The Complexity of the Covering Radius Problem on Lattices and Codes

We initiate the study of the computational complexity of the covering radius problem for point lattices, and approximation versions of the problem for both lattices and linear codes. We also investigate the computational complexity of the shortest linearly independent vectors problem, and its relation to the covering radius problem for lattices. For the covering radius on n-dimensional lattices, we show that the problem can be approximated within any constant factor (n) > 1 in random exponential time 2O(n) , it is in AM for (n) = 2, in coAM for (n) = n/ log n, and in NP coNP for (n) = n. For the covering radius on n-dimensional linear codes, we show that the problem can be solved in deterministic polynomial time for approximation factor (n) = log n, but cannot be solved in polynomial time for some (n) = (log log n) unless NP can be simulated in deterministic nO(log log log n) time. Moreover, we prove that the problem is NP-hard for every constant approximation factor, it is 2-har...

• Type: Conference
• Year: 2004
• Where: COCO
• Authors: Venkatesan Guruswami, Daniele Micciancio, Oded Regev
• Keywords: Algorithms, Approximation Factor, COCO 2004, Covering Radius, Independent Vectors Problem