On the Theory of Pfaffian Orientations.  I. Perfect Matchings and Permanents

Galluccio, Anna; Loebl, Martin

doi:10.37236/1438

Cited by 86 publications

(96 citation statements)

References 9 publications

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“…In order to obtain a more refined view on the complexity of the permanent, and to cope with its hardness in view of practical applications, various relaxations of this problem were studied: A celebrated randomized approximation scheme [34,33] allows one to approximate the permanent on matrices with non-negative entries. Furthermore, on some restricted graph classes, PerfMatch can be solved in time O(n 3 ): This includes the above-mentioned planar graphs, and in fact, all graph classes of bounded genus [29,49,44]. (We will present more classes in the remainder of the introduction.)…”

mentioning

confidence: 99%

Parameterizing the Permanent: Genus, Apices, Minors, Evaluation Mod 2k

Curticapean

Xia

2015

2015 IEEE 56th Annual Symposium on Foundations of Computer Science

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We identify and study relevant structural parameters for the problem PerfMatch of counting perfect matchings in a given input graph G. These generalize the well-known tractable planar case, and they include the genus of G, its apex number (the minimum number of vertices whose removal renders G planar), and its Hadwiger number (the size of a largest clique minor).To study these parameters, we first introduce the notion of combined matchgates, a general technique that bridges parameterized counting problems and the theory of so-called Holants and matchgates: Using combined matchgates, we can simulate certain non-existing gadgets F as linear combinations of t = O(1) existing gadgets. If a graph G features k occurrences of F , we can then reduce G to t k graphs that feature only existing gadgets, thus enabling parameterized reductions.As applications of this technique, we simplify known 4 g n O(1) time algorithms for PerfMatch on graphs of genus g. Orthogonally to this, we show #W[1]-hardness of the permanent on k-apex graphs, implying its #W[1]-hardness under the Hadwiger number. Additionally, we rule out n o(k/ log k) time algorithms under the counting exponential-time hypothesis #ETH.Finally, we use combined matchgates to prove ⊕W[1]-hardness of evaluating the permanent modulo 2 k , complementing an O(n 4k−3 ) time algorithm by Valiant and answering an open question of Björklund. We also obtain a lower bound of n Ω(k/ log k) under the parity version ⊕ETH of the exponential-time hypothesis.

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confidence: 99%

Parameterizing the Permanent: Genus, Apices, Minors, Evaluation Mod 2k

Curticapean

Xia

2015

2015 IEEE 56th Annual Symposium on Foundations of Computer Science

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“…Observe that equations (3.1), (3.2) imply (1.3). The proof of the Kasteleyn identities (3.2) can be found in the works of Galluccio and Loebl [4], Tesler [13], and Cimasoni and Reshetikhin [2]. It follows from formula (2.12) that the Pfaffians Pf A i are multivariate polynomials with respect to the weights z h , z v , z d .…”

Section: Preliminary Resultsmentioning

confidence: 95%

“…As shown by Galluccio and Loebl [4], Tesler [13], and Cimasoni and Reshetikhin [2], the dimer models on an orientable Riemann surface of genus g is expressed as a linear algebraic combination of 2 2g Pfaffians. It is extended to non-orientable surfaces in the work of Tesler [13].…”

Section: Resultsmentioning

confidence: 97%

The Pfaffian Sign Theorem for the Dimer Model on a Triangular Lattice

2018

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We prove the Pfaffian Sign Theorem for the dimer model on a triangular lattice embedded in the torus. More specifically, we prove that the Pfaffian of the Kasteleyn periodic-periodic matrix is negative, while the Pfaffians of the Kasteleyn periodicantiperiodic, antiperiodic-periodic, and antiperiodic-antiperiodic matrices are all positive. The proof is based on the Kasteleyn identities and on small weight expansions. As an application, we obtain an asymptotic behavior of the dimer model partition function with an exponentially small error term.

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“…He also stated without further detail that the partition function for a graph of genus g requires 2 2g Pfaffians [13]. Such a formula was found much later by Tesler [21] and Gallucio-Loebl [9], independently. (See also [6].)…”

Section: Kasteleyn's Theoremmentioning

confidence: 92%

The geometry of dimer models

Cimasoni¹

2015

Winter Braids Lecture Notes

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This is an expanded version of a three-hour minicourse given at the winterschool Winterbraids IV held in Dijon in February 2014. The aim of these lectures was to present some aspects of the dimer model to a geometrically minded audience. We spoke neither of braids nor of knots, but tried to show how several geometric tools that we know and love (e.g. (co)homology, spin structures, real algebraic curves) can be applied to very natural problems in combinatorics and statistical physics.

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On the Theory of Pfaffian Orientations. I. Perfect Matchings and Permanents

Abstract: Kasteleyn stated that the generating function of the perfect matchings of a graph of genus $g$ may be written as a linear combination of $4^g$ Pfaffians. Here we prove this statement. As a consequence we present a combinatorial way to compute the permanent of a square matrix.

Cited by 86 publications

References 9 publications

Parameterizing the Permanent: Genus, Apices, Minors, Evaluation Mod 2k

Parameterizing the Permanent: Genus, Apices, Minors, Evaluation Mod 2k

The Pfaffian Sign Theorem for the Dimer Model on a Triangular Lattice

The geometry of dimer models

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