Abstract. We show that approximating the shortest vector problem (in any ℓp norm) to within any constant factor less than p √ 2 is hardfor NP under reverse unfaithful random reductions with inverse polynomial error probability. In particular, approximating the shortest vector problem is not in RP (random polynomial time), unless NP equals RP. We also prove a proper NP-hardness result (i.e., hardness under deterministic many-one reductions) under a reasonable number theoretic conjecture on the distribution of square-free smooth numbers. As part of our proof, we give an alternative construction of Ajtai’s constructive variant of Sauer’s lemma that greatly simplifies Ajtai’s original proof. Key words. NP-hardness, shortest vector problem, point lattices, geometry of numbers, sphere packing

Abstract We give a new simple proof of the NP-hardness of the closest vector problem. In addition to being much simpler than all previously known proofs, the new proof yields new interesting results about the complexity of the closest vector problem with preprocessing. This is a variant of the closest vector problem in which the lattice is specified in advance, and can be preprocessed for an arbitrarily long amount of time before the target vector is revealed. We show that there are lattices for which the closest vector problem remains hard, regardless of the amount of preprocessing.

We consider a single perceptron N with synaptic delays which generalizes a simpli ed model for a spiking neuron where not only the time that a pulse needs to travel through a synapse is taken into account but also the input ring rates may have more dierent levels. A synchronization technique is introduced so that the results concerning the learnability of spiking neurons with binary delays also apply to N with arbitrary delays. In particular, the consistency problem for N with programmable delays and its approximation version prove to be NP-hard. It follows that the perceptrons with programmable synaptic delays are not properly PAC-learnable and the spiking neurons with arbitrary delays do not allow robust learning unless RP = NP . In addition, we show that the representation problem for N which is an issue whether an n-variable Boolean function given in DNF (or as a disjunction of O(n) threshold gates) can be computed by a spiking neuron is co-NP-hard.

. We rst present a brief survey of hardness results for training feedforward neural networks. These results are then completed by the proof that the simplest architecture containing only a single neuron that applies the standard (logistic) activation function to the weighted sum of n inputs is hard to train. In particular, the problem of nding the weights of such a unit that minimize the relative quadratic training error within 1 or its average (over a training set) within 13=(31n) of its inmum proves to be NP-hard. Hence, the well-known back-propagation learning algorithm appears to be not ecient even for one neuron which has negative consequences in constructive learning. 1 The Complexity of Neural Network Loading Neural networks establish an important class of learning models that are widely applied in practical applications to solving articial intelligence tasks [13]. The most prominent position among successful neural learning heuristics is occupied by the back-prop...