Attraction Radii in Binary Hopfield Nets are Hard to Compute

We prove that it is an NP-hard problem to determine the attraction radius of a stable vector in a binary Hoppeld memory network, and even that the attraction radius is hard to approximate. Under synchronous updating, the problems are already NP-hard for two-step attraction radii; direct (one-step) attraction radii can be computed in polynomial time. A Hoppeld memory network 6] consists of n binary valued nodes, or \neurons." We index the nodes by f1; : : :; ng, and choose f?1; +1g as their possible states (the values f0; 1g could be chosen equally well). Associated to each pair of nodes i; j is an interconnection weight w ij. The interconnections are symmetric, so that w ij = w ji for each i; j; moreover, w ii = 0 for each i. In addition, each node i has an internal threshold value t i. We denote the matrix of interconnection weights by W = (w ij), and the vector of threshold values by t = (t 1 ; t 2 ; : : :; t n). At any given moment, each node i in the network has a state x i , which is either ?1 or +1. The state at the next moment is determined as a function of the states of the other nodes as Work supported by the Academy of Finland.