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Approximating Bin Packing with Conflict Graphs via Maximization Techniques

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Graph-Theoretic Concepts in Computer Science (WG 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14093))

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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.

A full version of the paper is available in [6].

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Notes

  1. 1.

    See the formal definitions of graph coloring and independent sets in Sect. 2.

  2. 2.

    We give a detailed description of Algorithm FFD in [6].

  3. 3.

    Recently, Huang et al. [15] obtained a \(\frac{5}{3}\)-approximation for bipartite graphs, simultaneously and independently of our work. We note that the techniques of [15] are different than ours, and their algorithm is more efficient in terms of running time.

  4. 4.

    For more details on algorithms FFD and AsymptoticBP see, e.g., [27].

  5. 5.

    An efficient PTAS is a PTAS \(\{ A_{{\varepsilon }} \}\) where, for all \({\varepsilon }>0\), the running time of \(A_{{\varepsilon }}\) is given by \(f(1/ {\varepsilon })\) times a polynomial of the input size.

  6. 6.

    We note that a maximum matching based technique for BPC is used also in [8, 15].

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Authors and Affiliations

  1. Computer Science Department, Technion, Haifa, 3200003, Israel

    Ilan Doron-Arad & Hadas Shachnai

Authors
  1. Ilan Doron-Arad
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  2. Hadas Shachnai
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Correspondence to Hadas Shachnai .

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Editors and Affiliations

  1. Durham University, Durham, UK

    Daniël Paulusma

  2. University of Fribourg, Fribourg, Fribourg, Switzerland

    Bernard Ries

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Doron-Arad, I., Shachnai, H. (2023). Approximating Bin Packing with Conflict Graphs via Maximization Techniques. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_19

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  • DOI: https://doi.org/10.1007/978-3-031-43380-1_19

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