Solving Packing Problems with Few Small Items Using Rainbow Matchings

TL;DR

This paper develops fixed-parameter algorithms for packing problems like Bin Packing, parameterized by the number of small items, using a novel colored matching reduction, advancing the understanding of their parameterized complexity.

Contribution

Introduces randomized and deterministic fixed-parameter algorithms for packing problems based on small item count, via a new colored matching problem reduction.

Findings

Algorithms run in fixed-parameter time with respect to small items

Reduction to colored matching problem is effective for multiple packing variants

Deterministic algorithm complements randomized solutions

Abstract

An important area of combinatorial optimization is the study of packing and covering problems, such as Bin Packing, Multiple Knapsack, and Bin Covering. Those problems have been studied extensively from the viewpoint of approximation algorithms, but their parameterized complexity has only been investigated barely. For problem instances containing no "small" items, classical matching algorithms yield optimal solutions in polynomial time. In this paper we approach them by their distance from triviality, measuring the problem complexity by the number $k$ of small items. Our main results are fixed-parameter algorithms for vector versions of Bin Packing, Multiple Knapsack, and Bin Covering parameterized by $k$. The algorithms are randomized with one-sided error and run in time $4^{k} \cdot k! \cdot n^{O(1)}$. To achieve this, we introduce a colored matching problem to which we reduce all these packing problems. The colored matching problem is natural in itself and we expect it to be useful for other applications. We also present a deterministic fixed-parameter for Bin Packing with run time $(k!)^{2}\cdot k \cdot 2^{k}\cdot n\cdot \log(n)$.