An FPTAS for the $\Delta$-modular multidimensional knapsack problem

TL;DR

This paper introduces a Fully Polynomial-Time Approximation Scheme (FPTAS) for the elta-modular multidimensional knapsack problem, providing a complexity that depends linearly on elta and generalizes the classical FPTAS for the 1-dimensional case.

Contribution

The paper presents the first FPTAS for the elta-modular multidimensional knapsack problem with complexity linear in elta, extending classical methods to a broader class of problems.

Findings

The FPTAS has a complexity of O(T_{LP} elta (1/psilon)^{m+3} (2m)^{2m+6}].

For fixed m, the complexity simplifies to O(n elta (1/psilon)^{m+3}].

An exact polynomial-time algorithm exists when m is fixed and elta grows polynomially with n.

Abstract

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 $\Delta$-modular constraints matrix admits an FPTAS, whose complexity bound depends on $\Delta$ linearly. More precisely, the proposed algorithm complexity is $$O(T_{LP} \cdot (1/\varepsilon)^{m+3} \cdot (2m)^{2m + 6} \cdot \Delta),$$ where $T_{LP}$ is the linear programming complexity bound. In particular, for fixed $m$ the arithmetical complexity bound becomes $$ O(n \cdot (1/\varepsilon)^{m+3} \cdot \Delta). $$ 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 $\Delta$ 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 $$ O(n \cdot \Delta^{m + 1}), \quad \text{ for $m$ being fixed}. $$ Note that the last bound is non-linear by $\Delta$ with respect to the given FPTAS.