Algorithms for the Pagination Problem, a Bin Packing with Overlapping Items

TL;DR

This paper introduces the NP-complete pagination problem, an extension of bin packing with overlapping items, and evaluates various algorithms to approximate solutions, providing insights into their performance and difficulty measures.

Contribution

It formulates the pagination problem as an integer linear program and compares multiple algorithms, including heuristics and genetic algorithms, for solving it.

Findings

Algorithms vary in effectiveness depending on instance difficulty.

A predictive measure for instance difficulty is proposed.

Recommendations for algorithm selection based on constraints are provided.

Abstract

We introduce the strongly NP-complete pagination problem, an extension of BIN PACKING where packing together two items may make them occupy less volume than the sum of their individual sizes. To achieve this property, an item is defined as a finite set of symbols from a given alphabet: while, in BIN PACKING, any two such sets would be disjoint, in PAGINATION, they can share zero, one or more symbols. After formulating the problem as an integer linear program, we try to approximate its solutions with several families of algorithms: from straightforward adaptations of classical BIN PACKING heuristics, to dedicated algorithms (greedy and non-greedy), to standard and grouping genetic algorithms. All of them are studied first theoretically, then experimentally on an extensive random test set. Based upon these data, we propose a predictive measure of the statistical difficulty of a given instance, and finally recommend which algorithm should be used in which case, depending on either time constraints or quality requirements.