Bin Packing/Covering with Delivery: Some variations, theoretical results and efficient offline algorithms

TL;DR

This paper studies the offline Bin Packing/Covering with Delivery problem, introducing new theoretical insights and algorithms, including polynomial-time solutions under certain conditions and hardness results for general cases.

Contribution

It introduces a novel view on offline optimal solutions and provides polynomial algorithms for restricted cases, along with hardness bounds for unrestricted instances.

Findings

Polynomial-time solution when item sizes are bounded away from zero

No polynomial approximation better than 6/7 for unrestricted instances unless P=NP

New theoretical framework for ordering-dependent problems

Abstract

In the recent paper \cite{BDT10} we introduced a new problem that we call Bin Packing/Covering with Delivery, or BP/CD for short. Mainly we mean under this expression that we look for not only a good, but a "good and fast" packing or covering. In that paper we mainly dealt with only one possible online BP/CD model, and proposed a new method that we call the Evolution of Algorithms. In case of such methods a neighborhood structure is defined among algorithms, and using a metaheuristic (for example simulated annealing) in some sense the best algorithm is chosen to solve the problem. Now we turn to investigate the offline case. We define several ways to treat such a BP/CD problem, although we investigate only one of them here. For the analysis, a novel view on "offline optimum" is introduced, which appears to be relevant concerning all problems where a final solution is ordering-dependent. We prove that if the item sizes are not allowed to be arbitrarily close to zero, then an optimal offline solution can be found in polynomial time. On the other hand, for unrestricted problem instances, no polynomial-time algorithm can achieve an approximation ratio better than 6/7 if $P\ne NP$.