Computing Minimal Multi-Homogeneous B\'ezout Numbers is Hard
Klaus Meer and Gregorio Malajovich
Theory of Computing Systems
The multi-homogeneous B\'ezout number is a bound for
the number of solutions of a system of multi-homogeneous
polynomial equations, in a suitable product of projective
Given an arbitrary, not necessarily multi-homogeneous
system, one can ask for the optimal multi-homogenization
that would minimize the B\'ezout number.
In this paper, it is proved that the problem of computing,
or even estimating the optimal multi-homogeneous B\'ezout
number is actually $NP$-hard.
In terms of approximation theory for combinatorial
optimization, the problem of computing the best
multi-homogeneous structure does not belong to $APX$,
unless $P = NP$.
Moreover, polynomial time algorithms for estimating
the minimal multi-homo\-ge\-neous B\'ezout number up to a
fixed factor cannot exist even in a randomized setting,
unless $BPP \supseteq NP$.
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|Project Keyword:||Project Keyword UNSPECIFIED|
|Subjects:||Theory & Algorithms|
|Deposited By:||Klaus Meer|
|Deposited On:||28 November 2005|