Normal Factor Graphs and Holographic Transformations

This paper stands at the intersection of two distinct lines of research. One line is "holographic algorithms," a powerful approach introduced by Valiant for solving various counting problems in computer science; the other is "normal factor graphs," an elegant framework proposed by Forney for representing codes defined on graphs. We introduce the notion of holographic transformations for normal factor graphs, and establish a very general theorem, called the generalized Holant theorem, which relates a normal factor graph to its holographic transformation. We show that the generalized Holant theorem on the one hand underlies the principle of holographic algorithms, and on the other hand reduces to a general duality theorem for normal factor graphs, a special case of which was first proved by Forney. In the course of our development, we formalize a new semantics for normal factor graphs, which highlights various linear algebraic properties that potentially enable the use of normal factor graphs as a linear algebraic tool.Comment: To appear IEEE Trans. Inform. Theor

Holographic Algorithms Beyond Matchgates

Holographic algorithms based on matchgates were introduced by Valiant. These algorithms run in polynomial-time and are intrinsically for planar problems. We introduce two new families of holographic algorithms, which work over general, i.e., not necessarily planar, graphs. The two underlying families of constraint functions are of the affine and product types. These play the role of Kasteleyn’s algorithm for counting planar perfect matchings. The new algorithms are obtained by transforming a problem to one of these two families by holographic reductions. We present a polynomial-time algorithm to decide if a given counting problem has a holographic algorithm using these constraint families. When the constraints are symmetric, we give a polynomial-time decision procedure in the size of the succinct presentation of symmetric constraint functions. This procedure shows that the recent dichotomy theorem for Holant problems with symmetric constraints is polynomial-time decidable