1

In my problem of interest I have an arbitrary, finite set $S \subset \mathbb{Z}^3$. And I would like to cover $S$ with a set $C \subset \{T | T \subset \mathbb{Z}^3 \textrm{ is an axis-aligned cube of size } \ell\}$ of minimal cardinality.

I know there exist approximation algorithms with logarithmic-ish approximation factors for the covering problem in general, but I was hoping to squeeze some extra power out of the structure I have here (having the choice of any $\ell$-sized cubes $\subset \mathbb{Z}^3$ I like, as well as having $S$ sitting on the integer lattice). But I haven't had any success. Do these restrictions give us any extra advantage?

EarlyGame
  • 33
  • 4

1 Answers1

2

The problem is NP-hard. Fowler et al. proved it is NP-hard in two dimensions; it follows that it is NP-hard in three dimensions as well. See the following paper:

Optimal Packing and Covering in the Plane are NP-complete. Robert J. Fowler, Michael S. Paterson, and Steven L. Tanimoto. Information processing letters 12.3 (1981): 133-137.

See also Matrix covering by squares.

D.W.
  • 167,959
  • 22
  • 232
  • 500