An APTAS for Bin Packing with Clique-graph Conflicts

TL;DR

This paper presents an asymptotic polynomial time approximation scheme (APTAS) for the bin packing problem with clique-graph conflicts, achieving near-optimal solutions in polynomial time.

Contribution

It introduces a novel Shift & Swap technique and extends linear shifting methods to handle conflicts, improving the approximation ratio from 2 to near-optimal.

Findings

Developed an APTAS with ratio (1+ε) for the problem.

Applied a new Shift & Swap technique for conflict scenarios.

Combined enumeration and greedy methods for small item packing.

Abstract

We study the following variant of the classic {\em bin packing} problem. Given a set of items of various sizes, partitioned into groups, find a packing of the items in a minimum number of identical (unit-size) bins, such that no two items of the same group are assigned to the same bin. This problem, known as {\em bin packing with clique-graph conflicts}, has natural applications in storing file replicas, security in cloud computing and signal distribution. Our main result is an {\em asymptotic polynomial time approximation scheme (APTAS)} for the problem, improving upon the best known ratio of $2$. %In particular, for any instance $I$ and a fixed $\eps \in (0,1)$, the items are packed in at most $(1+\eps)OPT(I) +1$ bins, where $OPT(I)$ is the minimum number of bins required for packing the instance. As a key tool, we apply a novel {\em Shift \& Swap} technique which generalizes the classic linear shifting technique to scenarios allowing conflicts between items. The major challenge of packing {\em small} items using only a small number of extra bins is tackled through an intricate combination of enumeration and a greedy-based approach that utilizes the rounded solution of a {\em linear program}.