It is known that there is no EPTAS for the m-dimensional knapsack problem
unless W[1]=FPT. It is true already for the case, when m=2. But, an
FPTAS still can exist for some other particular cases of the problem.
In this note, we show that the m-dimensional knapsack problem with a
Δ-modular constraints matrix admits an FPTAS, whose complexity bound
depends on Δ linearly. More precisely, the proposed algorithm complexity
is O(TLP​⋅(1/ε)m+3⋅(2m)2m+6⋅Δ),
where TLP​ is the linear programming complexity bound. In particular, for
fixed m the arithmetical complexity bound becomes O(n⋅(1/ε)m+3⋅Δ). Our algorithm is actually a
generalisation of the classical FPTAS for the 1-dimensional case.
Strictly speaking, the considered problem can be solved by an exact
polynomial-time algorithm, when m is fixed and Δ grows as a polynomial
on n. This fact can be observed combining previously known results. In this
paper, we give a slightly more accurate analysis to present an exact algorithm
with the complexity bound O(n⋅Δm+1), for m being fixed. Note that
the last bound is non-linear by Δ with respect to the given FPTAS