Matchings in lattice graphs

We study the problem of counting the number of matchings of given cardinalitg in a d-dimensional rectangular lattice. This problem arises in several models in statistical phgsics, including monomerdimer systems and cell-cluster theory. A classical algorithm due to Fisher, Kasteleyn and Temperley counts perfect matchings exactly in two dimensions, but is not applicable in higher dimensions and does not allow one to count matchings of arbitrary cardinality. In this paper, we present the first eficient approximation algorithms for counting matchings of arbitrary cardinality in (i) d-dimensional ‘>en”odic” lattices (i. e., with wrap-around edges) in any fixed dimension d; and (ii) two-dimensional lattices with “fixed boundary conditions” (i. e., no wrap-around edges). Our technique generalizes to approximately counting matchings in any bipartite graph that is the Cayley graph of some finite group. t CNRS, Ecole Normale Sup&ieure de Lyon, France. Part of this work was done while this author was visiting ICSI, Berkeley. E-mail: kenyon@lip. ens-lyon. f r, $Department of Computer Science, University of California at Berkeley. Supported in part by an AT&T PhD Fellowship and NSF grant CCR88-13632. E-mail: randall@cs. berkeley. edu. $University of Edinburgh and International Computer Science Institute, Berkeley. Supported in part by grant GR/F 90363 of the UK Science and Engineering Research Council. E-mail: sinclairOicsi. berkeley. edu. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed fc,r “direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of tha Association for Computing Machinery. To copy otherwise, or to rapublish, requires a fea and/or specific permission. 25th ACM STOC ‘93-51931CA,WA 01993 ACM 0-89791 -591 -71931000510738 . ..$1 .50 1 Summary 1.1 Background and mot ivation This paper is concerned with the following computational problem: given a finite lattice graph in some fixed number of dimensions, and some number of dominoes, determine the number of ways of placing dominoes on the edges of the graph so that no two dominoes overlap at a vertex. Equivalently, we can think of dominoes as covering a pair of adjacent squares (cubes) in the dual lattice. This is a classical problem in statistical physics, first introduced by Fowler and Rushbrooke in 1937 [3], and is the earliest example of a large class of problems concerned with computing the number of nonoverlapping arrangements of figures of various shapes on a lattice (see, e.g., [11, 16] for a survey). The problem arises in several physical models. For example, in two dimensions the lattice represents the surface of a crystal and the dominoes diatomic molecules (or dimers), and the number of domino arrangements is the number of ways in which a given number of dimers can attach themselves onto the surface; from this information, most of the thermodynamical properties of the system can be computed. In three dimensions, the same problem occurs in the theory of mixtures of molecules of different sizes and in the cell-cluster theory of the liquid state. For further background information, see [4, 11] and the references given there. The problem also has inherent combinatorial interest: clearly a domino arrangement is simply a mat thing, so we are actually being asked for the number of matchings of specified cardinality in the lattice graph. Counting matchings is a central problem in computer science and has received much attention since the seminal work of Valiant [15], who proved that it is #P-complete for general graphs. The enumeration of perfect matchings (where the dominoes are required to completely cover the graph) is equivalent to computing the permanent of a O-1 matrix, a long-studied problem in its own right [12]. This paper investigates