5,885 research outputs found
FPTAS for Weighted Fibonacci Gates and Its Applications
Fibonacci gate problems have severed as computation primitives to solve other
problems by holographic algorithm and play an important role in the dichotomy
of exact counting for Holant and CSP frameworks. We generalize them to weighted
cases and allow each vertex function to have different parameters, which is a
much boarder family and #P-hard for exactly counting. We design a fully
polynomial-time approximation scheme (FPTAS) for this generalization by
correlation decay technique. This is the first deterministic FPTAS for
approximate counting in the general Holant framework without a degree bound. We
also formally introduce holographic reduction in the study of approximate
counting and these weighted Fibonacci gate problems serve as computation
primitives for approximate counting. Under holographic reduction, we obtain
FPTAS for other Holant problems and spin problems. One important application is
developing an FPTAS for a large range of ferromagnetic two-state spin systems.
This is the first deterministic FPTAS in the ferromagnetic range for two-state
spin systems without a degree bound. Besides these algorithms, we also develop
several new tools and techniques to establish the correlation decay property,
which are applicable in other problems
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
Holant Problems for Regular Graphs with Complex Edge Functions
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 Pairs, which allow us to prove that a pair of combinatorial gadgets in
combination succeed in proving #P-hardness; and (3) algebraic symmetrization,
which significantly lowers the symbolic complexity of the proof for
computational complexity. With holographic reductions the classification
theorem also applies to problems beyond the basic model.Comment: 19 pages, 4 figures, added proofs for full versio
Counting degree-constrained subgraphs and orientations
The goal of this short paper to advertise the method of gauge transformations
(aka holographic reduction, reparametrization) that is well-known in
statistical physics and computer science, but less known in combinatorics. As
an application of it we give a new proof of a theorem of A. Schrijver asserting
that the number of Eulerian orientations of a --regular graph on
vertices with even is at least
. We also show that a
--regular graph with even has always at least as many Eulerian
orientations as --regular subgraphs
Holographic Algorithm with Matchgates Is Universal for Planar CSP Over Boolean Domain
We prove a complexity classification theorem that classifies all counting
constraint satisfaction problems (CSP) over Boolean variables into exactly
three categories: (1) Polynomial-time tractable; (2) P-hard for general
instances, but solvable in polynomial-time over planar graphs; and (3)
P-hard over planar graphs. The classification applies to all sets of local,
not necessarily symmetric, constraint functions on Boolean variables that take
complex values. It is shown that Valiant's holographic algorithm with
matchgates is a universal strategy for all problems in category (2).Comment: 94 page
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