84 research outputs found
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
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
New Planar P-time Computable Six-Vertex Models and a Complete Complexity Classification
We discover new P-time computable six-vertex models on planar graphs beyond
Kasteleyn's algorithm for counting planar perfect matchings. We further prove
that there are no more: Together, they exhaust all P-time computable six-vertex
models on planar graphs, assuming #P is not P. This leads to the following
exact complexity classification: For every parameter setting in
for the six-vertex model, the partition function is either (1) computable in
P-time for every graph, or (2) #P-hard for general graphs but computable in
P-time for planar graphs, or (3) #P-hard even for planar graphs. The
classification has an explicit criterion. The new P-time cases in (2) provably
cannot be subsumed by Kasteleyn's algorithm. They are obtained by a non-local
connection to #CSP, defined in terms of a "loop space".
This is the first substantive advance toward a planar Holant classification
with not necessarily symmetric constraints. We introduce M\"obius
transformation on as a powerful new tool in hardness proofs for
counting problems.Comment: 61 pages, 16 figures. An extended abstract appears in SODA 202
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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