distributive lattices from graphs
TRANSCRIPT
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Distributive Lattices from Graphs
VI Jornadas de Matematica
Discreta y Algorıtmica
Universitat de Lleida
21-23 de julio de 2008
Stefan Felsner y Kolja Knauer
Technische Universitat [email protected]
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The Talk
Lattices from Graphs
Proving Distributivity: ULD-Lattices
Embedded Lattices and D-Polytopes
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Contents
Lattices from Graphs
Proving Distributivity: ULD-Lattices
Embedded Lattices and D-Polytopes
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Lattices from Planar Graphs
Definition. Given G = (V, E) and α : V → IN.
An α-orientation of G is an orientation with
outdeg(v) = α(v) for all v.
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Lattices from Planar Graphs
Definition. Given G = (V, E) and α : V → IN.
An α-orientation of G is an orientation with
outdeg(v) = α(v) for all v.
• Reverting directed cycles preserves α-orientations.
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Lattices from Planar Graphs
Definition. Given G = (V, E) and α : V → IN.
An α-orientation of G is an orientation with
outdeg(v) = α(v) for all v.
• Reverting directed cycles preserves α-orientations.
Theorem. The set of α-orientations of a planar graph G has
the structure of a distributive lattice.
• Diagram edge ∼ revert a directed essential/facial cycle.
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Example 1: Spanning Trees
Spanning trees are in bijection with αT orientations of a rooted
primal-dual completion G of G
• αT(v) = 1 for a non-root vertex v and αT(ve) = 3 for an
edge-vertex ve and αT(vr) = 0 and αT(v∗r ) = 0.
v∗r
vr
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Lattice of Spanning TreesGilmer and Litheland 1986, Propp 1993
e
e ′
v
v
vr
vr
v∗r
v∗r
T ′
T
G
Question. How does a change of roots affect the lattice?
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Example2: Matchings and f-Factors
Let G be planar and bipartite with parts (U, W). There is
bijection between f-factors of G and αf orientations:
• Define αf such that indeg(u) = f(u) for all u ∈ U and
outdeg(w) = f(w) for all w ∈ W.
Example. A matching and the corresponding orientation.
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Example 3: Eulerian Orientations
• Orientations with outdeg(v) = indeg(v) for all v,
i.e. α(v) =d(v)
2
7
41
26 5
3
5
2
2 ′ 3 ′
1 ′′
4 ′
7 1 ′6
3 4
1
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Example 4: Schnyder Woods
G a plane triangulation with outer triangle F = {a1,a2,a3}.
A coloring and orientation of the interior edges of G with
colors 1,2,3 is a Schnyder wood of G iff
• Inner vertex condition:
• Edges {v, ai} are oriented v → ai in color i.
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Digression: Schnyder’s Theorem
The incidence order PG of a graph G
PG
G
Theorem [Schnyder 1989 ].
A Graph G is planar ⇐⇒ dim(PG) ≤ 3.
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Schnyder Woods and 3-Orientations
Theorem. Schnyder wood and 3-orientation are in bijection.
Proof.• All edges incident to ai are oriented → ai.
Prf: G has 3n − 9 interior edges and n − 3 interior
vertices.
• Define the path of an edge:
• The path is simple (Euler), hence, ends at some ai.
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The Lattice of Schnyder Woods
Theorem. The set of Schnyder woods of a plane triangulation
G has the structure of a distributive lattice.
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A Dual Construction:c-Orientations
• Reorientations of directed cuts preserve flow-difference
(#forward arcs − #backward arcs) along cycles.
Theorem [Propp 1993 ]. The set of all orientations of a
graph with prescribed flow-difference for all cycles has the
structure of a distributive lattice.
• Diagram edge ∼ push a vertex ( 6= v†).
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Circulations in Planar Graphs
Theorem [Khuller, Naor and Klein 1993 ].The set of all integral flows respecting capacity constraints
(`(e) ≤ f(e) ≤ u(e)) of a planar graph has the structure of a
distributive lattice.
0 ≤ f(e) ≤ 1
• Diagram edge ∼ add or subtract a unit of flow in ccw
oriented facial cycle.
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∆-Bonds
G = (V, E) a connected graph with a prescribed orientation.
With x ∈ ZZE and C cycle we define the circular flow difference
∆x(C) :=∑e∈C+
x(e) −∑e∈C−
x(e).
With ∆ ∈ ZZC and `, u ∈ ZZE let BG(∆, `, u) be the set of
x ∈ ZZE such that ∆x = ∆ and ` ≤ x ≤ u.
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The Lattice of ∆-Bonds
Theorem [Felsner, Knauer 2007 ].BG(∆, `, u) is a distributive lattice.
The cover relation is vertex pushing.
u
`
u
`
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∆-Bonds as Generalization
BG(∆, `, u) is the set of x ∈ IRE such that
• ∆x = ∆ (circular flow difference)
• ` ≤ x ≤ u (capacity constraints).
Special cases:
• c-orientations are BG(∆, 0, 1)
(∆(C) = |C+| − c(C)).
• Circular flows on planar G are BG∗(0, `, u)
(G∗ the dual of G).
• α-orientations.
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Contents
Lattices from Graphs
Proving Distributivity: ULD-Lattices
Embedded Lattices and D-Polytopes
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ULD Lattices
Definition. [Dilworth ]A lattice is an upper locally distributive lattice (ULD) if each
element has a unique minimal representation as meet of meet-
irreducibles, i.e., there is a unique mapping x → Mx such that
• x =∧
Mx (representation.) and
• x 6=∧
A for all A $ Mx (minimal).
c
b
ae
d
c ∧ ea ∧ d
0 = a ∧ e =∧
{a, b, c, d, e}
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ULD vs. Distributive
Proposition.A lattice it is ULD and LLD ⇐⇒ it is distributive.
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Diagrams of ULD lattices:A Characterization
A coloring of the edges of a digraph is a U-coloring iff
• arcs leaving a vertex have different colors.
• completion property:
Theorem.A digraph D is acyclic, has a unique source and admits a
U-coloring ⇐⇒ D is the diagram of an ULD lattice.
↪→ Unique 1.
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Examples of U-colorings
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Examples of U-colorings
• Chip firing game with a fixed starting position (the
source), colors are the names of fired vertices.
• ∆-bond lattices, colors are the names of pushed vertices.
(Connected, unique 0).
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More Examples
Some LLD lattices with respect to inclusion order:
• Subtrees of a tree (Boulaye ’67).
• Convex subsets of posets (Birkhoff and Bennett ’85).
• Convex subgraphs of acyclic digraphs (Pfaltz ’71).
(C is convex if with x, y all directed (x, y)-paths are in C).
• Convex sets of an abstract convex geometry, this is an
universal family of examples (Edelman ’80).
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Contents
Lattices from Graphs
Proving Distributivity: ULD-Lattices
Embedded Lattices and D-Polytopes
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Embedded Lattices
A U-coloring of a distributive lattice L yields a cover preserving
embedding φ : L → ZZ#colors.
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Embedded Lattices
A U-coloring of a distributive lattice L yields a cover preserving
embedding φ : L → ZZ#colors.
In the case of ∆-bond lattices there is a
polytope P = conv(φ(L) in IRn−1 such that
φ(L) = P ∩ ZZn−1
• This is a special property:
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D-Polytopes
Definition. A polytope P is a D-polytope if with x, y ∈ P
also max(x, y),min(x, y) ∈ P.
• A D-polytope is a (infinite!) distributive lattice.
• Every subset of a D-polytope generates a distributive
lattice in P. E.g. Integral points in a D-polytope are a
distributive lattice.
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D-Polytopes
Remark. Distributivity is preserved under
• scaling
• translation
• intersection
Theorem. A polytope P is a D-polytope iff every facet
inducing hyperplane of P is a D-hyperplane, i.e., closed under
max and min.
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D-Hyperplanes
Theorem. An hyperplane is a D-hyperplane iff it has a normal
ei − λijej with λij ≥ 0.
( ⇐)λijei + ej together with ek with k 6= i, j is a basis. The
coefficient of max(x, y) is the max of the coefficients
of x and y.
( ⇒) Let n =∑
i aiei be the normal vector. If ai > 0 and
aj > 0, then x = ajei − aiej and y = −x are in n⊥ but
max(x, y) is not.
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A First Graph Model for D-Polytopes
Consider `, u ∈ ZZm and a Λ-weighted network matrix NΛ of
a connected graph. (Rows of NΛ are of type ei − λijej with λij ≥ 0.)
• [Strong case, rank(NΛ) = n]
The set of p ∈ ZZn with ` ≤ N>Λp ≤ u is a distributive
lattice.
• [Weak case, rank(NΛ) = n − 1]
The set of p ∈ ZZn−1 with ` ≤ N>Λ(0, p) ≤ u is a
distributive lattice.
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A Second Graph Model forD-Polytopes
(Rows of NΛ are of type ei − λijej with λij ≥ 0.)
Theorem [Felsner, Knauer 2008 ].Let Z = ker(NΛ) be the space of Λ-circulations. The set of
x ∈ ZZm with
• ` ≤ x ≤ u (capacity constraints)
• 〈x, z〉 = 0 for all z ∈ Z
(weighted circular flow difference).
is a distributive lattice DG(Λ, `, u).
• Lattices of ∆-bonds are covered by the case λij = 1.
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The Strong Case
For a cycle C let
γ(C) :=∏e∈C+
λe
∏e∈C−
λ−1e .
A cycle with γ(C) 6= 1 is strong.
Proposition. rank(NΛ) = n iff it contains a strong cycle.
Remark.C strong =⇒ there is no circulation with support C.
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Fundamental Basis
A fundamental basis for the space of Λ-circulations:
• Fix a 1-tree T , i.e, a unicyclic set of n edges. With e 6∈ T
there is a circulation in T + e
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Fundamental Basis
A fundamental basis for the space of Λ-circulations:
• Fix a 1-tree T , i.e, a unicyclic set of n edges. With e 6∈ T
there is a circulation in T + e
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Fundamental Basis
In the theory of generalized flows ,i,e, flows with multiplicative
losses and gains, these objects are known as bicycles.
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Fundamental Basis
In the theory of generalized flows ,i,e, flows with multiplicative
losses and gains, these objects are known as bicycles.
=⇒ Further topic: D-polytopes and optimization.
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Conclusion
• ∆-bond lattices generalize previously known distributive
lattices from graphs.
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Conclusion
• ∆-bond lattices generalize previously known distributive
lattices from graphs.
• U-colorings yield pretty proves for UL-distributivity and
distributivity.
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Conclusion
• ∆-bond lattices generalize previously known distributive
lattices from graphs.
• U-colorings yield pretty proves for UL-distributivity and
distributivity.
• D-polytopes are related to generalized network matrices.
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Conclusion
• ∆-bond lattices generalize previously known distributive
lattices from graphs.
• U-colorings yield pretty proves for UL-distributivity and
distributivity.
• D-polytopes are related to generalized network matrices.
Finally:
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Conclusion
• ∆-bond lattices generalize previously known distributive
lattices from graphs.
• U-colorings yield pretty proves for UL-distributivity and
distributivity.
• D-polytopes are related to generalized network matrices.
Finally: Don’t forget Schnyder’s Theorem.
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Conclusion
• ∆-bond lattices generalize previously known distributive
lattices from graphs.
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Conclusion
• ∆-bond lattices generalize previously known distributive
lattices from graphs.
• U-colorings yield pretty proves for UL-distributivity and
distributivity.
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Conclusion
• ∆-bond lattices generalize previously known distributive
lattices from graphs.
• U-colorings yield pretty proves for UL-distributivity and
distributivity.
• D-polytopes are related to generalized network matrices.
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Conclusion
• ∆-bond lattices generalize previously known distributive
lattices from graphs.
• U-colorings yield pretty proves for UL-distributivity and
distributivity.
• D-polytopes are related to generalized network matrices.
Finally:
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Conclusion
• ∆-bond lattices generalize previously known distributive
lattices from graphs.
• U-colorings yield pretty proves for UL-distributivity and
distributivity.
• D-polytopes are related to generalized network matrices.
Finally: Don’t forget Schnyder’s Theorem.
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The End