Inapproximability of the Tutte polynomial

• Authors: Leslie Ann Goldberg;Mark Jerrum
• Venue: Information and Computation
• Year: 2008

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 interested in determining for which points (x,y) there is a fully polynomial randomised approximation scheme (FPRAS) for T(G;x,y). Under the assumption RPNP, we prove that there is no FPRAS at (x,y) if (x,y) is in one of the half-planes xNP, there is no FPRAS at the point (x,y)=(0,1-@l) when @l2 is a positive integer. Thus, there is no FPRAS for counting nowhere-zero @l flows for @l2. This is an interesting consequence of our work since the corresponding decision problem is in P for example for @l=6. Although our main concern is to distinguish regions of the Tutte plane that admit an FPRAS from those that do not, we also note that the latter regions exhibit different levels of intractability. At certain points (x,y), for example the integer points on the x-axis, or any point in the positive quadrant, there is a randomised approximation scheme for T(G;x,y) that runs in polynomial time using an oracle for an NP predicate. On the other hand, we identify a region of points (x,y) at which even approximating T(G;x,y) is as hard as #P.