SLIDE 1
On the Unique-Lifting Property
Amitabh Basu Joint Work with Gennadiy Averkov
18th Aussois Combinatorial Optimization Workshop
SLIDE 2 Cut Generating Functions for Mixed-Integer Linear Programs
◮ Solve problem as Linear Program and obtain optimal simplex
tableau.
◮ If some basic variables are non-integral, “apply” Cut
Generating Functions to one or more rows of the tableau
◮ Add these cutting planes and re-iterate (usually combined
with some enumeration scheme).
SLIDE 3
General Framework for Cut Generating Functions
x = f + k
j=1 rjsj
+ ℓ
j=1 pjyj
x ∈ Zq s ∈ Rk
+
y ∈ Zℓ
+
SLIDE 4 General Framework for Cut Generating Functions
x = f + k
j=1 rjsj
+ ℓ
j=1 pjyj
x ∈ Zq s ∈ Rk
+
y ∈ Zℓ
+
We seek pair of functions ψf : Rq → R πf : Rq → R such that the inequality
k
ψf (rj)sj +
ℓ
πf (pj)yj ≥ 1 is valid for any k, ℓ, rj, pj.
SLIDE 5 General Framework for Cut Generating Functions
x = f + k
j=1 rjsj
+ ℓ
j=1 pjyj
x ∈ Zq s ∈ Rk
+
y ∈ Zℓ
+
We seek pair of functions ψf : Rq → R πf : Rq → R such that the inequality
k
ψf (rj)sj +
ℓ
πf (pj)yj ≥ 1 is valid for any k, ℓ, rj, pj. Want minimal valid pairs to remove redundancies.
SLIDE 6 What are we after ?
- Johnson’s theorem. Structural characterization of all minimal
valid pairs.
SLIDE 7 What are we after ?
- Johnson’s theorem. Structural characterization of all minimal
valid pairs.
- MAIN GOAL: Find closed form formulas for these functions.
SLIDE 8 What are we after ?
- Johnson’s theorem. Structural characterization of all minimal
valid pairs.
- MAIN GOAL: Find closed form formulas for these functions.
- Let B ∈ Rq be a maximal lattice-free convex polytope with
f ∈ int(B): B = {x ∈ Rq : ai(x − f ) ≤ 1, i ∈ I}. Define the function φf ,B(r) = max
i∈I air,
∀r ∈ Rq. Then, ψf = πf = φf ,B is a valid pair, but not minimal.
SLIDE 9 What are we after ?
- Johnson’s theorem. Structural characterization of all minimal
valid pairs.
- MAIN GOAL: Find closed form formulas for these functions.
- Let B ∈ Rq be a maximal lattice-free convex polytope with
f ∈ int(B): B = {x ∈ Rq : ai(x − f ) ≤ 1, i ∈ I}. Define the function φf ,B(r) = max
i∈I air,
∀r ∈ Rq. Then, ψf = πf = φf ,B is a valid pair, but not minimal.
- The idea is to start with some maximal lattice-free B, define
ψf = φf ,B and find a minimal function πf such that it forms a valid pair with ψf . These are called minimal liftings of ψf .
SLIDE 10 What are we after ?
- Johnson’s theorem. Structural characterization of all minimal
valid pairs.
- MAIN GOAL: Find closed form formulas for these functions.
- Let B ∈ Rq be a maximal lattice-free convex polytope with
f ∈ int(B): B = {x ∈ Rq : ai(x − f ) ≤ 1, i ∈ I}. Define the function φf ,B(r) = max
i∈I air,
∀r ∈ Rq. Then, ψf = πf = φf ,B is a valid pair, but not minimal.
- The idea is to start with some maximal lattice-free B, define
ψf = φf ,B and find a minimal function πf such that it forms a valid pair with ψf . These are called minimal liftings of ψf .
- For some special f , B’s, minimal liftings are unique. Gives us
formulas for minimal valid pairs.
SLIDE 11 Recognizing pairs f , B with unique minimal liftings
f
SLIDE 12 Recognizing pairs f , B with unique minimal liftings
- 1. Given f , B.
- 2. For every facet
F, construct PF := conv(F ∪ {f }).
f F1 F2 F3 PF1 PF2 PF3
SLIDE 13 Recognizing pairs f , B with unique minimal liftings
- 1. Given f , B.
- 2. For every facet
F, construct PF := conv(F ∪ {f }).
z ∈ Zq ∩ relint(F), construct SF,z(f ) := PF(f ) ∩ (z + f − PF(f )).
f F1 F2 F3 Sz1,F1 Sz2,F2 Sz3,F3 z1 z2 z3
SLIDE 14 Recognizing pairs f , B with unique minimal liftings
- 1. Given f , B.
- 2. For every facet
F, construct PF := conv(F ∪ {f }).
z ∈ Zq ∩ relint(F), construct SF,z(f ) := PF(f ) ∩ (z + f − PF(f )).
f F1 F2 F3 Sz1,F1 Sz2,F2 Sz3,F3 z1 z2 z3
R(f , B) =
Sz,F(f )
SLIDE 15 Recognizing pairs f , B with unique minimal liftings
- 1. Given f , B.
- 2. For every facet
F, construct PF := conv(F ∪ {f }).
z ∈ Zq ∩ relint(F), construct SF,z(f ) := PF(f ) ∩ (z + f − PF(f )).
z1 xz2
z3
Sz1,F1 Sz2,F2 Sz3,F3 f
F3 F1 F2
R(f , B) =
Sz,F(f )
SLIDE 16 Recognizing pairs f , B with unique minimal liftings
R(f , B) =
Sz,F(f ) THEOREM Basu, Campelo, Conforti, Cornu´
ejols, Zambelli 2011
f , B has the unique-lifting property if and only if R(f , B) + Zq = Rq.
SLIDE 17 Recognizing pairs f , B with unique minimal liftings
R(f , B) =
Sz,F(f ) THEOREM Basu, Campelo, Conforti, Cornu´
ejols, Zambelli 2011
f , B has the unique-lifting property if and only if R(f , B) + Zq = Rq. Main Credit for sparking this line of research: Santanu Dey and Laurence Wolsey 2009.
SLIDE 18 x4 x3 x5 x6 x1 x2 f
SLIDE 19 x4 x3 x5 x6 x1 x2 f R(x2)
SLIDE 20 x4 x3 x5 x6 x1 x2 f R(x2)
SLIDE 21 x4 x3 x5 x6 x1 x2 f R(x2)
SLIDE 22 Rψ + Zq = Rq
x4 x3 x5 x6 x1 x2 f R(x2)
SLIDE 23 Rψ + Zq = Rq
x1 x2 x3 R(x1) R(x2) R(x3) f Bψ
SLIDE 24 Rψ + Zq = Rq
x1 x2 x3 R(x1) R(x2) R(x3) f Bψ
SLIDE 25 MORAL :
- 1. If the pair f , B has the unique-lifting property, then we get
closed form formulas for a minimal valid pair.
- 2. The question of deciding if f , B has the unique-lifting property
is equivalent to deciding if R(f , B) + Zq = Rq. Potentially connects with a lot of research on coverings and tilings by star-shaped bodies, extensively studied in Geometry of Numbers.
SLIDE 26
Invariance of the Unique-lifting property
For a fixed maximal lattice-free convex polytope B, R(f , B) (in fact ψf itself) depends on the position of f in the interior. So, a priori, the same lattice-free set B might have the unique-lifting property when paired with one f1, and have the multiple-lifting property when paired with a different f2.
SLIDE 27
Invariance of the Unique-lifting property
For a fixed maximal lattice-free convex polytope B, R(f , B) (in fact ψf itself) depends on the position of f in the interior. So, a priori, the same lattice-free set B might have the unique-lifting property when paired with one f1, and have the multiple-lifting property when paired with a different f2. THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope in Rq(q ≥ 2). Then B has the unique-lifting property for all f ∈ int(B), or B has the multiple-lifting property for all f ∈ int(B).
SLIDE 28
Invariance of the Unique-lifting property
THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope in Rq(q ≥ 2). Then B has the unique-lifting property for all f ∈ int(B), or B has the multiple-lifting property for all f ∈ int(B). OBSERVATION Deciding if Rψ + Zq = Rq is the same as deciding if volTq(Rψ/Zq) = 1.
SLIDE 29
Invariance of the Unique-lifting property
THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope in Rq(q ≥ 2). Then B has the unique-lifting property for all f ∈ int(B), or B has the multiple-lifting property for all f ∈ int(B). OBSERVATION Deciding if Rψ + Zq = Rq is the same as deciding if volTq(Rψ/Zq) = 1. THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope and let f ∈ B. Then volTq(Rψ/Zq) is an affine function of the coordinates of f .
SLIDE 30 Lattice Volume is an affine function
THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope and let f ∈ B. Then volTq(Rψ/Zq) is an affine function
SLIDE 31 Lattice Volume is an affine function
THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope and let f ∈ B. Then volTq(Rψ/Zq) is an affine function
Proof Ingredient 1 : The volume of each Sz,F(f ) is an affine function of f .
f1
SLIDE 32 Lattice Volume is an affine function
THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope and let f ∈ B. Then volTq(Rψ/Zq) is an affine function
Proof Ingredient 1 : The volume of each Sz,F(f ) is an affine function of f .
f1 f2
SLIDE 33 Lattice Volume is an affine function
THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope and let f ∈ B. Then volTq(Rψ/Zq) is an affine function
Proof Ingredient 1 : The volume of each Sz,F(f ) is an affine function of f . Proof Ingredient 2 : Take care of intersections due to lattice translations: We find a closed form integral expression for volTq(Rψ/Zq).
SLIDE 34 Operations that preserve that Unique-lifting property
Pyramid Construction. Let B ⊆ Rq be a maximal lattice-free
- polytope. Consider B as embedded in Rq+1, i.e.,
B ⊆ Rq × {0} ⊆ Rq+1. Let v ∈ Rq+1 \ (Rq × {0}). Let C(B, v) be the cone formed with B − v as base. We define Pyr(B, v) = (C(B, v) + v) ∩ {x ∈ Rq+1 : xq+1 ≥ −1}.
SLIDE 35 Operations that preserve that Unique-lifting property
Pyramid Construction. Let B ⊆ Rq be a maximal lattice-free
- polytope. Consider B as embedded in Rq+1, i.e.,
B ⊆ Rq × {0} ⊆ Rq+1. Let v ∈ Rq+1 \ (Rq × {0}). Let C(B, v) be the cone formed with B − v as base. We define Pyr(B, v) = (C(B, v) + v) ∩ {x ∈ Rq+1 : xq+1 ≥ −1}. THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope and v ∈ Rq+1 be such that Pyr(B, v) is a maximal lattice-free polytope. Suppose further that the base of Pyr(B, v) contains an integer translate of B. If B is a body with unique lifting, then Pyr(B, v) is a body with unique lifting.
SLIDE 36
Operations that preserve that Unique-lifting property
THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope and v ∈ Rq+1 be such that Pyr(B, v) is a maximal lattice-free polytope. Suppose further that the base of Pyr(B, v) contains an integer translate of B. If B is a body with unique lifting, then Pyr(B, v) is a body with unique lifting.
Pyr(B, v) v B
SLIDE 37
Operations that preserve that Unique-lifting property
Free Sum Construction. Let B1 ⊆ Rq1 and B2 ⊆ Rq2. Let ci ∈ int(Bi), i = 1, 2. For any 0 < µ < 1, define the “free sum” as a polytope in Rq1+q2: B1 ⊕ B2 := conv(({B1 − c1 1 − µ × {o2}) ∪ ({o1} × B2 − c2 µ )) + (c1, c2).
SLIDE 38
Operations that preserve that Unique-lifting property
Free Sum Construction. Let B1 ⊆ Rq1 and B2 ⊆ Rq2. Let ci ∈ int(Bi), i = 1, 2. For any 0 < µ < 1, define the “free sum” as a polytope in Rq1+q2: B1 ⊕ B2 := conv(({B1 − c1 1 − µ × {o2}) ∪ ({o1} × B2 − c2 µ )) + (c1, c2).
q1 = q2 = 1 B1 B2 c1 c2
SLIDE 39 Operations that preserve that Unique-lifting property
Free Sum Construction. Let B1 ⊆ Rq1 and B2 ⊆ Rq2. Let ci ∈ int(Bi), i = 1, 2. For any 0 < µ < 1, define the “free sum” as a polytope in Rq1+q2: B1 ⊕ B2 := conv(({B1 − c1 1 − µ × {o2}) ∪ ({o1} × B2 − c2 µ )) + (c1, c2).
q1 = q2 = 1 B1 B2 c1 c2
B1−c1 1−µ × {o2}
{o1} × B2−c2
µ
(c1, c2) B1 ⊕ B2
SLIDE 40
Operations that preserve that Unique-lifting property
Free Sum Construction. Let B1 ⊆ Rq1 and B2 ⊆ Rq2. Let ci ∈ int(Bi), i = 1, 2. For any 0 < µ < 1, define the “free sum” as a polytope in Rq1+q2: B1 ⊕ B2 := conv(({B1 − c1 1 − µ × {o2}) ∪ ({o1} × B2 − c2 µ )) + (c1, c2). THEOREM Averkov, Basu 2013 Let B1 ⊆ Rq1, B2 ⊆ Rq2 be a maximal lattice-free polytopes. Then B1 ⊕ B2 ⊆ Rq1+q2 is a maximal lattice-free polytope. Moreover, if B1, B2 both have the unique-lifting property, then so does B1 ⊕ B2.
SLIDE 41
Unique-Lifting Property of Pyramids
THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope and v ∈ Rq+1 be such that Pyr(B, v) is a maximal lattice-free polytope. Suppose further that the base of Pyr(B, v) contains an integer translate of B. If B is a body with unique lifting, then Pyr(B, v) is a body with unique lifting.
SLIDE 42
Unique-Lifting Property of Pyramids
THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope and v ∈ Rq+1 be such that Pyr(B, v) is a maximal lattice-free polytope. Suppose further that the base of Pyr(B, v) contains an integer translate of B. If B is a body with unique lifting, then Pyr(B, v) is a body with unique lifting. Let a = (a1, . . . , aq) be an q-tuple of real numbers such that
1 a1 + . . . + 1 aq = 1. Then
S(a) := conv{0, a1e1, a2e2, . . . , aqeq} ⊆ Rq is a maximal lattice-free simplex.
SLIDE 43
Unique-Lifting Property of Pyramids
THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope and v ∈ Rq+1 be such that Pyr(B, v) is a maximal lattice-free polytope. Suppose further that the base of Pyr(B, v) contains an integer translate of B. If B is a body with unique lifting, then Pyr(B, v) is a body with unique lifting. Let a = (a1, . . . , aq) be an q-tuple of real numbers such that
1 a1 + . . . + 1 aq = 1. Then
S(a) := conv{0, a1e1, a2e2, . . . , aqeq} ⊆ Rq is a maximal lattice-free simplex. COROLLARY Averkov, Basu 2013 S(a) is a body with unique-lifting for any q-tuple a = (a1, . . . , aq) such that
1 a1 + . . . + 1 aq = 1.
SLIDE 44
Unique-Lifting Property of Pyramids
Let a = (a1, . . . , aq) be an q-tuple of real numbers such that
1 a1 + . . . + 1 aq = 1. Then
S(a) := conv{0, a1e1, a2e2, . . . , aqeq} ⊆ Rq is a maximal lattice-free simplex. COROLLARY Averkov, Basu 2013 S(a) is a body with unique-lifting for any q-tuple a = (a1, . . . , aq) such that
1 a1 + . . . + 1 aq = 1.
THEOREM Averkov, Basu 2013 Let P be a maximal lattice-free pyramid in Rq such that every facet of P contains exactly one integer point in its relative interior. P has the unique-lifting property if and only if P is an affine unimodular transformation of conv{0, qe1, . . . , qeq}.
SLIDE 45 Unique-Lifting Property of Pyramids
THEOREM Averkov, Basu 2013 Let P be a maximal lattice-free pyramid in Rq such that every facet of P contains exactly one integer point in its relative interior. P has the unique-lifting property if and only if P is an affine unimodular transformation of conv{0, qe1, . . . , qeq}. PROOF:
- 1. Let f be the apex of P and consider Sz,F(f ).
SLIDE 46 Unique-Lifting Property of Pyramids
THEOREM Averkov, Basu 2013 Let P be a maximal lattice-free pyramid in Rq such that every facet of P contains exactly one integer point in its relative interior. P has the unique-lifting property if and only if P is an affine unimodular transformation of conv{0, qe1, . . . , qeq}. PROOF:
- 1. Let f be the apex of P and consider Sz,F(f ).
- 2. Venkov-Alexandrov-McMullen theorem ⇒ Sz,F(f ) is centrally
symmetric with centrally symmetric facets.
SLIDE 47 Unique-Lifting Property of Pyramids
THEOREM Averkov, Basu 2013 Let P be a maximal lattice-free pyramid in Rq such that every facet of P contains exactly one integer point in its relative interior. P has the unique-lifting property if and only if P is an affine unimodular transformation of conv{0, qe1, . . . , qeq}. PROOF:
- 1. Let f be the apex of P and consider Sz,F(f ).
- 2. Venkov-Alexandrov-McMullen theorem ⇒ Sz,F(f ) is centrally
symmetric with centrally symmetric facets.
- 3. McMullen’s characterization of zonotopes ⇒ Sz,F(f ) is a
zonotope whose every belt is of length 4.
SLIDE 48 Unique-Lifting Property of Pyramids
THEOREM Averkov, Basu 2013 Let P be a maximal lattice-free pyramid in Rq such that every facet of P contains exactly one integer point in its relative interior. P has the unique-lifting property if and only if P is an affine unimodular transformation of conv{0, qe1, . . . , qeq}. PROOF:
- 1. Let f be the apex of P and consider Sz,F(f ).
- 2. Venkov-Alexandrov-McMullen theorem ⇒ Sz,F(f ) is centrally
symmetric with centrally symmetric facets.
- 3. McMullen’s characterization of zonotopes ⇒ Sz,F(f ) is a
zonotope whose every belt is of length 4.
- 4. Combinatorial geoemtry of zonotopes ⇒ Sz,F(f ) is a
- parallelotope. This implies P is a simplex.
SLIDE 49 Unique-Lifting Property of Pyramids
THEOREM Averkov, Basu 2013 Let P be a maximal lattice-free pyramid in Rq such that every facet of P contains exactly one integer point in its relative interior. P has the unique-lifting property if and only if P is an affine unimodular transformation of conv{0, qe1, . . . , qeq}. PROOF:
- 1. Let f be the apex of P and consider Sz,F(f ).
- 2. Venkov-Alexandrov-McMullen theorem ⇒ Sz,F(f ) is centrally
symmetric with centrally symmetric facets.
- 3. McMullen’s characterization of zonotopes ⇒ Sz,F(f ) is a
zonotope whose every belt is of length 4.
- 4. Combinatorial geoemtry of zonotopes ⇒ Sz,F(f ) is a
- parallelotope. This implies P is a simplex.
- 5. Minkowski’s Convex Body theorem + Minkowski-Haj´
- s theorem
⇒ P is an affine unimodular transformation of conv{0, qe1, . . . , qeq}.
SLIDE 50 Conclusions
- Maximal Lattice-free polytopes with the unique-lifting
property give closed form formulas for minimal cut generating functions.
- Detecting the unique-lifting property can be coverted into a
geometric question, i.e., covering by lattice translates. Can leverage a lot of research done in Geometry of Numbers and Discrete Geometry.
- Characterize maximal lattice-free polytopes with the
unique-lifting property.
- Invariance of Unique-lifting property for S-free sets.
SLIDE 51
THANK YOU ! Questions/Comments ?
