相关论文: Normal Factor Graphs and Holographic Transformatio…
The Holant theorem is a powerful tool for studying the computational complexity of counting problems in the Holant framework. Due to the great expressiveness of the Holant framework, a converse to the Holant theorem would itself be a very…
Holographic algorithms are a recent breakthrough in computer science and has found applications in information theory. This paper provides a proof to the central component of holographic algorithms, namely, the Holant theorem. Compared with…
Holographic algorithms introduced by Valiant are composed of two ingredients: matchgates, which are gadgets realizing local constraint functions by weighted planar perfect matchings, and holographic reductions, which show equivalences among…
Recently, a general tool called a holographic transformation, which transforms an expression of the partition function to another form, has been used for polynomial-time algorithms and for improvement and understanding of the belief…
Holant problems are a framework for the analysis of counting complexity problems on graphs. This framework is simultaneously general enough to encompass many other counting problems on graphs and specific enough to allow the derivation of…
Holant problems are a family of counting problems parameterised by sets of algebraic-complex valued constraint functions, and defined on graphs. They arise from the theory of holographic algorithms, which was originally inspired by concepts…
We prove a complexity dichotomy theorem for Holant Problems on 3-regular graphs with an arbitrary complex-valued edge function. Three new techniques are introduced: (1) higher dimensional iterations in interpolation; (2) Eigenvalue Shifted…
Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate…
We present a new probabilistic modelling framework based on the recent notion of normal factor graph (NFG). We show that the proposed NFG models and their transformations unify some existing models such as factor graphs, convolutional…
A conceptual framework involving partition functions of normal factor graphs is introduced, paralleling a similar recent development by Al-Bashabsheh and Mao. The partition functions of dual normal factor graphs are shown to be a Fourier…
Valiant's Holant theorem is a powerful tool for algorithms and reductions for counting problems. It states that if two sets $\mathcal{F}$ and $\mathcal{G}$ of tensors (a.k.a. constraint functions or signatures) are related by a…
The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, yields insight into the hierarchy of complexity classes. In particular, the theory produces algebraic tests for a…
Holant problems are a general framework to study the algorithmic complexity of counting problems. Both counting constraint satisfaction problems and graph homomorphisms are special cases. All previous results of Holant problems are over the…
We explore the intricate interdependent relationship among counting problems, considered from three frameworks for such problems: Holant Problems, counting CSP and weighted H-colorings. We consider these problems for general complex valued…
The dual normal factor graph and the factor graph duality theorem have been considered for discrete graphical models. In this paper, we show an application of the factor graph duality theorem to continuous graphical models. Specifically, we…
Inspired by some new advances on normal factor graphs (NFGs), we introduce NFGs as a simple and intuitive diagrammatic approach towards encoding some concepts from linear algebra. We illustrate with examples the workings of such an approach…
The theory of holographic algorithms introduced by Valiant represents a novel approach to achieving polynomial-time algorithms for seemingly intractable counting problems via a reduction to counting planar perfect matchings and a linear…
We prove a complexity dichotomy for complex-weighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. This dichotomy is specifically to answer the question: Is the FKT algorithm under a…
We present fully polynomial-time (deterministic or randomised) approximation schemes for Holant problems, defined by a non-negative constraint function satisfying a generalised second order recurrence modulo a couple of exceptional cases.…
Graphs constructed to translate some graph problem into another graph problem are usually called auxiliary graphs. Specifically total graphs of simple graphs are used to translate the total colouring problem of the original graph into a…