Inapproximability of the Tutte polynomial

@article{Goldberg2008InapproximabilityOT,
  title={Inapproximability of the Tutte polynomial},
  author={Leslie Ann Goldberg and Mark Jerrum},
  journal={Information \& Computation},
  year={2008},
  volume={206},
  pages={908-929}
}

Figures from this paper

The Complexity of Computing the Sign of the Tutte Polynomial (and Consequent #P-hardness of Approximation)
TLDR
This work completely resolve the complexity of computing the sign of the chromatic polynomial -- this is easily computable at q=2 and when q≤32/27, and is NP-hard to compute for all other values of the parameter q.
Exponential Time Complexity of the Permanent and the Tutte Polynomial
TLDR
The Exponential Time Hypothesis is relaxed by introducing its counting version, namely that every algorithm that counts the satisfying assignments requires time exp(Ω(n), and the sparsification lemma for d-CNF formulas is transferred to the counting setting, which makes #ETH robust.
ON THE QUANTUM COMPLEXITY OF EVALUATING THE TUTTE POLYNOMIAL
We show that the problem of approximately evaluating the Tutte polynomial of triangular graphs at the points (q, 1/q) of the Tutte plane is BQP-complete for (most) roots of unity q. We also consider
Complexity and Approximability of the Cover Polynomial
TLDR
Under the reasonable complexity assumptions RP ≠ NP and RFP ≠ #P, a succinct characterization of a large class of points at which approximating the geometric cover polynomials within any polynomial factor is not possible is given.
Matroids, Complexity and Computation
TLDR
The hierarchy of all matroid descriptions is created and studied and generalize this to all descriptions of countable objects and shows 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.
The Complexity of the Cover Polynomials for Planar Graphs of Bounded Degree
TLDR
It is shown that almost the same dichotomy holds when restricting the evaluation to planar graphs, and algorithms that allow for polynomial-time evaluation of the cover polynomials at certain new points by utilizing Valiant's holographic framework are provided.
The BQP-hardness of approximating the Jones Polynomial
TLDR
The universality proof of Freedman et al (2002) is extended to ks that grow polynomially with the number of strands and crossings in the link, thus extending the BQP-hardness of Jones polynomial approximations to all values to which the AJL algorithm applies, proving that for all those values, the problems are B QP-complete.
The complexity of approximating the complex-valued Potts model
We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the
The Ising Partition Function: Zeros and Deterministic Approximation
TLDR
A tight version of the Lee–Yang theorem is established for the Ising model on hypergraphs, improving a classical result of Suzuki and Fisher.
...
1
2
3
...

References

SHOWING 1-10 OF 37 REFERENCES
On the computational complexity of the Jones and Tutte polynomials
We show that determining the Jones polynomial of an alternating link is #P-hard. This is a special case of a wide range of results on the general intractability of the evaluation of the Tutte
Polynomial Time Randomised Approximation Schemes for Tutte-Gröthendieck Invariants: The Dense Case
TLDR
A general technique is developed that supplies fully polynomial randomised approximation schemes for approximating the value of T(G; x, y) for any dense graph G, that is, any graph on n vertices whose minimum.
A contribution to the theory of chromatic polynomials
Summary Two polynomials θ(G, n) and ϕ(G, n) connected with the colourings of a graph G or of associated maps are discussed. A result believed to be new is proved for the lesser-known polynomial ϕ(G,
The Tutte polynomial
$q$-Matroids are defined on complemented modular support lattices. Minors of length 2 are of four types as in a "classical" matroid. Tutte polynomials $\tau(x,y)$ of matroids are calculated either by
On Unapproximable Versions of NP-Complete Problems
  • D. Zuckerman
  • Computer Science, Mathematics
    SIAM J. Comput.
  • 1996
TLDR
All of Karp's 21 original $NP$-complete problems have a version that is hard to approximate, and it is shown that it is even harder to approximate two counting problems: counting the number of satisfying assignments to a monotone 2SAT formula and computing the permanent of $-1, $0, $1$ matrices.
The complexity of approximate counting
TLDR
The complexity of computing approximate solutions to problems in #P is classified in terms of the polynomial-time hierarchy (for short, P-hierarchy) in order to study a class of restricted, but very natural, probabilistic sampling methods motivated by the particular counting problems.
The Complexity of Multiterminal Cuts
TLDR
It is shown that the problem becomes NP-hard as soon as $k=3$, but can be solved in polynomial time for planar graphs for any fixed $k$, if the planar problem is NP- hard, however, if £k$ is not fixed.
The complexity of counting graph homomorphisms
The problem of counting homomorphisms from a general graph G to a fixed graph H is a natural generalization of graph coloring, with important applications in statistical physics. The problem of
The complexity of counting graph homomorphisms
TLDR
The theorems provide the first proof of #P-completeness of the partition function of certain models from statistical physics, such as the Widom–Rowlinson model, even in graphs of maximum degree 3.
On the relative complexity of approximate counting problems
TLDR
This work describes and investigates a third class of counting problems, of intermediate complexity, that is not known to be identical to (i) or (ii), and can be characterised as the hardest problems in a logically defined subclass of #P.
...
1
2
3
4
...