Approximation Schemes for Packing Splittable Items with Cardinality Constraints

Abstract

We continue the study of bin packing with splittable items and cardinality constraints. In this problem, a set of items must be packed into as few bins as possible. Items may be split, but each bin may contain at most k (parts of) items, where k is some fixed constant. Complicating the problem further is the fact that items may be larger than 1, which is the size of a bin. We close this problem by providing a polynomial-time approximation scheme for it. We first present a scheme for the case k = 2 and then for the general case of constant k.Additionally, we present dual approximation schemes for k = 2 and constant k. Thus we show that for any ε > 0, it is possible to pack the items into the optimal number of bins in polynomial time, if the algorithm may use bins of size 1 + ε.

Heterogeneous Resource Allocation under Degree Constraints

“…Epstein and van Stee [16] showed that Bin Packing with splittable items and cardinality constraints is NP-Hard for any fixed value of k, and that the simple NEXT-FIT algorithm achieves an approximation ratio of 2 − 1/k. They also design a PTAS and a dual PTAS [15] for the general case where k is a constant.…”

Improved Results for a Memory Allocation Problem

“…An interesting question is whether it is possible to give an efficient algorithm with a better approximation ratio for k = 2 or for larger k. In a companion paper [6] we present polynomial time approximation schemes (PTAS) for these problems. Note that using an approximation scheme as a (1 + δ)-approximation for moderate values of δ leads to large running times.…”

“…Items are not only rounded but also split into sizes of at most 1 δ (before the rounding). For instance, for δ = 1/2, the PTAS from [6] assumes that there are four rounded sizes of items, all sizes are no larger than 2, and all trees have at most 4 nodes. A tree is called a pattern, if it is maximal in the sense that splitting it into two trees would increase the number of bins.…”

“…To design an approximation algorithm of absolute approximation ratio (1 + ε)OPT. Otherwise, using the techniques from [6], only constant time is required for enumerating all packings, so an optimal solution can be found. In particular, to achieve an absolute approximation ratio of 10 7 it is sufficient to find an optimal packing for the case of at most 54 items that have total size at most 27.…”

Approximation of the k-batch consolidation problem

“…Another related model is the packing splittable items with cardinality constraints, or PSIC [5]. PSIC is a generalization of the bin packing: the items can be split over bins but a bin cannot contain more than k items.…”