Approximating Bin Packing with Conflict Graphs via Maximization Techniques

TL;DR

This paper studies bin packing with conflicts, showing it is harder than classic bin packing even on simple graphs, and provides new approximation algorithms for specific graph classes using a novel maximization framework.

Contribution

It proves the non-existence of an APTAS for BPC on easy graph classes and introduces a new maximization-based framework for approximation.

Findings

No APTAS for BPC on bipartite and split graphs.

Provides a 1.391-approximation for bipartite graphs.

Offers a 2.445-approximation for perfect graphs.

Abstract

We give a comprehensive study of bin packing with conflicts (BPC). The input is a set $I$ of items, sizes $s:I \rightarrow [0,1]$, and a conflict graph $G = (I,E)$. The goal is to find a partition of $I$ into a minimum number of independent sets, each of total size at most $1$. Being a generalization of the notoriously hard graph coloring problem, BPC has been studied mostly on polynomially colorable conflict graphs. An intriguing open question is whether BPC on such graphs admits the same best known approximation guarantees as classic bin packing. We answer this question negatively, by showing that (in contrast to bin packing) there is no asymptotic polynomial-time approximation scheme (APTAS) for BPC already on seemingly easy graph classes, such as bipartite and split graphs. We complement this result with improved approximation guarantees for BPC on several prominent graph classes. Most notably, we derive an asymptotic $1.391$-approximation for bipartite graphs, a $2.445$-approximation for perfect graphs, and a $\left(1+\frac{2}{e}\right)$-approximation for split graphs. To this end, we introduce a generic framework relying on a novel interpretation of BPC allowing us to solve the problem via maximization techniques. Our framework may find use in tackling BPC on other graph classes arising in applications.