT x ≤ 1}
a
Rd
b
b
b
b
b
b
sT x ≤ 1
b
b
b
b
b
b
b
b
b
b
b
1
b
2
b
b
b
P
b
b
b
b
Problems:
◮ Points in P exponentially many
A geometric view
◮
Define P = {x ∈ Rd≥0 | sT x ≤ 1}
a
Rd
b
b
b
b
b
b
sT x ≤ 1
b
b
b
b
b
b
b
b
b
b
b
1
b
2
b
b
b
P
b
b
b
b
Problems:
◮ Points in P exponentially many
◮ Weights can be exponential
Main results
Theorem (Goemans, R. ’13)
Bin Packing with d = O(1) item sizes can be solved in
poly-time.
Solves question by
◮ [McCormick, Smallwood, Spieksma ’97]:
“might be NP-hard for d = 3”
◮ [Eisenbrand & Shmonin ’06]
◮ [Filippi ’07]: “hard open problem for general d”
Main results (2)
◮
P
Def.: int.cone(X) := { x∈X λx · x | λx ∈ Z≥0 }
Main results (2)
◮
P
Def.: int.cone(X) := { x∈X λx · x | λx ∈ Z≥0 }
Theorem (Goemans, R. ’13)
For fixed-dim. polytopes P, Q ⊆ Rd , testing
int.cone(P ∩ Zd ) ∩ Q 6= ∅
O(d)
is doable in poly-time (actually inputlength2
).
b
b
b
Q
b
b
b
P
b
0
b
b
b
b
Main results (2)
◮
P
Def.: int.cone(X) := { x∈X λx · x | λx ∈ Z≥0 }
Theorem (Goemans, R. ’13)
For fixed-dim. polytopes P, Q ⊆ Rd , testing
int.cone(P ∩ Zd ) ∩ Q 6= ∅
O(d)
is doable in poly-time (actually inputlength2
).
b
b
b
Q
b
b
b
P
b
0
b
2
b
b
1
b
Main results (2)
◮
P
Def.: int.cone(X) := { x∈X λx · x | λx ∈ Z≥0 }
Theorem (Goemans, R. ’13)
For fixed-dim. polytopes P, Q ⊆ Rd , testing
int.cone(P ∩ Zd ) ∩ Q 6= ∅
O(d)
is doable in poly-time (actually inputlength2
).
b
b
b
Q
b
b
b
P
b
0
b
2
b
b
1
b
Main results (2)
◮
P
Def.: int.cone(X) := { x∈X λx · x | λx ∈ Z≥0 }
Theorem (Goemans, R. ’13)
For fixed-dim. polytopes P, Q ⊆ Rd , testing
int.cone(P ∩ Zd ) ∩ Q 6= ∅
O(d)
is doable in poly-time (actually inputlength2
).
b
b
b
Q
b
b
b
P
b
0
b
2
b
b
1
b
Main results (2)
◮
P
Def.: int.cone(X) := { x∈X λx · x | λx ∈ Z≥0 }
Theorem (Goemans, R. ’13)
For fixed-dim. polytopes P, Q ⊆ Rd , testing
int.cone(P ∩ Zd ) ∩ Q 6= ∅
O(d)
is doable in poly-time (actually inputlength2
).
b
b
b
Q
b
b
b
P
b
0
◮
b
2
b
b
b
1
For Bin Packing:
P := { x1 | sT x ≤ 1, x ≥ 0} and Q := {
a
OP T
}
Int. conic combinations
Theorem (Eisenbrand & Shmonin ’06)
If P ⊆ Rd convex, then any integer conic combination
X
a=
λx · x
x∈P ∩Zd
d
needs at most 2 points.
P
b
b
b
b
b
b
b
b
b
b
b
b
b
b
Int. conic combinations
Theorem (Eisenbrand & Shmonin ’06)
If P ⊆ Rd convex, then any integer conic combination
X
a=
λx · x
x∈P ∩Zd
d
needs at most 2 points.
◮
Suppose |supp(λ)| > 2d
P
b
b
b
b
b
b
b
b
b
b
b
b
b
b
Int. conic combinations
Theorem (Eisenbrand & Shmonin ’06)
If P ⊆ Rd convex, then any integer conic combination
X
a=
λx · x
x∈P ∩Zd
d
needs at most 2 points.
◮
◮
Suppose |supp(λ)| > 2d
Take points x, y ∈ supp(λ) of
same parity
P
xb
b
b
b
b
b
b
b
b
b
y
b
b
b
b
Int. conic combinations
Theorem (Eisenbrand & Shmonin ’06)
If P ⊆ Rd convex, then any integer conic combination
X
a=
λx · x
x∈P ∩Zd
d
needs at most 2 points.
◮
◮
◮
Suppose |supp(λ)| > 2d
Take points x, y ∈ supp(λ) of
same parity
Midpoint z = 21 (x + y) ∈ P ∩ Zd
P
xb
b
b
b
b
b
zb
b
b
b
y
b
b
b
b
Int. conic combinations
Theorem (Eisenbrand & Shmonin ’06)
If P ⊆ Rd convex, then any integer conic combination
X
a=
λx · x
x∈P ∩Zd
d
needs at most 2 points.
◮
◮
◮
◮
Suppose |supp(λ)| > 2d
Take points x, y ∈ supp(λ) of
same parity
Midpoint z = 21 (x + y) ∈ P ∩ Zd
Move weight from x, y to z
P
xb
b
b
b
b
zb
b
−1
b
b
b
+2
b
b
y
b
−1
b
Int. conic combinations
Theorem (Eisenbrand & Shmonin ’06)
If P ⊆ Rd convex, then any integer conic combination
X
a=
λx · x
x∈P ∩Zd
d
needs at most 2 points.
◮
◮
◮
◮
◮
Suppose |supp(λ)| > 2d
Take points x, y ∈ supp(λ) of
same parity
Midpoint z = 21 (x + y) ∈ P ∩ Zd
Move weight from x, y to z
P
Potential function x λx f (x)
decreases (f strictly convex)
P
xb
b
b
b
b
zb
b
−1
b
b
b
+2
b
b
y
b
−1
b
Int. conic combinations
Theorem (Eisenbrand & Shmonin ’06)
If P ⊆ Rd convex, then any integer conic combination
X
a=
λx · x
x∈P ∩Zd
d
needs at most 2 points.
◮
◮
◮
◮
◮
◮
Suppose |supp(λ)| > 2d
Take points x, y ∈ supp(λ) of
same parity
Midpoint z = 21 (x + y) ∈ P ∩ Zd
Move weight from x, y to z
P
Potential function x λx f (x)
decreases (f strictly convex)
P
xb
b
b
b
b
zb
b
−1
b
b
b
+2
b
Problem: Still don’t know which points to take!
b
y
b
−1
b
Redistributing weight
P
Redistributing weight
x
P
Redistributing weight
◮
Consider parallelepiped Π ∋ x
with integral vertices
x
P
Π
Redistributing weight
◮
◮
Consider parallelepiped Π ∋ x
with integral vertices
Let y vertex of Π, in quadrant of x
y
x
P
Π
Redistributing weight
◮
◮
◮
Consider parallelepiped Π ∋ x
with integral vertices
Let y vertex of Π, in quadrant of x
Let z ∈ Π ∩ Zd be mirrored point
y
Π
x
z
P
Redistributing weight
◮
◮
◮
◮
Consider parallelepiped Π ∋ x
with integral vertices
Let y vertex of Π, in quadrant of x
Let z ∈ Π ∩ Zd be mirrored point
If λx ≥ 2 ⇒ redistribute weight
y
+1
Π
x
−2
+1
P
z
Redistributing weight
◮
◮
◮
◮
◮
Consider parallelepiped Π ∋ x
with integral vertices
Let y vertex of Π, in quadrant of x
Let z ∈ Π ∩ Zd be mirrored point
If λx ≥ 2 ⇒ redistribute weight
At most 2d points left inside Π
Π
1
1
1
P
Redistributing weight
Lemma
For x in parallelepiped Π and λx ∈ N, one can write
X
λx x = int.cone(vertices of Π) +
of 2d points in Π ∩ Zd
◮
◮
◮
◮
◮
Consider parallelepiped Π ∋ x
with integral vertices
Let y vertex of Π, in quadrant of x
Let z ∈ Π ∩ Zd be mirrored point
If λx ≥ 2 ⇒ redistribute weight
At most 2d points left inside Π
Π
1
1
1
P
Covering a polytope with parallelepipeds
Lemma
For fixed-dim P ⊆ Rd , we can cover P ∩ Zd with poly-many
parallelepipeds (with int. vertices and ⊆ P ).
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
P
b
Covering a polytope with parallelepipeds
Lemma
For fixed-dim P ⊆ Rd , we can cover P ∩ Zd with poly-many
parallelepipeds (with int. vertices and ⊆ P ).
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
P
b
Covering a polytope w. parallelep. (2)
P
Covering a polytope w. parallelep. (2)
bi − Ai x = 0
Split P = {x | Ax ≤ b}
into poly many cells
C = {x | αj(i) ≤ Ai x ≤ αj(i)+1 }
◮
P
bi − Ai x
=
(1 + d1 )Z
Covering a polytope w. parallelep. (2)
cell C
Split P = {x | Ax ≤ b}
into poly many cells
C = {x | αj(i) ≤ Ai x ≤ αj(i)+1 }
◮
P
Covering a polytope w. parallelep. (2)
cell C
Split P = {x | Ax ≤ b}
into poly many cells
C = {x | αj(i) ≤ Ai x ≤ αj(i)+1 }
◮ Consider int.hull CI
(poly many vertices)
CI
◮
b
b
b
b
P
b
b
b
Covering a polytope w. parallelep. (2)
cell C
Split P = {x | Ax ≤ b}
into poly many cells
C = {x | αj(i) ≤ Ai x ≤ αj(i)+1 }
◮ Consider int.hull CI
(poly many vertices)
◮ Extend any d + 1 vertices
of CI to parallelepiped
CI
◮
x1
b
b
b
b
P
b
b
b
x2
x0
Covering a polytope w. parallelep. (2)
cell C
Split P = {x | Ax ≤ b}
into poly many cells
C = {x | αj(i) ≤ Ai x ≤ αj(i)+1 }
◮ Consider int.hull CI
(poly many vertices)
◮ Extend any d + 1 vertices
of CI to parallelepiped
CI
◮
x1
b
b
b
b
P
b
b
b
x0
x2
parallelepiped
Covering a polytope w. parallelep. (2)
cell C
Split P = {x | Ax ≤ b}
into poly many cells
C = {x | αj(i) ≤ Ai x ≤ αj(i)+1 }
◮ Consider int.hull CI
(poly many vertices)
◮ Extend any d + 1 vertices
of CI to parallelepiped
CI
◮
x1
b
b
b
x0
x2
parallelepiped
2
◮
b
b
b
P
b
2
Each cell covered with dO(d ) mO(d ) (log ∆)O(d
parallelepipeds
2)
Covering a polytope w. parallelep. (2)
CI
Split P = {x | Ax ≤ b}
into poly many cells
C = {x | αj(i) ≤ Ai x ≤ αj(i)+1 }
◮ Consider int.hull CI
(poly many vertices)
◮ Extend any d + 1 vertices
of CI to parallelepiped
◮
2
◮
◮
b
b
b
b
2
P
b
b
b
2
Each cell covered with dO(d ) mO(d ) (log ∆)O(d )
parallelepipeds
Improvement: Approximate CI by polytope with d2
vertices using John’s ellipsoid
Covering a polytope w. parallelep. (2)
CI
Split P = {x | Ax ≤ b}
into poly many cells
C = {x | αj(i) ≤ Ai x ≤ αj(i)+1 }
◮ Consider int.hull CI
(poly many vertices)
◮ Extend any d + 1 vertices
of CI to parallelepiped
◮
2
◮
◮
b
b
b
b
2
P
b
b
b
2
Each cell covered with dO(d ) mO(d ) (log ∆)O(d )
parallelepipeds
Improvement: Approximate CI by polytope with d2
vertices using John’s ellipsoid
The algorithm
◮
◮
Input: polytopes P, Q in ineq. description
Output: Coefficients for int.cone(P ∩ Zd ) ∩ Q 6= ∅
The algorithm
◮
◮
Input: polytopes P, Q in ineq. description
Output: Coefficients for int.cone(P ∩ Zd ) ∩ Q 6= ∅
Algorithm:
(1) Compute poly many parallelepipeds covering P
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
P
b
The algorithm
◮
◮
Input: polytopes P, Q in ineq. description
Output: Coefficients for int.cone(P ∩ Zd ) ∩ Q 6= ∅
Algorithm:
(1) Compute poly many parallelepipeds covering P
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
P
b
The algorithm
◮
◮
Input: polytopes P, Q in ineq. description
Output: Coefficients for int.cone(P ∩ Zd ) ∩ Q 6= ∅
Algorithm:
(1) Compute poly many parallelepipeds covering P
(2) Guess the 2d parallelepipeds containing solution
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
P
b
The algorithm
◮
◮
Input: polytopes P, Q in ineq. description
Output: Coefficients for int.cone(P ∩ Zd ) ∩ Q 6= ∅
Algorithm:
(1) Compute poly many parallelepipeds covering P
(2) Guess the 2d parallelepipeds containing solution
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
P
b
The algorithm
◮
◮
Input: polytopes P, Q in ineq. description
Output: Coefficients for int.cone(P ∩ Zd ) ∩ Q 6= ∅
Algorithm:
(1) Compute poly many parallelepipeds covering P
(2) Guess the 2d parallelepipeds containing solution
→ X := vertices
bb
bb
b
b
bb
bb
bb
b
b
b
b
b
b
b
b
b
bb
b
b
b
b
b
bb
b
P
b
The algorithm
◮
◮
Input: polytopes P, Q in ineq. description
Output: Coefficients for int.cone(P ∩ Zd ) ∩ Q 6= ∅
Algorithm:
(1) Compute poly many parallelepipeds covering P
(2) Guess the 2d parallelepipeds containing solution
→ X := vertices
(3) Solve ILP with 2O(d) variables
X
X
λx · x +
1· x ∈Q
x∈X
some x∈P ∩Zd
bb
bb
b
b
bb
bb
bb
b
b
b
b
b
b
b
b
b
bb
b
b
b
b
b
bb
b
P
b
The algorithm
◮
◮
Input: polytopes P, Q in ineq. description
Output: Coefficients for int.cone(P ∩ Zd ) ∩ Q 6= ∅
Algorithm:
(1) Compute poly many parallelepipeds covering P
(2) Guess the 2d parallelepipeds containing solution
→ X := vertices
(3) Solve ILP with 2O(d) variables
X
X
λx · x +
1· x ∈Q
x∈X
some x∈P ∩Zd
variables
bb
bb
b
b
bb
bb
bb
b
b
b
b
b
b
b
b
b
bb
b
b
b
b
b
bb
b
P
b
A Structure Theorem
Structure Theorem
For polytope P
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
P
b
A Structure Theorem
Structure Theorem
For polytope P
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
P
b
A Structure Theorem
Structure Theorem
For polytope P , ∃ poly-time comp. set X ⊆ P ∩ Zd
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A Structure Theorem
Structure Theorem
For polytope P , ∃ poly-time comp. set X ⊆ P ∩ Zd
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A Structure Theorem
Structure Theorem
For polytope P , ∃ poly-time comp. set X ⊆ P ∩ Zd s.t. for
all a ∈ int.cone(P ∩ Zd ) one can express
X
of 22d points in P ∩ Zd
a = int.cone(22d points in X) +
a
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More applications (1)
Theorem
Cutting stock with d = O(1) item types and m = O(1) of bin
types can be solved in polytime.
Items:
Bins:
a1 many
cost c1
s1
w1
item 1 . . . item d
bin
type 1
...
bin
type m
More applications (1)
Theorem
Cutting stock with d = O(1) item types and m = O(1) of bin
types can be solved in polytime.
Items:
Bins:
a1 many
cost c1
s1
w1
item 1 . . . item d
◮
P = P1 × . . . × Pm with Pj := {
bin
type 1
x
cj
...
| sT x ≤ w j }
bin
type m
More applications (2)
High Multiplicity Scheduling:
◮ Input: d job types, each with release time, deadline,
processing time, multiplicity
◮ Goal: Schedule jobs to minimize number of used machines
More applications (2)
High Multiplicity Scheduling:
◮ Input: d job types, each with release time, deadline,
processing time, multiplicity
◮ Goal: Schedule jobs to minimize number of used machines
Theorem
High Multiplicity Scheduling can be solved in poly-time for
fixed d.
The end
Thanks for your attention