Polynomial-time decidability of comparing sums of square roots

Determine whether comparing two sums of square roots of integers can be decided in polynomial time in the bit-complexity model.

Background

The difficulty of comparing sums of square roots is a classical obstacle in computational geometry and has implications for problems such as Euclidean matching and shortest paths.

The authors highlight this as a core bottleneck for exact computations involving Euclidean distances in the bit-complexity setting.

References

In fact, the problem of comparing a two sums of square roots is not known to be in P (see, for example,).

— Geometric Bipartite Matching is in NC (2405.18833 - Bhore et al., 29 May 2024) in Introduction