54 research outputs found
The Complexity of Holant Problems over Boolean Domain with Non-Negative Weights
Holant problem is a general framework to study the computational complexity of counting problems. We prove a complexity dichotomy theorem for Holant problems over the Boolean domain with non-negative weights. It is the first complete Holant dichotomy where constraint functions are not necessarily symmetric.
Holant problems are indeed read-twice #CSPs. Intuitively, some #CSPs that are #P-hard become tractable when restricted to read-twice instances. To capture them, we introduce the Block-rank-one condition. It turns out that the condition leads to a clear separation. If a function set F satisfies the condition, then F is of affine type or product type. Otherwise (a) Holant(F) is #P-hard; or (b) every function in F is a tensor product of functions of arity at most 2; or (c) F is transformable to a product type by some real orthogonal matrix. Holographic transformations play an important role in both the hardness proof and the characterization of tractability
Approximating Holant problems by winding
We give an FPRAS for Holant problems with parity constraints and
not-all-equal constraints, a generalisation of the problem of counting
sink-free-orientations. The approach combines a sampler for near-assignments of
"windable" functions -- using the cycle-unwinding canonical paths technique of
Jerrum and Sinclair -- with a bound on the weight of near-assignments. The
proof generalises to a larger class of Holant problems; we characterise this
class and show that it cannot be extended by expressibility reductions.
We then ask whether windability is equivalent to expressibility by matchings
circuits (an analogue of matchgates), and give a positive answer for functions
of arity three
The complexity of approximating conservative counting CSPs
We study the complexity of approximately solving the weighted counting
constraint satisfaction problem #CSP(F). In the conservative case, where F
contains all unary functions, there is a classification known for the case in
which the domain of functions in F is Boolean. In this paper, we give a
classification for the more general problem where functions in F have an
arbitrary finite domain. We define the notions of weak log-modularity and weak
log-supermodularity. We show that if F is weakly log-modular, then #CSP(F)is in
FP. Otherwise, it is at least as difficult to approximate as #BIS, the problem
of counting independent sets in bipartite graphs. #BIS is complete with respect
to approximation-preserving reductions for a logically-defined complexity class
#RHPi1, and is believed to be intractable. We further sub-divide the #BIS-hard
case. If F is weakly log-supermodular, then we show that #CSP(F) is as easy as
a (Boolean) log-supermodular weighted #CSP. Otherwise, we show that it is
NP-hard to approximate. Finally, we give a full trichotomy for the arity-2
case, where #CSP(F) is in FP, or is #BIS-equivalent, or is equivalent in
difficulty to #SAT, the problem of approximately counting the satisfying
assignments of a Boolean formula in conjunctive normal form. We also discuss
the algorithmic aspects of our classification.Comment: Minor revisio
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
A Dichotomy Theorem for the Approximate Counting of Complex-Weighted Bounded-Degree Boolean CSPs
We determine the computational complexity of approximately counting the total
weight of variable assignments for every complex-weighted Boolean constraint
satisfaction problem (or CSP) with any number of additional unary (i.e., arity
1) constraints, particularly, when degrees of input instances are bounded from
above by a fixed constant. All degree-1 counting CSPs are obviously solvable in
polynomial time. When the instance's degree is more than two, we present a
dichotomy theorem that classifies all counting CSPs admitting free unary
constraints into exactly two categories. This classification theorem extends,
to complex-weighted problems, an earlier result on the approximation complexity
of unweighted counting Boolean CSPs of bounded degree. The framework of the
proof of our theorem is based on a theory of signature developed from Valiant's
holographic algorithms that can efficiently solve seemingly intractable
counting CSPs. Despite the use of arbitrary complex weight, our proof of the
classification theorem is rather elementary and intuitive due to an extensive
use of a novel notion of limited T-constructibility. For the remaining degree-2
problems, in contrast, they are as hard to approximate as Holant problems,
which are a generalization of counting CSPs.Comment: A4, 10pt, 20 pages. This revised version improves its preliminary
version published under a slightly different title in the Proceedings of the
4th International Conference on Combinatorial Optimization and Applications
(COCOA 2010), Lecture Notes in Computer Science, Springer, Vol.6508 (Part I),
pp.285--299, Kailua-Kona, Hawaii, USA, December 18--20, 201
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