170 research outputs found
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
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
Approximation Complexity of Complex-Weighted Degree-Two Counting Constraint Satisfaction Problems
Constraint satisfaction problems have been studied in numerous fields with
practical and theoretical interests. In recent years, major breakthroughs have
been made in a study of counting constraint satisfaction problems (or #CSPs).
In particular, a computational complexity classification of bounded-degree
#CSPs has been discovered for all degrees except for two, where the "degree" of
an input instance is the maximal number of times that each input variable
appears in a given set of constraints. Despite the efforts of recent studies,
however, a complexity classification of degree-2 #CSPs has eluded from our
understandings. This paper challenges this open problem and gives its partial
solution by applying two novel proof techniques--T_{2}-constructibility and
parametrized symmetrization--which are specifically designed to handle
"arbitrary" constraints under randomized approximation-preserving reductions.
We partition entire constraints into four sets and we classify the
approximation complexity of all degree-2 #CSPs whose constraints are drawn from
two of the four sets into two categories: problems computable in
polynomial-time or problems that are at least as hard as #SAT. Our proof
exploits a close relationship between complex-weighted degree-2 #CSPs and
Holant problems, which are a natural generalization of complex-weighted #CSPs.Comment: A4, 10pt, 23 pages. This is a complete version of the paper that
appeared in the Proceedings of the 17th Annual International Computing and
Combinatorics Conference (COCOON 2011), Lecture Notes in Computer Science,
vol.6842, pp.122-133, Dallas, Texas, USA, August 14-16, 201
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