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Parameterized Complexity of Factorization Problems

Markus Lohrey, Andreas Rosowski

TL;DR

Several new upper bounds and fpt-equivalences with more restricted problems (subset sum and knapsack) with variants of the parameterized change-making problems are shown.

Abstract

We study the parameterized complexity of the following factorization problem: given elements $a,a_1, \ldots, a_m$ of a monoid and a parameter $k$, can $a$ be written as the product of at most (or exactly) $k$ elements from $a_1, \ldots, a_m$. Several new upper bounds and fpt-equivalences with more restricted problems (subset sum and knapsack) are shown. Finally, some new upper bounds for variants of the parameterized change-making problems are shown.

Parameterized Complexity of Factorization Problems

TL;DR

Several new upper bounds and fpt-equivalences with more restricted problems (subset sum and knapsack) with variants of the parameterized change-making problems are shown.

Abstract

We study the parameterized complexity of the following factorization problem: given elements of a monoid and a parameter , can be written as the product of at most (or exactly) elements from . Several new upper bounds and fpt-equivalences with more restricted problems (subset sum and knapsack) are shown. Finally, some new upper bounds for variants of the parameterized change-making problems are shown.
Paper Structure (16 sections, 25 theorems, 65 equations)

This paper contains 16 sections, 25 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Related work.
  3. General notations
  4. Parameterized complexity theory
  5. Parameterized complexity of factorization problems
  6. Generic variants of factorization problems
  7. Factorization in transformation monoids
  8. Factorization in symmetric groups
  9. Factorization in the commutative setting
  10. Simple lemmas
  11. Commutative subset sum problems
  12. Abelian groups
  13. Change making problems
  14. NP-hardness of the approximative change-making problems for a>0 and b=0
  15. Conclusion
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $n_1, n_2, \ldots, n_l \geq 2$ be pairwise coprime integers and let $N = \prod_{1 \le i \le l} n_i$. Then there is a ring isomorphism Moreover, given the binary representations of $n_1, \ldots, n_l \in \naturals$ and $x_i \in \integers_{n_i}$ for $1 \le i \le l$, one can compute in polynomial time the binary representation of $h(x_1, \ldots, x_l) \in \integers_N$.This follows from the standar

Theorems & Definitions (52)

  • Theorem 2.1: see e.g. ireland
  • Definition 4.4
  • Remark 4.5
  • Remark 4.6
  • Lemma 4.7
  • proof
  • Theorem 4.9
  • proof
  • Corollary 4.10
  • Theorem 4.11
  • ...and 42 more