Theory of Kasteleyn Orientations Martin Loebl Sep. 10, 2018 - - PowerPoint PPT Presentation

Theory of Kasteleyn Orientations

Martin Loebl

• Sep. 10, 2018

Matching theory became very rich and complex, beautiful part of discrete mathematics and I want to introduce one of its jewels: the theory of Kasteleyn

• rientations.

Martin Loebl Theory of Kasteleyn Orientations

• 1. Eulerian closed tour.

We denote graphs as G = (V , E) where V is the set of vertices and E is the set of edges. In the graph G, an eulerian tour is a sequence of adjacent edges which contains each edge exactly once. An eulerian tour is closed if it starts and ends in the same vertex. Theorem (Euler 1736) Graph G has a closed eulerian tour if and only if it is connected and each vertex-degree is even.

Martin Loebl Theory of Kasteleyn Orientations

• 2. Even edge-sets and the cycle space of graph G.

A set of edges E ′ ⊂ E is even if the graph (V , E ′) has all degrees even. An example is the empty set ∅: it induces degree zero (even) at each vertex.

• Claim. Each even edge-set can be partitioned into edge-sets of cycles.

This easily implies the Euler’s theorem, and provides an efficient algorithm to find a closed eulerian tour, if it exists. The incidence matrix IG of the graph G is V × E matrix defined by (IG)v,e = 1 if v ∈ e and is zero otherwise. We consider IG over the field F2: counting is modulo 2. Theorem The Kernel of IG, i.e., the set {u ∈ F E

2 : IGu = 0}, is the set of the incident

vectors of the even edge-sets. The Kernel of IG is a vector space over F2 called the cycle space of G.

Martin Loebl Theory of Kasteleyn Orientations

• 3. Edge-cuts and the cut space of graph G.

A set of edges E ′ ⊂ E is edge-cut if there is V ′ ⊂ V so that E ′ = {e ∈ E : |e ∩ V ′| = 1}. Theorem The set of all sums (modulo 2) of subsets of rows of IG is the set of the incident vectors of the edge-cuts. This set is called the cut space of G. MAX CUT PROBLEM: Find an edge-cut with maximum number of edges (maximum total weight). Max Cut is a basic extensively studied NP-complete problem. Compare: Min Cut is polynomial: max flow min cut theorem.

Martin Loebl Theory of Kasteleyn Orientations

• 4. Max Cut is polynomial for the planar graphs.

Let G = (V , E) be a planar graph properly drawn in the plane. Let G ∗ be its geometric dual, i.e., G ∗ = (F, E, f ) where F is the set of the faces of G and f : E → V ∪ V

2

• .

Lemma A ⊂ E is an edge-cut of G if and only if A∗ is an even set of G ∗. Hence MAX CUT problem for G is equivalent to MAX EVEN SET of G ∗. Theorem (Fisher 60’s) Let G be any graph (not necessarily planar). There is a graph G∆ and a natural bijection between the set of the even sets of G and the set of the perfect matchings of G∆. Moreover, G and G∆ have the same genus. Theorem (Edmonds 60’s) Perfect matching of max weight can be found in strongly polynomial time.

Martin Loebl Theory of Kasteleyn Orientations

• 5. Fisher’s construction.

Theorem (Fisher 60’s) Let G be any graph (not necessarily planar). There is a graph G∆ and a natural bijection between the set of the even sets of G and the set of the perfect matchings of G∆. Moreover, G and G∆ have the same genus. Replace each vertex of graph G by a path of triangles. Summarising, MAX CUT problem for the planar graphs can be found is strongly polynomial time. This is still open for toroidal graphs. Only weakly polynomial algorithm is

• known. It is based on enumeration.

Martin Loebl Theory of Kasteleyn Orientations

• 6. Optimisation by enumeration I.

We consider square real matrix A = (Ai,j). Per(A) =

π

• i Ai,π(i),

det(A) =

π sign(π) i Ai,π(i),.

It iS HARD (Sharp P) to calculate permanents. It is EASY (polynomial) to calculate determinants. If G is bipartite graph then the permanent of its adjacency matrix is equal to the number of perfect matchings of G. !! Generating function of signed perfect matchings can be calculated as a determinant (efficiently) !! This is true for the general graphs: determinants are replaced by Pfaffians. Pfaffians introduced into discrete mathematics by Tutte in the proof of his seminal Perfect matchings theorem. From there into theoretical physics. W.T. Tutte, The factorisation of linear graphs, J. London Mathematical Society 22 (2) 1947

Martin Loebl Theory of Kasteleyn Orientations

• 7. Optimisation by enumeration II.

Let G = (V , E) be a graph, and w : E → Q be a (rational) weight function. If P ⊂ E then let w(P) =

e∈P w(e).

Generating function of perfect matchings: P(G, w, x) =

perfect matching P xw(P).

Let D be its orientation and let M be its perfect matching. We let the Pfaffian be Pfaf (D.M, w, x) =

perfect matching P sign(D, M; P)xw(P).

The sign(D, M; P) is defined as (−1)z where z is the number of D-clockwise even cycles of M∆P. P(G, w, x) is HARD to calculate; Pfaf (D.M, w, x) is EASY to calculate.

Martin Loebl Theory of Kasteleyn Orientations

• 8. Optimisation by enumeration III: Kasteleyn’s theorem

Theorem (Kasteleyn; Fisher, Temperley 61) Each planar graph G has an orientation D so that for each perfect matching M, P(G, w, x) = Pfaf (D.M, w, x). How to construct D? Make each inner face clockwise odd. As a corollary, for the planar graphs, the generating functions of perfect matchings, even sets, edge-cuts can be calculated efficiently as a single determinant-type expression (the Pfaffian). How about toroidal graphs? Higher genus graphs?

Martin Loebl Theory of Kasteleyn Orientations

• 9. Optimisation by enumeration IV: Aditive determinant complexity

Theorem (Kasteleyn 61; Galluccio, Loebl 89; Tessler 90; Cimasoni, Reshetikhin 2002) Per(A) = 2−g

4g

• i=1

si det(Ai), where si ∈ {1, −1} and each Ai is obtained from A by change of sign of some

• entries. Here g is genus of the bipartite graph whose adjacency matrix is A.

Aditive determinantal complexity: What is minimum number of signings Ai so that Per(A) is linear combination of their determinants? Norine made conjecture in 2004 that the answer is always power of 4 (4g) but it was disproved by Miranda and Lucchesi. It is not known whether aditive determinant complexity of the permanent is exponential in the size of the matrix.

Martin Loebl Theory of Kasteleyn Orientations

• 9. Optimisation by enumeration V: Even sets of edges and Ising partition

function

A set A of edges of graph G = (V , E) is even if graph (V , A) has all degrees

• even. For example, the empty set is even.

Graph G = (V , E) variable xe associated with each edge e, x = (xe)e∈E. Ising partition function is E(G, x) =

• A⊂E even
• e∈A

xe. There is a natural way to define basic sign s(A) for each even set of edges; we let Es(G, x) =

• A⊂E even

s(A)

• e∈A

xe.

Martin Loebl Theory of Kasteleyn Orientations

• 10. Optimisation by enumeration VI: Aditive determinant complexity of

Ising partition function

Aditive determinant complexity of E(G, x): minimum number c of sets of edges Si, i = 1, . . . , c of G so that E(G, x) is linear combination of Ei(G, x) =

• A⊂E even

s(A)(−1)|A∩Si |

e∈A

xe. Theorem (Loebl, Masbaum, 2011) Aditive determinat complexity of Ising partition function is 4g. Is there a relation of determinant and aditive determinat complexity?

Martin Loebl Theory of Kasteleyn Orientations

• 11. Optimisation by enumeration VII: Conjecture: Permanent is

exponentially harder than determinant.

We consider matrix A = (Ai,j) as matrix of variables; det(A), Per(A) are thus multivariable polynomials with each coefficient 1 or −1. Per(A) =

π

• i Ai,π(i).

If G is bipartite graph then the permanent of its adjacency matrix is equal to the number of perfect matchings of G. Formula size of a polynomial: minimum number of additions and multiplications needed to get the polynomial startring from the variables. Valiant: Determinant complexity of a polynomial: min size of a matrix A so that the polynomial equals det(A) after substitution of some Ai,j’s by

• ther variables or real constants.

Theorem (Valiant). Determinant complexity is at most twice formula size. Does permanent have exponential determinant complexity?

Martin Loebl Theory of Kasteleyn Orientations

• 12. Optimisation by enumeration VIII: Summary and open problems

There is a weakly polynomial algorithm to solve MAX CUT problem for the graphs of any fixed genus. Is there a strongly polynomial or ’direct’ algorithm for MAX CUT for subgraphs

• f toroidal square grids?

Is optimization of edge-cuts easier than enumeration of edge-cuts for the embedded graphs? Is it possible to relate the additive determinant complexity and the Strong Exponential Time Hypothesis?

Martin Loebl Theory of Kasteleyn Orientations

• 13. Kasteleyn theory for hypergraphs: Feynman, Sherman formal product

Earlier attempt (50’s) to write E(G, x), G planar, as a determinant, by Kac and Ward. They used the matrix of transitions M between orientations of edges of G determined by the ’rotation contribution’. The formula correct, the

• riginal proof wrong.

Feynman provided a way towards a correct proof of the formula. Feynman noticed and Sherman proved in the beginning of 60’ that E(G, x)2 for planar graph G is equal to a formal (infinite) product: E(G, x)2 =

• p

(1 −

• t transition of p

Mt), where the product ranges over aperiodic reduced closed walks p on G. This infinite product is equal to the determinant of Kac, Ward. The product expression is useful for studying the logarithm of the Ising partition function.

Martin Loebl Theory of Kasteleyn Orientations

• 14. Kasteleyn theory for hypergraphs: Bass’ theorem

D = (V , A) directed graph with no loops, x = (xa)a∈A vector of variables. Let Z(D) denote the set of all aperiodic closed walks of D. Discrete Ihara Selberg function ISD(x) =

• W ∈Z(D)

(1 −

• a∈W

xa). Theorem (Bass 80’s) ISD(x) = det(I − A(D, x)) where A(D, x)(u,v) = x(u,v) if (u, v) belongs to D, and it is zero otherwise. Lemma det(I − A(D, x)) =

• C1,...,Ck vertex-disjoint dicycles

(−1)k

a∈∪Ci

xa

Martin Loebl Theory of Kasteleyn Orientations

• 15. Kasteleyn theory for hypergraphs: Geometric representations I (jointly

with Pavel Rytir)

Motivating questions: Can the Theory of Kasteleyn orientations and formal products be generalised from graphs to hypergraphs? Can the Theory of Kasteleyn orientations and formal products be generalised from the cycle space of graphs to general binary linear codes? View a graph as a geometric representation of its cycle space. Is there a higher dimensional geometric representation which is more advantageous? Theorem (Rytir 2010 master thesis) Let C be a vector space over F2 (i.e., a binary linear code). Then there is an almost disjoint 3-hypergraph H so that the weight enumerator of C is equal to the cycle space of H.

Martin Loebl Theory of Kasteleyn Orientations

• 16. Kasteleyn theory for hypergraphs. Geometric representations II; jointly

with Pavel Rytir

The permanent of a (k + 1)−matrix A is defined to be Per(A) =

• σ1,...,σk ∈Sn

n

• i=1

aσ1(i)...σk (i)i. The determinant of a (k + 1)−matrix A is defined to be det(A) =

• σ1,...,σk ∈Sn

sign(σ1) . . . sign(σk)

n

• i=1

aσ1(i)...σk (i)i. Theorem (ML, Rytir) Let C be a vector space over F2 (i.e., a binary linear code). Then there is an almost disjoint 3-partite 3-hypergraph H so that the weight enumerator of C is equal to Per(T(H, x), where T(H, x) is the incidence 3-matrix of H.

Martin Loebl Theory of Kasteleyn Orientations

• 17. Kasteleyn 4-matrices, 4-dimensional Bass’ theorem

We say that a 4-matrix A is Kasteleyn if there is 4-matrix A′ obtained from A by changing signs of some entries so that Per(A) = det(A′). Theorem (ML, 2018) Let C be a vector space over F2 (i.e., a binary linear code). Then there is a directed 4-hypergraph D so that the weight enumerator of C is equal to det(I − A(D, x)); matrix A(D, x) is the adjacency matrix of D. Theorem (ML, 2018) The theorem of Bass holds for the determinant of 4-matrices as well. Summarising, the weight enumerator of each binary linear code is expressed as a single formal product. Work in progress: formulas for the log of the 3-dimensional Ising partition function.

Martin Loebl Theory of Kasteleyn Orientations