Dice Question Streamline Icon: https://streamlinehq.com

FPRAS for Tutte polynomial without density assumption

Develop a fully polynomial randomized approximation scheme (FPRAS) for computing the Tutte polynomial T_G(x,y) at points with x,y ≥ 1 for all finite graphs G, removing the requirement that G be α-dense (|E| ≥ α|V|).

Information Square Streamline Icon: https://streamlinehq.com

Background

Alon, Frieze, and Welsh established an FPRAS for Tutte–Grothendieck invariants (including the Tutte polynomial) on the domain x, y ≥ 1 for the class of α-dense graphs, with stronger results when α > 1/2.

Welsh conjectured that this density condition could be dropped, extending the FPRAS to arbitrary graphs. The memoir states that this conjecture remains unsettled.

This problem targets the existence of such an FPRAS across all graphs in the specified parameter regime.

References

With Noga Alon and Alan Frieze, Dominic showed in [MR1368847] the existence of an FPRAS when x,y≥1 for the class G_α with α>0 (with a stronger conclusion when α>1/2). Never lacking in bravery, Dominic has conjectured that the condition on α may be removed, but the jury is still out on that (see [handbook10]).

— Dominic Welsh (1938-2023) (2404.13942 - Grimmett, 22 Apr 2024) in Section 8.2 (Tutte polynomials)