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On tropical knapsack-type problems

I. M. Buchinskiy, M. V. Kotov, A. V. Treier

TL;DR

The paper analyzes the subset sum and knapsack problems in two tropical matrix semigroups, $M_{max,+}^k$ and $M_{max,×}^k$, establishing NP-completeness for all four problem variants. It provides pseudo-polynomial dynamic-programming algorithms and, importantly, shows polynomial generic-time algorithms for SSP and KP in the max-times semigroup via generic-case complexity arguments and density analyses. The work clarifies the computational landscape of tropical knapsack-type problems and highlights potential for efficient generic algorithms in specific tropical settings, while outlining several open questions for the max-plus case and other tropical algebras. These results contribute to non-commutative discrete optimization by connecting tropical matrix behavior with classical combinatorial problems and reductions.

Abstract

In this paper, we investigate the computational complexity of the knapsack problem and subset sum problem for the following tropical algebraic structures. We consider the semigroup of square matrices of size $k \times k$ with non-negative entries over the max-plus algebra and the semigroup square matrices of size $k \times k$ with positive entries over the max-times algebra. We prove that the knapsack problem and subset sum problem for these structures are $\textsf{NP}$-complete. We demonstrate that there are pseudo-polynomial algorithms to solve these problems. Also, we show that for the latter semigroup, there are polynomial generic algorithms to solve the knapsack problem and the subset sum problem.

On tropical knapsack-type problems

TL;DR

The paper analyzes the subset sum and knapsack problems in two tropical matrix semigroups, and , establishing NP-completeness for all four problem variants. It provides pseudo-polynomial dynamic-programming algorithms and, importantly, shows polynomial generic-time algorithms for SSP and KP in the max-times semigroup via generic-case complexity arguments and density analyses. The work clarifies the computational landscape of tropical knapsack-type problems and highlights potential for efficient generic algorithms in specific tropical settings, while outlining several open questions for the max-plus case and other tropical algebras. These results contribute to non-commutative discrete optimization by connecting tropical matrix behavior with classical combinatorial problems and reductions.

Abstract

In this paper, we investigate the computational complexity of the knapsack problem and subset sum problem for the following tropical algebraic structures. We consider the semigroup of square matrices of size with non-negative entries over the max-plus algebra and the semigroup square matrices of size with positive entries over the max-times algebra. We prove that the knapsack problem and subset sum problem for these structures are -complete. We demonstrate that there are pseudo-polynomial algorithms to solve these problems. Also, we show that for the latter semigroup, there are polynomial generic algorithms to solve the knapsack problem and the subset sum problem.

Paper Structure

This paper contains 6 sections, 18 theorems, 56 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. NP-completeness of the problems
  4. Pseudo-polynomial algorithms to solve the problems
  5. Generic polynomial algorithms to solve SSP(M_max, ×k) and KP(M_max, ×k)
  6. Conclusion and questions for further study

Key Result

Lemma 1

Let $\mathcal{S}$ be $\mathcal{M}_{\max, {+}}^k$ or $\mathcal{M}_{\max, {\times}}^k$. Let $A, B, X$ be matrices of size $k \times k$. $X$ is a solution to $A \otimes X = B$ over $\mathcal{S}$ if and only if $X \leq X^{*}(A, B)$ and where $M_{ij}(A, B) = \{(l, j) : x^{*}_{ij} = b_{lj} \otimes a_{li}^{\otimes {-1}}\}$.

Theorems & Definitions (37)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 1
  • Remark 2
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 27 more