Minimal Reachability is Hard To Approximate

TL;DR

This paper proves that selecting actuated nodes in a linear dynamical system for state transfer is computationally hard to approximate, under standard complexity assumptions.

Contribution

It establishes the computational intractability of the minimal reachability problem, showing no efficient approximation algorithms exist under common complexity hypotheses.

Findings

The problem is NP-hard to approximate.

No polynomial-time algorithms can solve the problem unless P=NP.

The hardness holds even for quasi-polynomial time algorithms.

Abstract

In this note, we consider the problem of choosing which nodes of a linear dynamical system should be actuated so that the state transfer from the system's initial condition to a given final state is possible. Assuming a standard complexity hypothesis, we show that this problem cannot be efficiently solved or approximated in polynomial, or even quasi-polynomial, time.