A New Facet Generating Procedurefor the Stable Set Polytope
Alinson S. Xaviera Manoel Campelob
a Mestrado e Doutorado em Ciencia da ComputacaoUniversidade Federal do Ceara
Fortaleza, Brazil
b Departamento de Estatıstica e Matematica AplicadaUniversidade Federal do Ceara
Fortaleza, Brazil
LAGOS’11
1 / 22
Outline
1. Introduction
2. Preliminaries
3. The Procedure
4. Example
5. Concluding Remarks
2 / 22
Stable Set Polytope: Definition
� Given a simple, undirected graph G = (V ,E)
Definition
A stable set is a set of pairwise non-adjacent vertices.
Example Not an example
Definition
The stable set polytope is the convex hull of the incidence vectors of all thestable sets:
STAB(G) = conv{x ∈ {0, 1}V : xu + xv ≤ 1, ∀{u, v} ∈ E}
3 / 22
Stable Set Polytope: Definition
� Given a simple, undirected graph G = (V ,E)
Definition
A stable set is a set of pairwise non-adjacent vertices.
Example Not an example
Definition
The stable set polytope is the convex hull of the incidence vectors of all thestable sets:
STAB(G) = conv{x ∈ {0, 1}V : xu + xv ≤ 1, ∀{u, v} ∈ E}
3 / 22
Stable Set Polytope: Applications
� Applications in various fields (Bomze, Budinich, Pardalos & Pedillo (1999)):
� Coding theory
� Computer vision and pattern matching
� Molecular biology
� Scheduling
� Model other important combinatorial problems:
� Set packing and set partitioning (Padberg (1973))
� Graph coloring (Cornaz & Jost (2008))
4 / 22
Stable Set Polytope: Applications
� Applications in various fields (Bomze, Budinich, Pardalos & Pedillo (1999)):
� Coding theory
� Computer vision and pattern matching
� Molecular biology
� Scheduling
� Model other important combinatorial problems:
� Set packing and set partitioning (Padberg (1973))
� Graph coloring (Cornaz & Jost (2008))
4 / 22
Previous Research: Facet Producing Subgraphs
� Facet producing subgraphs:
� Cliques (Padberg (1973))
� Odd holes (Padberg (1973))
� Odd anti-holes (Nemhauser & Trotter (1974))
� Webs and anti-webs (Trotter (1975))
� Subdivided wheels (Cheng & Cunningham (1996))
� others...
Example
1
2
34
5
P5i=1 xi ≤ 1
5 / 22
Previous Research: Facet Producing Subgraphs
� Facet producing subgraphs:
� Cliques (Padberg (1973))
� Odd holes (Padberg (1973))
� Odd anti-holes (Nemhauser & Trotter (1974))
� Webs and anti-webs (Trotter (1975))
� Subdivided wheels (Cheng & Cunningham (1996))
� others...
Example
12
3
45
6
7
P7i=1 xi ≤ 3
5 / 22
Previous Research: Facet Producing Subgraphs
� Facet producing subgraphs:
� Cliques (Padberg (1973))
� Odd holes (Padberg (1973))
� Odd anti-holes (Nemhauser & Trotter (1974))
� Webs and anti-webs (Trotter (1975))
� Subdivided wheels (Cheng & Cunningham (1996))
� others...
Example
12
3
45
6
7
P7i=1 xi ≤ 2
5 / 22
Previous Research: Facet Producing Subgraphs
� Facet producing subgraphs:
� Cliques (Padberg (1973))
� Odd holes (Padberg (1973))
� Odd anti-holes (Nemhauser & Trotter (1974))
� Webs and anti-webs (Trotter (1975))
� Subdivided wheels (Cheng & Cunningham (1996))
� others...
Example
12
3
45
6
7
8
3x8 +P7
i=1 xi ≤ 3
5 / 22
Previous Research: Facet Producing Subgraphs
� Facet producing subgraphs:
� Cliques (Padberg (1973))
� Odd holes (Padberg (1973))
� Odd anti-holes (Nemhauser & Trotter (1974))
� Webs and anti-webs (Trotter (1975))
� Subdivided wheels (Cheng & Cunningham (1996))
� others...
Example
12
3
45
6
7
8
3x8 +P7
i=1 xi ≤ 3
5 / 22
Previous Research: Facet Generating Procedures
� Facet generating procedures:
� Subdividing edges (Wolsey (1976))
� Subdividing stars (Barahona & Mahjoub (1994))
� Replacing vertices with stars (Canovas, Landete & Marın (2003))
� Replacing edges with gears (Galluccio, Gentile & Ventura (2008))
� others...
Example
1
2
34
56
1
2
34
5
6
8
7
2x6 +P5
i=1 xi ≤ 2 2x6 +P5
i=1 xi + x7 + x8 ≤ 3
6 / 22
Previous Research: Facet Generating Procedures
� Facet generating procedures:
� Subdividing edges (Wolsey (1976))
� Subdividing stars (Barahona & Mahjoub (1994))
� Replacing vertices with stars (Canovas, Landete & Marın (2003))
� Replacing edges with gears (Galluccio, Gentile & Ventura (2008))
� others...
Example
1
2
34
56
1
2
34
56
78
910
11
2x6 +P5
i=1 xi ≤ 2 3x6 +P5
i=1 xi +P11
i=7 xi ≤ 5
6 / 22
Previous Research: Facet Generating Procedures
� Facet generating procedures:
� Subdividing edges (Wolsey (1976))
� Subdividing stars (Barahona & Mahjoub (1994))
� Replacing vertices with stars (Canovas, Landete & Marın (2003))
� Replacing edges with gears (Galluccio, Gentile & Ventura (2008))
� others...
Example
1
6 7 8
4 3
5 2
1
4 3
5 2
x1 +P5
i=2 xi ≤ 1 2x1 +P5
i=2 xi +P8
i=6 xi ≤ 3
6 / 22
Previous Research: Facet Generating Procedures
� Facet generating procedures:
� Subdividing edges (Wolsey (1976))
� Subdividing stars (Barahona & Mahjoub (1994))
� Replacing vertices with stars (Canovas, Landete & Marın (2003))
� Replacing edges with gears (Galluccio, Gentile & Ventura (2008))
� others...
Example
7
2
1 3 1 3
6
11
8
14 5
2
10
5 4
P3i=1 xi + x4 + x5 ≤ 2
P3i=1 xi + 2x4 + 2x5 +
P11i=6 xi ≤ 4
6 / 22
Previous Research: Facet Generating Procedures
� Facet generating procedures:
� Subdividing edges (Wolsey (1976))
� Subdividing stars (Barahona & Mahjoub (1994))
� Replacing vertices with stars (Canovas, Landete & Marın (2003))
� Replacing edges with gears (Galluccio, Gentile & Ventura (2008))
� others...
Example
7
2
1 3 1 3
6
11
8
14 5
2
10
5 4
P3i=1 xi + x4 + x5 ≤ 2
P3i=1 xi + 2x4 + 2x5 +
P11i=6 xi ≤ 4
6 / 22
The New Procedure
� Replace (k − 1)-cliques with certain k-partite graphs
� Particular case: replace vertices with bipartite graphs
� Unifies and generalizes many previous procedures:
� Subdividing edges
� Subdividing stars
� Replacing vertices with stars
� Generates many new facet-defining inequalities
� The idea that leads to the procedure can be used with other polytopes
7 / 22
The New Procedure
� Replace (k − 1)-cliques with certain k-partite graphs
� Particular case: replace vertices with bipartite graphs
� Unifies and generalizes many previous procedures:
� Subdividing edges
� Subdividing stars
� Replacing vertices with stars
� Generates many new facet-defining inequalities
� The idea that leads to the procedure can be used with other polytopes
7 / 22
The New Procedure
� Replace (k − 1)-cliques with certain k-partite graphs
� Particular case: replace vertices with bipartite graphs
� Unifies and generalizes many previous procedures:
� Subdividing edges
� Subdividing stars
� Replacing vertices with stars
� Generates many new facet-defining inequalities
� The idea that leads to the procedure can be used with other polytopes
7 / 22
The New Procedure
� Replace (k − 1)-cliques with certain k-partite graphs
� Particular case: replace vertices with bipartite graphs
� Unifies and generalizes many previous procedures:
� Subdividing edges
� Subdividing stars
� Replacing vertices with stars
� Generates many new facet-defining inequalities
� The idea that leads to the procedure can be used with other polytopes
7 / 22
Preliminaries
8 / 22
Affine Isomorphism
Definition (Ziegler (1994))
Two polytopes are affinely isomorphic (not. P1∼= P2) if there is an affine map
that is a bijection between their points.
� Implies combinatorial equivalence
� Easy to transform facets of P1 into facets of P2
9 / 22
Affine Isomorphism
Definition (Ziegler (1994))
Two polytopes are affinely isomorphic (not. P1∼= P2) if there is an affine map
that is a bijection between their points.
� Implies combinatorial equivalence
� Easy to transform facets of P1 into facets of P2
9 / 22
Affine Isomorphism
Definition (Ziegler (1994))
Two polytopes are affinely isomorphic (not. P1∼= P2) if there is an affine map
that is a bijection between their points.
� Implies combinatorial equivalence
� Easy to transform facets of P1 into facets of P2
9 / 22
Facets from Faces: Overview
� Convert facets of a face F into facets of the polytope P:
� Find a sequence F1, . . . ,Fk such that:
� F1 = F and Fk = P
� Fi is a facet of Fi+1
� Repeatedly apply the following theorem:
Theorem
Let P be a convex polytope and S a finite set such that P = conv(S). If cx ≤ dis facet-defining for P and πx ≤ π∗ is facet-defining for {x ∈ P : cx = d}, thenπx + α(cx − d) ≤ π∗ is facet-defining for P, where
α = max
π∗ − πx
cx − d: x ∈ S , cx < d
ff
� Extension of the sequential lifting procedure
� Different sequences may yield different facets
10 / 22
Facets from Faces: Overview
� Convert facets of a face F into facets of the polytope P:
� Find a sequence F1, . . . ,Fk such that:
� F1 = F and Fk = P
� Fi is a facet of Fi+1
� Repeatedly apply the following theorem:
Theorem
Let P be a convex polytope and S a finite set such that P = conv(S). If cx ≤ dis facet-defining for P and πx ≤ π∗ is facet-defining for {x ∈ P : cx = d}, thenπx + α(cx − d) ≤ π∗ is facet-defining for P, where
α = max
π∗ − πx
cx − d: x ∈ S , cx < d
ff
� Extension of the sequential lifting procedure
� Different sequences may yield different facets
10 / 22
Facets from Faces: Overview
� Convert facets of a face F into facets of the polytope P:
� Find a sequence F1, . . . ,Fk such that:
� F1 = F and Fk = P
� Fi is a facet of Fi+1
� Repeatedly apply the following theorem:
Theorem
Let P be a convex polytope and S a finite set such that P = conv(S). If cx ≤ dis facet-defining for P and πx ≤ π∗ is facet-defining for {x ∈ P : cx = d}, thenπx + α(cx − d) ≤ π∗ is facet-defining for P, where
α = max
π∗ − πx
cx − d: x ∈ S , cx < d
ff
� Extension of the sequential lifting procedure
� Different sequences may yield different facets
10 / 22
Facets from Faces: Overview
� Convert facets of a face F into facets of the polytope P:
� Find a sequence F1, . . . ,Fk such that:
� F1 = F and Fk = P
� Fi is a facet of Fi+1
� Repeatedly apply the following theorem:
Theorem
Let P be a convex polytope and S a finite set such that P = conv(S). If cx ≤ dis facet-defining for P and πx ≤ π∗ is facet-defining for {x ∈ P : cx = d}, thenπx + α(cx − d) ≤ π∗ is facet-defining for P, where
α = max
π∗ − πx
cx − d: x ∈ S , cx < d
ff
� Extension of the sequential lifting procedure
� Different sequences may yield different facets
10 / 22
Facets from Faces: Geometric View
Example
� P = conv{(0, 0), (2, 0), (2, 1), (1, 2), (0, 1)}
� F = {x ∈ P : x1 + x2 = 3}� −x1 ≤ −1 is facet-defining for F
� α = maxn−x1+1
3−x1−x2: x ∈ P, x1 + x2 < 3
o= 1
2
� −x1 + 12(x1 + x2 − 3) ≤ −1
� −x1 + x2 ≤ 1 is facet-defining for P
11 / 22
Facets from Faces: Geometric View
Example
� P = conv{(0, 0), (2, 0), (2, 1), (1, 2), (0, 1)}� F = {x ∈ P : x1 + x2 = 3}
� −x1 ≤ −1 is facet-defining for F
� α = maxn−x1+1
3−x1−x2: x ∈ P, x1 + x2 < 3
o= 1
2
� −x1 + 12(x1 + x2 − 3) ≤ −1
� −x1 + x2 ≤ 1 is facet-defining for P
11 / 22
Facets from Faces: Geometric View
Example
� P = conv{(0, 0), (2, 0), (2, 1), (1, 2), (0, 1)}� F = {x ∈ P : x1 + x2 = 3}� −x1 ≤ −1 is facet-defining for F
� α = maxn−x1+1
3−x1−x2: x ∈ P, x1 + x2 < 3
o= 1
2
� −x1 + 12(x1 + x2 − 3) ≤ −1
� −x1 + x2 ≤ 1 is facet-defining for P
11 / 22
Facets from Faces: Geometric View
Example
� P = conv{(0, 0), (2, 0), (2, 1), (1, 2), (0, 1)}� F = {x ∈ P : x1 + x2 = 3}� −x1 ≤ −1 is facet-defining for F
� α = maxn−x1+1
3−x1−x2: x ∈ P, x1 + x2 < 3
o= 1
2
� −x1 + 12(x1 + x2 − 3) ≤ −1
� −x1 + x2 ≤ 1 is facet-defining for P
11 / 22
Facets from Faces: Geometric View
Example
� P = conv{(0, 0), (2, 0), (2, 1), (1, 2), (0, 1)}� F = {x ∈ P : x1 + x2 = 3}� −x1 ≤ −1 is facet-defining for F
� α = maxn−x1+1
3−x1−x2: x ∈ P, x1 + x2 < 3
o= 1
2
� −x1 + 12(x1 + x2 − 3) ≤ −1
� −x1 + x2 ≤ 1 is facet-defining for P
11 / 22
Facets from Faces: Geometric View
Example
� P = conv{(0, 0), (2, 0), (2, 1), (1, 2), (0, 1)}� F = {x ∈ P : x1 + x2 = 3}� −x1 ≤ −1 is facet-defining for F
� α = maxn−x1+1
3−x1−x2: x ∈ P, x1 + x2 < 3
o= 1
2
� −x1 + 12(x1 + x2 − 3) ≤ −1
� −x1 + x2 ≤ 1 is facet-defining for P
11 / 22
Strong Hypertrees: Definitions
Definition
Two hyperedges are strongly adjacent if they have same size k and share exactlyk − 1 vertices.
Example Not an example
12 / 22
Strong Hypertrees: Definitions
Definition
A hypergraph H = (V , E) is a strong hypertree either if E = {V } or there is aleaf v incident to an hyperedge e such that:
� e is strongly adjacent to some other hyperedge,
� (V \ {v}, E \ {e}) is also a strong hypertree.
Example
(hyperedges are represented as cliques)
Example
12 / 22
Strong Hypertrees: Definitions
Definition
A hypergraph H = (V , E) is a strong hypertree either if E = {V } or there is aleaf v incident to an hyperedge e such that:
� e is strongly adjacent to some other hyperedge,
� (V \ {v}, E \ {e}) is also a strong hypertree.
Example
(hyperedges are represented as cliques)
Example
12 / 22
Strong Hypertrees: Definitions
Definition
A hypergraph H = (V , E) is a strong hypertree either if E = {V } or there is aleaf v incident to an hyperedge e such that:
� e is strongly adjacent to some other hyperedge,
� (V \ {v}, E \ {e}) is also a strong hypertree.
Example
(hyperedges are represented as cliques)
Example
12 / 22
Strong Hypertrees: Definitions
Definition
A hypergraph H = (V , E) is a strong hypertree either if E = {V } or there is aleaf v incident to an hyperedge e such that:
� e is strongly adjacent to some other hyperedge,
� (V \ {v}, E \ {e}) is also a strong hypertree.
Example
(hyperedges are represented as cliques)
Example
12 / 22
Strong Hypertrees: Definitions
Definition
A hypergraph H = (V , E) is a strong hypertree either if E = {V } or there is aleaf v incident to an hyperedge e such that:
� e is strongly adjacent to some other hyperedge,
� (V \ {v}, E \ {e}) is also a strong hypertree.
Example
(hyperedges are represented as cliques)
Example
12 / 22
Strong Hypertrees: Definitions
Definition
A hypergraph H = (V , E) is a strong hypertree either if E = {V } or there is aleaf v incident to an hyperedge e such that:
� e is strongly adjacent to some other hyperedge,
� (V \ {v}, E \ {e}) is also a strong hypertree.
Example
(hyperedges are represented as cliques)
Example
12 / 22
Strong Hypertrees: Definitions
Definition
A hypergraph H = (V , E) is a strong hypertree either if E = {V } or there is aleaf v incident to an hyperedge e such that:
� e is strongly adjacent to some other hyperedge,
� (V \ {v}, E \ {e}) is also a strong hypertree.
Example
(hyperedges are represented as cliques)
Example
12 / 22
Strong Hypertrees: Definitions
Definition
A hypergraph H = (V , E) is a strong hypertree either if E = {V } or there is aleaf v incident to an hyperedge e such that:
� e is strongly adjacent to some other hyperedge,
� (V \ {v}, E \ {e}) is also a strong hypertree.
Example
(hyperedges are represented as cliques)
Example
12 / 22
Strong Hypertrees: Definitions
Definition
A hypergraph H = (V , E) is a strong hypertree either if E = {V } or there is aleaf v incident to an hyperedge e such that:
� e is strongly adjacent to some other hyperedge,
� (V \ {v}, E \ {e}) is also a strong hypertree.
Example
(hyperedges are represented as cliques)
Example
12 / 22
Strong Hypertrees: Definitions
Definition
A hypergraph H = (V , E) is a strong hypertree either if E = {V } or there is aleaf v incident to an hyperedge e such that:
� e is strongly adjacent to some other hyperedge,
� (V \ {v}, E \ {e}) is also a strong hypertree.
Example
(hyperedges are represented as cliques)
Example
12 / 22
Strong Hypertrees: Definitions
Definition
A hypergraph H = (V , E) is a strong hypertree either if E = {V } or there is aleaf v incident to an hyperedge e such that:
� e is strongly adjacent to some other hyperedge,
� (V \ {v}, E \ {e}) is also a strong hypertree.
Example
(hyperedges are represented as cliques)
Example
12 / 22
Strong Hypertrees: Definitions
Definition
A hypergraph H = (V , E) is a strong hypertree either if E = {V } or there is aleaf v incident to an hyperedge e such that:
� e is strongly adjacent to some other hyperedge,
� (V \ {v}, E \ {e}) is also a strong hypertree.
Example
(hyperedges are represented as cliques)
Example
12 / 22
Strong Hypertrees: Definitions
Definition
A hypergraph H = (V , E) is a strong hypertree either if E = {V } or there is aleaf v incident to an hyperedge e such that:
� e is strongly adjacent to some other hyperedge,
� (V \ {v}, E \ {e}) is also a strong hypertree.
Not an exampleNot an example
12 / 22
Strong Hypertrees: Definitions
Definition
A strong hypertree is a strong hyperpath if it has exactly two leaves.
Example
12 / 22
Strong Hypertrees: Properties
� Similarly to ordinary trees:
Proposition
Every strong hypertree with n vertices:
� is a k-uniform hypergraph, for some k
� has n − k + 1 hyperedges
� has a full rank incidence matrix
� contains a strong hyperpath between any pair of non-adjacent vertices
Example
k = 3 |V | = 10 |E| = 8
13 / 22
Strong Hypertrees: Properties
� Similarly to ordinary trees:
Proposition
Every strong hypertree with n vertices:
� is a k-uniform hypergraph, for some k
� has n − k + 1 hyperedges
� has a full rank incidence matrix
� contains a strong hyperpath between any pair of non-adjacent vertices
Example
k = 3 |V | = 10 |E| = 8
13 / 22
Strong Hypertrees: Properties
� Similarly to ordinary trees:
Proposition
Every strong hypertree with n vertices:
� is a k-uniform hypergraph, for some k
� has n − k + 1 hyperedges
� has a full rank incidence matrix
� contains a strong hyperpath between any pair of non-adjacent vertices
Example
k = 3 |V | = 10 |E| = 8
13 / 22
The Procedure
14 / 22
The Procedure: Context
� Q: set of maximal cliques of G
� C(G) = (V ,Q): clique hypergraph of G
� T = (VT ,QT ) ⊆ C(G): A k-uniform strong hypertree such that:
� G [VT ] is k-partite with classes V1, . . . ,Vk
� No vertex in V0 := V \ VT has neighbors in all classes V1, . . . ,Vk
� Consider the face FT = {x ∈ STAB(G) : xQ = 1,∀Q ∈ QT}
Example
7
6 8
1
2 53 4
V0 = {6, 7, 8}V1 = {2, 4}V2 = {3, 5}V3 = {1}
FT =
8<:x ∈ STAB(G) :x1 + x2 + x3 = 1x1 + x3 + x4 = 1x1 + x4 + x5 = 1
9=;
15 / 22
The Procedure: Context
� Q: set of maximal cliques of G
� C(G) = (V ,Q): clique hypergraph of G
� T = (VT ,QT ) ⊆ C(G): A k-uniform strong hypertree such that:
� G [VT ] is k-partite with classes V1, . . . ,Vk
� No vertex in V0 := V \ VT has neighbors in all classes V1, . . . ,Vk
� Consider the face FT = {x ∈ STAB(G) : xQ = 1, ∀Q ∈ QT}
Example
7
6 8
1
2 53 4
V0 = {6, 7, 8}V1 = {2, 4}V2 = {3, 5}V3 = {1}
FT =
8<:x ∈ STAB(G) :x1 + x2 + x3 = 1x1 + x3 + x4 = 1x1 + x4 + x5 = 1
9=;15 / 22
The Procedure: Lemmas 1 and 2
Lemma
The dimension of FT is |V | − |QT |.
� Each hyperedge we remove from T increases the dimension of FT byexactly one (if the remaining hypergraph is still a strong hypertree)
� Useful when building the sequence of faces
Lemma
If x ∈ FT , then xu = xv , for all u, v ∈ Vi , i ∈ {1, . . . , k}.
� Each class can be seen as a single vertex
� Vk is selected if and only if no other class is selected
16 / 22
The Procedure: Lemmas 1 and 2
Lemma
The dimension of FT is |V | − |QT |.
� Each hyperedge we remove from T increases the dimension of FT byexactly one (if the remaining hypergraph is still a strong hypertree)
� Useful when building the sequence of faces
Lemma
If x ∈ FT , then xu = xv , for all u, v ∈ Vi , i ∈ {1, . . . , k}.
� Each class can be seen as a single vertex
� Vk is selected if and only if no other class is selected
16 / 22
The Procedure: Lemmas 1 and 2
Lemma
The dimension of FT is |V | − |QT |.
� Each hyperedge we remove from T increases the dimension of FT byexactly one (if the remaining hypergraph is still a strong hypertree)
� Useful when building the sequence of faces
Lemma
If x ∈ FT , then xu = xv , for all u, v ∈ Vi , i ∈ {1, . . . , k}.
� Each class can be seen as a single vertex
� Vk is selected if and only if no other class is selected
16 / 22
The Procedure: Lemmas 1 and 2
Lemma
The dimension of FT is |V | − |QT |.
� Each hyperedge we remove from T increases the dimension of FT byexactly one (if the remaining hypergraph is still a strong hypertree)
� Useful when building the sequence of faces
Lemma
If x ∈ FT , then xu = xv , for all u, v ∈ Vi , i ∈ {1, . . . , k}.
� Each class can be seen as a single vertex
� Vk is selected if and only if no other class is selected
16 / 22
The Procedure: Lemma 3
� Consider the graph GT obtained from G by:
� Contracting each class Vi into a vertex vi ∈ Vi , for i ∈ {1, . . . , k − 1}� Removing the vertices of Vk
Lemma
If V0 has no neighbors in VK then FT is affinely isomorphic to STAB(GT ).
Example
7
6 8
1
2 53 4
6
7
8
5 4
17 / 22
The Procedure: Lemma 3
� Consider the graph GT obtained from G by:
� Contracting each class Vi into a vertex vi ∈ Vi , for i ∈ {1, . . . , k − 1}� Removing the vertices of Vk
Lemma
If V0 has no neighbors in VK then FT is affinely isomorphic to STAB(GT ).
Example
17 / 22
The Procedure: Lemma 3
� Consider the graph GT obtained from G by:
� Contracting each class Vi into a vertex vi ∈ Vi , for i ∈ {1, . . . , k − 1}� Removing the vertices of Vk
Lemma
If V0 has no neighbors in VK then FT is affinely isomorphic to STAB(GT ).
Example
17 / 22
The Procedure: Lemma 3
� Consider the graph GT obtained from G by:
� Contracting each class Vi into a vertex vi ∈ Vi , for i ∈ {1, . . . , k − 1}� Removing the vertices of Vk
Lemma
If V0 has no neighbors in VK then FT is affinely isomorphic to STAB(GT ).
Example
17 / 22
The Procedure: Lemma 3
� Consider the graph GT obtained from G by:
� Contracting each class Vi into a vertex vi ∈ Vi , for i ∈ {1, . . . , k − 1}� Removing the vertices of Vk
Lemma
If V0 has no neighbors in VK then FT is affinely isomorphic to STAB(GT ).
Example
17 / 22
The Procedure: Lemma 3
� Consider the graph GT obtained from G by:
� Contracting each class Vi into a vertex vi ∈ Vi , for i ∈ {1, . . . , k − 1}� Removing the vertices of Vk
Lemma
If V0 has no neighbors in VK then FT is affinely isomorphic to STAB(GT ).
Example
17 / 22
The Procedure: Lemma 3
� Consider the graph GT obtained from G by:
� Contracting each class Vi into a vertex vi ∈ Vi , for i ∈ {1, . . . , k − 1}� Removing the vertices of Vk
Lemma
If V0 has no neighbors in VK then FT is affinely isomorphic to STAB(GT ).
Example
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The Procedure: Main Theorem
Theorem
Suppose that V0 has no neighbors in VK . IfXv∈V (GT )
πv xv ≤ π∗ (1)
is facet-defining for STAB(GT ), then there exists αQ , ∀Q ∈ QT , such thatXv∈V (GT )
πv xv +X
Q∈QT
αQ(xQ − 1) ≤ π∗ (2)
is facet-defining for STAB(G).
Sketch of the proof.
� The inequality (1) is facet-defining for FT (lemma 3)
� Remove one hyperedge at a time from T , but keeping it a strong hypertree
� Each removal increases dim(FT ) by exactly one (lemma 1)
� We have the sequence needed for the procedure outlined previously
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Example
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The Procedure: Example 1
Example
1
2 5
7
6 83 4
�P8
i=4 xi ≤ 1 is facet-defining for STAB(GT )
� α = max
8<:P8i=4 xi − 1 : x ∈ STAB(G),
x1 + x2 + x3 = 0,x1 + x3 + x4 = 1,x1 + x4 + x5 = 1
9=; = 1
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The Procedure: Example 1
Example
7
6 8
5 4
�P8
i=4 xi ≤ 1 is facet-defining for STAB(GT )
� α = max
8<:P8i=4 xi − 1 : x ∈ STAB(G),
x1 + x2 + x3 = 0,x1 + x3 + x4 = 1,x1 + x4 + x5 = 1
9=; = 1
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The Procedure: Example 1
Example
1
2 5
7
6 83 4
�P8
i=4 xi ≤ 1 is facet-defining for FT
� α = max
8<:P8i=4 xi − 1 : x ∈ STAB(G),
x1 + x2 + x3 = 0,x1 + x3 + x4 = 1,x1 + x4 + x5 = 1
9=; = 1
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The Procedure: Example 1
Example
1
2 5
7
6 83 4
�P8
i=4 xi ≤ 1 is facet-defining for FT
� α = max
8<:P8i=4 xi − 1 : x ∈ STAB(G),
x1 + x2 + x3 = 0,x1 + x3 + x4 = 1,x1 + x4 + x5 = 1
9=; = 1
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The Procedure: Example 1
Example
2
7
6 8
53 4
1
�P3
i=1 xi +P8
i=4 xi ≤ 2 is facet-defining for FT
� α = max
8<:P3i=1 xi +
P8i=4 xi − 2 : x ∈ STAB(G),
x1 + x2 + x3 = 0,
x1 + x3 + x4 = 0,x1 + x4 + x5 = 1
9=; = 1
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The Procedure: Example 1
Example
2 5
6 83 4
7
1
�P3
i=1 xi +P8
i=4 xi ≤ 2 is facet-defining for FT
� α = max
8<:P3i=1 xi +
P8i=4 xi − 2 : x ∈ STAB(G),
x1 + x2 + x3 = 0,
x1 + x3 + x4 = 0,x1 + x4 + x5 = 1
9=; = 1
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The Procedure: Example 1
Example
2
7
6 8
53 4
1
� 2x1 + x2 + 2x3 + 2x4 +P8
i=5 xi ≤ 3 is facet-defining for FT
� α = max
8<:2x1 + x2 + 2x3 + 2x4 +P8
i=5 xi − 3 : x ∈ STAB(G),
x1 + x2 + x3 = 0,x1 + x3 + x4 = 0,
x1 + x4 + x5 = 0
9=; = 0
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The Procedure: Example 1
Example
2 5
7
6 83 4
1
� 2x1 + x2 + 2x3 + 2x4 +P8
i=5 xi ≤ 3 is facet-defining for FT
� α = max
8<:2x1 + x2 + 2x3 + 2x4 +P8
i=5 xi − 3 : x ∈ STAB(G),
x1 + x2 + x3 = 0,x1 + x3 + x4 = 0,
x1 + x4 + x5 = 0
9=; = 0
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The Procedure: Example 1
Example
2
7
6 8
53 4
1
� 2x1 + x2 + 2x3 + 2x4 +P8
i=5 xi ≤ 3 is facet-defining for STAB(G)
� α = max
8<:x4 + x5 + x6 + x7 + x8 − 1 : x ∈ STAB(G),x1 + x2 + x3 = 0,x1 + x3 + x4 = 1,x1 + x4 + x5 = 1
9=; = 1
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Concluding Remarks
� We presented a new facet generating procedure which:
� Unifies and generalizes many previous procedures
� Generates new facet-defining inequalities
� We also introduced the concept of strong hypertree
� Future research:
� Use the procedure to find new facet producing subgraphs
� Apply the main idea to other polytopes:
� Find faces that are affinely isomorphic to known polytopes
� Convert their facets into facets of the polytope
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Concluding Remarks
� We presented a new facet generating procedure which:
� Unifies and generalizes many previous procedures
� Generates new facet-defining inequalities
� We also introduced the concept of strong hypertree
� Future research:
� Use the procedure to find new facet producing subgraphs
� Apply the main idea to other polytopes:
� Find faces that are affinely isomorphic to known polytopes
� Convert their facets into facets of the polytope
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Concluding Remarks
� We presented a new facet generating procedure which:
� Unifies and generalizes many previous procedures
� Generates new facet-defining inequalities
� We also introduced the concept of strong hypertree
� Future research:
� Use the procedure to find new facet producing subgraphs
� Apply the main idea to other polytopes:
� Find faces that are affinely isomorphic to known polytopes
� Convert their facets into facets of the polytope
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Thank you!
22 / 22