Bin Packing with Colocations

Abstract

Motivated by an assignment problem arising in MapReduce computations, we investigate a generalization of the Bin Packing problem which we call Bin Packing with Colocations Problem. We are given a weigthed graph $G=(V,E)$, where V represents the set of items with positive integer weights and E the set of related (to be colocated) items, and an integer q. The goal is to pack the items into a minimum number of bins so that (i) for each bin, the total weight of the items packed in this bin is at most q, and (ii) for each edge $(i,j) \in E$ there is at least one bin containing both items i and j.

We first point out that, when the graph is unweighted (i.e., all the items have equal weights), the problem is equivalent to the q-clique problem, and when furthermore the graph is a clique, optimal solutions are obtained from Covering Designs. We prove that the problem is strongly NP-hard even for paths and unweighted trees. Then, we propose approximation algorithms for particular families of graphs, including: a $(3+\sqrt{5})$-approximation algorithm for complete graphs (improving a previous ratio of 8), a 2-approximation algorithm for paths, a 5-approximation algorithm for trees, and an $(1+ O(\log q/q))$-approximation algorithm for unweighted trees. For general graphs, we propose a $3+2\lceil mad(G)/2\rceil$-approximation algorithm, where mad(G) is the maximum average degree of G. Finally, we show how to convert any approximation algorithm for Bin Packing (resp. Densest q-Subgraph) problem into an approximation algorithm for the problem on weighted (resp. unweighted) general graphs.

This work is partially supported by ANR project Stint under reference ANR-13-BS02-0007, ANR program “Investments for the Future” under reference ANR-11-LABX-0031-01, the Research Center of Athens University of Economics and Business (RC-AUEB), and the Special Account for Research Grants of National and Kapodistrian University of Athens.