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 orientations. 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 I G of the graph G is V × E matrix defined by ( I G ) v , e = 1 if v ∈ e and is zero otherwise. We consider I G over the field F 2 : counting is modulo 2. Theorem The Kernel of I G , i.e., the set { u ∈ F E 2 : I G u = 0 } , is the set of the incident vectors of the even edge-sets. The Kernel of I G is a vector space over F 2 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 I G 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 = ( A i , j ). Per ( A ) = � � i A i ,π ( i ) , π det( A ) = � π sign ( π ) � i A i ,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: perfect matching P x w ( P ) . P ( G , w , x ) = � Let D be its orientation and let M be its perfect matching. We let the Pfaffian be perfect matching P sign ( D , M ; P ) x w ( P ) . Pfaf ( D . M , w , x ) = � 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) 4 g � Per ( A ) = 2 − g s i det( A i ) , i =1 where s i ∈ { 1 , − 1 } and each A i 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 A i so that Per ( A ) is linear combination of their determinants? Norine made conjecture in 2004 that the answer is always power of 4 (4 g ) 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 x e associated with each edge e , x = ( x e ) e ∈ E . Ising partition function is � � E ( G , x ) = x e . A ⊂ E even e ∈ A There is a natural way to define basic sign s ( A ) for each even set of edges; we let E s ( G , x ) = � � s ( A ) x e . A ⊂ E even e ∈ A 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 S i , i = 1 , . . . , c of G so that E ( G , x ) is linear combination of E i ( G , x ) = � s ( A )( − 1) | A ∩ S i | � x e . A ⊂ E even e ∈ A Theorem (Loebl, Masbaum, 2011) Aditive determinat complexity of Ising partition function is 4 g . 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 = ( A i , j ) as matrix of variables; det( A ) , Per ( A ) are thus multivariable polynomials with each coefficient 1 or − 1. Per ( A ) = � � i A i ,π ( i ) . π 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 A i , j ’s by other 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 of 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
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