  Recently, due to the genomic sequence analysis in several types of cancer,
the genomic data based on {\em copy number profiles} ({\em CNP} for short) are
getting more and more popular. A CNP is a vector where each component is a
non-negative integer representing the number of copies of a specific gene or
segment of interest.
  In this paper, we present two streams of results. The first is the negative
results on two open problems regarding the computational complexity of the
Minimum Copy Number Generation (MCNG) problem posed by Qingge et al. in 2018.
It was shown by Qingge et al. that the problem is NP-hard if the duplications
are tandem and they left the open question of whether the problem remains
NP-hard if arbitrary duplications are used. We answer this question
affirmatively in this paper; in fact, we prove that it is NP-hard to even
obtain a constant factor approximation. We also prove that the parameterized
version is W[1]-hard, answering another open question by Qingge et al.
  The other result is positive and is based on a new (and more general) problem
regarding CNP's. The \emph{Copy Number Profile Conforming (CNPC)} problem is
formally defined as follows: given two CNP's $C_1$ and $C_2$, compute two
strings $S_1$ and $S_2$ with $cnp(S_1)=C_1$ and $cnp(S_2)=C_2$ such that the
distance between $S_1$ and $S_2$, $d(S_1,S_2)$, is minimized. Here,
$d(S_1,S_2)$ is a very general term, which means it could be any genome
rearrangement distance (like reversal, transposition, and tandem duplication,
etc). We make the first step by showing that if $d(S_1,S_2)$ is measured by the
breakpoint distance then the problem is polynomially solvable.
