# 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} }

The Tutte polynomial of a graph G is a two-variable polynomial T(G;x,y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: take as input a graph G, and output a value which is a good approximation to T(G;x,y). Jaeger et al. have completely mapped the complexity of exactly computing the Tutte polynomial. They have shown that this is #P-hard, except along the hyperbola (x-1)(y-1)=1 and at four special points. We are…

## 25 Citations

The Complexity of Computing the Sign of the Tutte Polynomial (and Consequent #P-hardness of Approximation)

- Mathematics, Computer ScienceICALP
- 2012

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

- Mathematics, Computer ScienceICALP
- 2010

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

- Mathematics
- 2010

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

- Mathematics, Computer Sciencecomputational complexity
- 2011

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

- Mathematics
- 2013

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…

The Complexity of the Cover Polynomials for Planar Graphs of Bounded Degree

- Mathematics, Computer ScienceMFCS
- 2011

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

- Computer Science, PhysicsArXiv
- 2006

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 Ising Partition Function: Zeros and Deterministic Approximation

- Mathematics, Computer ScienceJournal of Statistical Physics
- 2018

A tight version of the Lee–Yang theorem is established for the Ising model on hypergraphs, improving a classical result of Suzuki and Fisher.

Block interpolation: A framework for tight exponential-time counting complexity

- Computer Science, MathematicsInf. Comput.
- 2018

This framework allows to convert many known \(\mathsf {\#P}\)-hardness results for counting problems into results of the following type: if the given problem admits an algorithm with running time \(2^{o(n)}\) on graphs with \(n\) vertices and \(\mathcal {O}(n)\) edges, then the problem fails.

Interpretations of the Tutte and characteristic polynomials of matroids

- MathematicsJournal of Algebraic Combinatorics
- 2019

We study interpretations of the Tutte and characteristic polynomials of matroids. If M is a matroid with rank function r whose ground set E is given with a linear ordering <, then $$X\subseteq E$$ X…

## References

SHOWING 1-10 OF 37 REFERENCES

On the computational complexity of the Jones and Tutte polynomials

- Mathematics
- 1990

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

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 1994

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

- Mathematics
- 1954

Two polynomials 6(G, n) and <f>(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 <f>(G, n).…

The Tutte polynomial

- Mathematics
- 1969

$q$-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

- Mathematics, Computer ScienceSIAM J. Comput.
- 1996

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

- Computer ScienceSTOC
- 1983

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

- Mathematics, Computer ScienceSIAM J. Comput.
- 1994

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

- Mathematics
- 2000

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

- Computer ScienceRandom Struct. Algorithms
- 2000

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

- Computer ScienceAPPROX
- 2000

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.