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A constant time complexity algorithm for the unbounded knapsack problem with bounded coefficients

Yang Yang

TL;DR

An exact algorithm with time complexity of $O(R^4)$ and space complexity of $O(R^3)$ and the proof techniques required are primarily covered in the first-year mathematics curriculum, which is convenient for subsequent researchers to grasp.

Abstract

Benchmark instances for the unbounded knapsack problem are typically generated according to specific criteria within a given constant range $R$, and these instances can be referred to as the unbounded knapsack problem with bounded coefficients (UKPB). In order to increase the difficulty of solving these instances, the knapsack capacity $C$ is usually set to a very large value. Therefore, an exact algorithm that neither time complexity nor space complexity includes the capacity coefficient $C$ is highly anticipated. In this paper, we propose an exact algorithm with time complexity of $O(R^4)$ and space complexity of $O(R^3)$. The algorithm initially divides the multiset $N$ into two multisubsets, $N_1$ and $N_2$, based on the profit density of their types. For the multisubset $N_2$ composed of types with profit density lower than the maximum profit density type, we utilize a recent branch and bound (B\&B) result by Dey et al. (Math. Prog., pp 569-587, 2023) to determine the maximum selection number for types in $N_2$. We then employ the Unbounded-DP algorithm to exactly solve for the types in $N_2$. For the multisubset $N_1$ composed of the maximum profit density type and its counterparts with the same profit density, we transform it into a linear Diophantine equation and leverage relevant conclusions from the Frobenius problem to solve it efficiently. In particular, the proof techniques required by the algorithm are primarily covered in the first-year mathematics curriculum, which is convenient for subsequent researchers to grasp.

A constant time complexity algorithm for the unbounded knapsack problem with bounded coefficients

TL;DR

An exact algorithm with time complexity of and space complexity of and the proof techniques required are primarily covered in the first-year mathematics curriculum, which is convenient for subsequent researchers to grasp.

Abstract

Benchmark instances for the unbounded knapsack problem are typically generated according to specific criteria within a given constant range , and these instances can be referred to as the unbounded knapsack problem with bounded coefficients (UKPB). In order to increase the difficulty of solving these instances, the knapsack capacity is usually set to a very large value. Therefore, an exact algorithm that neither time complexity nor space complexity includes the capacity coefficient is highly anticipated. In this paper, we propose an exact algorithm with time complexity of and space complexity of . The algorithm initially divides the multiset into two multisubsets, and , based on the profit density of their types. For the multisubset composed of types with profit density lower than the maximum profit density type, we utilize a recent branch and bound (B\&B) result by Dey et al. (Math. Prog., pp 569-587, 2023) to determine the maximum selection number for types in . We then employ the Unbounded-DP algorithm to exactly solve for the types in . For the multisubset composed of the maximum profit density type and its counterparts with the same profit density, we transform it into a linear Diophantine equation and leverage relevant conclusions from the Frobenius problem to solve it efficiently. In particular, the proof techniques required by the algorithm are primarily covered in the first-year mathematics curriculum, which is convenient for subsequent researchers to grasp.
Paper Structure (13 sections, 11 theorems, 18 equations, 2 algorithms)

This paper contains 13 sections, 11 theorems, 18 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our results
  3. Sketch of proof techniques
  4. Organization of the paper
  5. Preliminary
  6. Dynamic programming
  7. Branch and bound
  8. linear Diophantine equation
  9. A constant time complexity algorithm for the UKPB
  10. An upper bound for the selected times of the type in set $N_2$
  11. An Algorithm for types of set $N_1$
  12. Complexity analysis
  13. Conclusion

Key Result

Theorem 2.1

Pisinger1997 For the $j$-th item, if we can conclude that $y_j = 0$, where $\textbf{Y}$ denotes the optimal solution.

Theorems & Definitions (15)

  • Definition 1
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • ...and 5 more