An impossible utopia in distance geometry

TL;DR

This paper investigates the computational complexity of counting solutions in distance geometry problems related to protein structures, proving that a certain combinatorial counting method cannot exist in general.

Contribution

It demonstrates the impossibility of developing a general combinatorial counting method for a variant of the distance geometry problem without contiguity constraints.

Findings

Proves the non-existence of a general combinatorial counting method

Analyzes the complexity of distance geometry problem variants

Highlights the impact of contiguity constraints on problem tractability

Abstract

The Distance Geometry Problem asks for a realization of a given weighted graph in $\mathbb{R}^K$. Two variants of this problem, both originating from protein conformation, are based on a given vertex order (which abstracts the protein backbone). Both variants involve an element of discrete decision in the realization of the next vertex in the order using $K$ preceding (already realized) vertices. The difference between these variants is that one requires the $K$ preceding vertices to be contiguous. The presence of this constraint allows one to prove, via a combinatorial counting of the number of solutions, that the realization algorithm is fixed-parameter tractable. Its absence, on the other hand, makes it possible to efficiently construct the vertex order directly from the graph. Deriving a combinatorial counting method without using the contiguity requirement would therefore be desirable. In this paper we prove that, unfortunately, such a counting method cannot be devised in general.