Abstract
We answer the question of existence of polynomial-time constant-factor approximation algorithms for the space of nonfixed dimension. We prove that, in Euclidean space the problem is solvable in polynomial time with accuracy $$\sqrt a $$ , where α = 2/π, and if P ≠ NP then there are no polynomial algorithms with better accuracy. It is shown that, in the case of the lp spaces, the problem is APX-complete if p ∈ [1, 2] and not approximable with constant accuracy if P ≠ NP and p ∈ (2,∞).
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