11 research outputs found
FPTAS for #BIS with Degree Bounds on One Side
Counting the number of independent sets for a bipartite graph (#BIS) plays a
crucial role in the study of approximate counting. It has been conjectured that
there is no fully polynomial-time (randomized) approximation scheme
(FPTAS/FPRAS) for #BIS, and it was proved that the problem for instances with a
maximum degree of is already as hard as the general problem. In this paper,
we obtain a surprising tractability result for a family of #BIS instances. We
design a very simple deterministic fully polynomial-time approximation scheme
(FPTAS) for #BIS when the maximum degree for one side is no larger than .
There is no restriction for the degrees on the other side, which do not even
have to be bounded by a constant. Previously, FPTAS was only known for
instances with a maximum degree of for both sides.Comment: 15 pages, to appear in STOC 2015; Improved presentations from
previous version
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
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 algebraic structure of the densification and the sparsification tasks for CSPs
The tractability of certain CSPs for dense or sparse instances is known from
the 90s. Recently, the densification and the sparsification of CSPs were
formulated as computational tasks and the systematical study of their
computational complexity was initiated.
We approach this problem by introducing the densification operator, i.e. the
closure operator that, given an instance of a CSP, outputs all constraints that
are satisfied by all of its solutions. According to the Galois theory of
closure operators, any such operator is related to a certain implicational
system (or, a functional dependency) . We are specifically interested
in those classes of fixed-template CSPs, parameterized by constraint languages
, for which the size of an implicational system is a
polynomial in the number of variables . We show that in the Boolean case,
is of polynomial size if and only if is of bounded width. For
such languages, can be computed in log-space or in a logarithmic time
with a polynomial number of processors. Given an implicational system ,
the densification task is equivalent to the computation of the closure of input
constraints. The sparsification task is equivalent to the computation of the
minimal key. This leads to -algorithm for
the sparsification task where is the number of non-redundant
sparsifications of an original CSP.
Finally, we give a complete classification of constraint languages over the
Boolean domain for which the densification problem is tractable
Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results
Recent results establish for 2-spin antiferromagnetic systems that the
computational complexity of approximating the partition function on graphs of
maximum degree D undergoes a phase transition that coincides with the
uniqueness phase transition on the infinite D-regular tree. For the
ferromagnetic Potts model we investigate whether analogous hardness results
hold. Goldberg and Jerrum showed that approximating the partition function of
the ferromagnetic Potts model is at least as hard as approximating the number
of independent sets in bipartite graphs (#BIS-hardness). We improve this
hardness result by establishing it for bipartite graphs of maximum degree D. We
first present a detailed picture for the phase diagram for the infinite
D-regular tree, giving a refined picture of its first-order phase transition
and establishing the critical temperature for the coexistence of the disordered
and ordered phases. We then prove for all temperatures below this critical
temperature that it is #BIS-hard to approximate the partition function on
bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to
approximate the number of k-colorings on bipartite graphs of maximum degree D
when k <= D/(2 ln D).
The #BIS-hardness result for the ferromagnetic Potts model uses random
bipartite regular graphs as a gadget in the reduction. The analysis of these
random graphs relies on recent connections between the maxima of the
expectation of their partition function, attractive fixpoints of the associated
tree recursions, and induced matrix norms. We extend these connections to
random regular graphs for all ferromagnetic models and establish the Bethe
prediction for every ferromagnetic spin system on random regular graphs. We
also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm
is torpidly mixing on random D-regular graphs at the critical temperature for
large q.Comment: To appear in SIAM J. Computin
Boolean approximate counting CSPs with weak conservativity, and implications for ferromagnetic two-spin
We analyse the complexity of approximate counting constraint satisfactions
problems , where is a set of
nonnegative rational-valued functions of Boolean variables. A complete
classification is known in the conservative case, where is
assumed to contain arbitrary unary functions. We strengthen this result by
fixing any permissive strictly increasing unary function and any permissive
strictly decreasing unary function, and adding only those to :
this is weak conservativity. The resulting classification is employed to
characterise the complexity of a wide range of two-spin problems, fully
classifying the ferromagnetic case. In a further weakening of conservativity,
we also consider what happens if only the pinning functions are assumed to be
in (instead of the two permissive unaries). We show that any set
of functions for which pinning is not sufficient to recover the two kinds of
permissive unaries must either have a very simple range, or must satisfy a
certain monotonicity condition. We exhibit a non-trivial example of a set of
functions satisfying the monotonicity condition.Comment: 37 page
Holant clones and the approximability of conservative holant problems
We construct a theory of holant clones to capture the notion of expressibility in the holant framework. Their role is analogous to the role played by functional clones in the study of weighted counting Constraint Satisfaction Problems. We explore the landscape of conservative holant clones and determine the situations in which a set F of functions is “universal in the conservative case”, which means that all functions are contained in the holant clone generated by F together with all unary functions. When F is not universal in the conservative case, we give concise generating sets for the clone. We demonstrate the usefulness of the holant clone theory by using it to give a complete complexity-theory classification for the problem of approximating the solution to conservative holant problems. We show that approximation is intractable exactly when F is universal in the conservative case