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Fixed-dimension NP-hardness for unitary SLPS

Establish whether there exists a fixed dimension d in the natural numbers such that the reachability problem in unitary d-dimensional Simple Linear Path Schemes (SLPS), where all counter updates are restricted to {-1,0,1}, is NP-hard.

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Background

The paper proves NP-completeness of reachability for unitary SLPS when the dimension is allowed to grow extremely slowly with the input size (on the order of the inverse Ackermann function). This leaves open whether a constant dimension already suffices for NP-hardness.

The authors conjecture a stronger statement than their main hardness result to pinpoint the precise fixed-dimensional frontier for unitary SLPS.

References

We conjecture that even a stronger version of this statement is true. Conjecture There exists a number d in N such that the reachability problem for unitary d-dimensional SLPS is #1{NP}-hard.

— The Tractability Border of Reachability in Simple Vector Addition Systems with States (2412.16612 - Chistikov et al., 21 Dec 2024) in Further research (Section: Further research)