10 research outputs found
Approximating the partition function of the ferromagnetic Potts model
We provide evidence that it is computationally difficult to approximate the
partition function of the ferromagnetic q-state Potts model when q>2.
Specifically we show that the partition function is hard for the complexity
class #RHPi_1 under approximation-preserving reducibility. Thus, it is as hard
to approximate the partition function as it is to find approximate solutions to
a wide range of counting problems, including that of determining the number of
independent sets in a bipartite graph. Our proof exploits the first order phase
transition of the "random cluster" model, which is a probability distribution
on graphs that is closely related to the q-state Potts model.Comment: Minor correction
The Complexity of Computing the Sign of the Tutte Polynomial
We study the complexity of computing the sign of the Tutte polynomial of a
graph. As there are only three possible outcomes (positive, negative, and
zero), this seems at first sight more like a decision problem than a counting
problem. Surprisingly, however, there are large regions of the parameter space
for which computing the sign of the Tutte polynomial is actually #P-hard. As a
trivial consequence, approximating the polynomial is also #P-hard in this case.
Thus, approximately evaluating the Tutte polynomial in these regions is as hard
as exactly counting the satisfying assignments to a CNF Boolean formula. For
most other points in the parameter space, we show that computing the sign of
the polynomial is in FP, whereas approximating the polynomial can be done in
polynomial time with an NP oracle. As a special case, we completely resolve the
complexity of computing the sign of the chromatic polynomial - this is easily
computable at q=2 and when q is less than or equal to 32/27, and is NP-hard to
compute for all other values of the parameter q.Comment: minor updates. This is the final version (to appear in SICOMP
The complexity of approximating the complex-valued Potts model
We study the complexity of approximating the partition function of the
-state Potts model and the closely related Tutte polynomial for complex
values of the underlying parameters. Apart from the classical connections with
quantum computing and phase transitions in statistical physics, recent work in
approximate counting has shown that the behaviour in the complex plane, and
more precisely the location of zeros, is strongly connected with the complexity
of the approximation problem, even for positive real-valued parameters.
Previous work in the complex plane by Goldberg and Guo focused on , which
corresponds to the case of the Ising model; for , the behaviour in the
complex plane is not as well understood and most work applies only to the
real-valued Tutte plane.
Our main result is a complete classification of the complexity of the
approximation problems for all non-real values of the parameters, by
establishing \#P-hardness results that apply even when restricted to planar
graphs. Our techniques apply to all and further complement/refine
previous results both for the Ising model and the Tutte plane, answering in
particular a question raised by Bordewich, Freedman, Lov\'{a}sz and Welsh in
the context of quantum computations.Comment: 58 pages. Changes on version 2: minor change
Matroids, Complexity and Computation
The node deletion problem on graphs is: given a graph and integer k, can we
delete no more than k vertices to obtain a graph that satisfies some property π.
Yannakakis showed that this problem is NP-complete for an infinite family of well-
defined properties. The edge deletion problem and matroid deletion problem are
similar problems where given a graph or matroid respectively, we are asked if we
can delete no more than k edges/elements to obtain a graph/matroid that satisfies
a property π. We show that these problems are NP-hard for similar well-defined
infinite families of properties.
In 1991 Vertigan showed that it is #P-complete to count the number of bases
of a representable matroid over any fixed field. However no publication has been
produced. We consider this problem and show that it is #P-complete to count
the number of bases of matroids representable over any infinite fixed field or finite
fields of a fixed characteristic.
There are many different ways of describing a matroid. Not all of these are
polynomially equivalent. That is, given one description of a matroid, we cannot
create another description for the same matroid in time polynomial in the size of
the first description. Due to this, the complexity of matroid problems can vary
greatly depending on the method of description used. Given one description a
problem might be in P while another description gives an NP-complete problem.
Based on these interactions between descriptions, we create and study the hierarchy
of all matroid descriptions and generalize this to all descriptions of countable
objects
The computational complexity of approximation of partition functions
This thesis studies the computational complexity of approximately evaluating partition functions. For various classes of partition functions, we investigate whether there is an FPRAS: a fully polynomial randomised approximation scheme. In many of these settings we also study “expressibility”, a simple notion of defining a constraint by combining other constraints, and we show that the results cannot be extended by expressibility reductions alone. The main contributions are: -� We show that there is no FPRAS for evaluating the partition function of the hard-core gas model on planar graphs at fugacity 312, unless RP = NP. -� We generalise an argument of Jerrum and Sinclair to give FPRASes for a large class of degree-two Boolean #CSPs. -� We initiate the classification of degree-two Boolean #CSPs where the constraint language consists of a single arity 3 relation. -� We show that the complexity of approximately counting downsets in directed acyclic graphs is not affected by restricting to graphs of maximum degree three. -� We classify the complexity of degree-two #CSPs with Boolean relations and weights on variables. -� We classify the complexity of the problem #CSP(F) for arbitrary finite domains when enough non-negative-valued arity 1 functions are in the constraint language. -� We show that not all log-supermodular functions can be expressed by binary logsupermodular functions in the context of #CSPs