Abstract
We study the following variant of the classic 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 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 asymptotic polynomial time approximation scheme (APTAS) for the problem, improving upon the best known ratio of 2. As a key tool, we apply a novel Shift & Swap technique which generalizes the classic linear shifting technique to scenarios allowing conflicts between items. The major challenge of packing 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 linear program.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
For the subclass of interval graphs the paper [7] gives a \(\frac{7}{3}\)-approximation algorithm.
- 3.
A graph G is d-inductive if the vertices of G can be numbered such that each vertex is connected by an edge to at most d lower numbered vertices.
- 4.
Recall that the number of medium/large items that fit in a single bin is at most \({\lfloor {\frac{1}{\varepsilon ^{k+1}}}\rfloor }\).
- 5.
For the definition of patterns see Sect. 3.2.
- 6.
Note that we do not need a label for the \(t(g_{opt})\)-insignificant groups, because their items are packed separately.
- 7.
An item is b-non-negligible w.r.t \(f^1_b\) in this iteration, or w.r.t \(f^h_b\) in iteration \(h+1, h \in \{0,\ldots ,\alpha -1\}\).
- 8.
Recall that we consider only items that were not discarded in previous steps, as discarded items are packed in a separate set of bins.
References
Adany, R., et al.: All-or-nothing generalized assignment with application to scheduling advertising campaigns. ACM Trans. Algorithms (TALG) 12(3), 1–25 (2016)
Alon, N., Azar, Y., Woeginger, G.J., Yadid, T.: Approximation schemes for scheduling on parallel machines. J. Sched. 1(1), 55–66 (1998)
Christensen, H.I., Khan, A., Pokutta, S., Tetali, P.: Approximation and online algorithms for multidimensional bin packing: a survey. Comput. Sci. Rev. 24, 63–79 (2017)
Coffman, E.G., Csirik, J., Galambos, G., Martello, S., Vigo, D.: Bin packing approximation algorithms: survey and classification. In: Handbook of Combinatorial Optimization, pp. 455–531 (2013)
Das, S., Wiese, A.: On minimizing the makespan when some jobs cannot be assigned on the same machine. In: 25th Annual European Symposium on Algorithms, ESA, pp. 31:1–31:14 (2017)
Doron-Arad, I., Kulik, A., Shachnai, H.: An APTAS for bin packing with clique-graph conflicts. arXiv preprint arXiv:2011.04273 (2020)
Epstein, L., Levin, A.: On bin packing with conflicts. SIAM J. Optim. 19(3), 1270–1298 (2008)
Fernandez de la Vega, W., Lueker, G.S.: Bin packing can be solved within 1 + \(\varepsilon \) in linear time. Combinatorica 1, 349–355 (1981)
Grage, K., Jansen, K., Klein, K.M.: An EPTAS for machine scheduling with bag-constraints. In: The 31st ACM Symposium on Parallelism in Algorithms and Architectures, pp. 135–144 (2019)
Hochbaum, D.S. (ed.): Approximation Algorithms for NP-Hard Problems. PWS Publishing Co., USA (1996)
Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems theoretical and practical results. J. ACM 34(1), 144–162 (1987)
Jansen, K.: An approximation scheme for bin packing with conflicts. J. Comb. Optim. 3(4), 363–377 (1999)
Jansen, K.: An EPTAS for scheduling jobs on uniform processors: using an MILP relaxation with a constant number of integral variables. SIAM J. Discret. Math. 24(2), 457–485 (2010)
Jansen, K., Klein, K., Verschae, J.: Closing the gap for makespan scheduling via sparsification techniques. In: 43rd International Colloquium on Automata, Languages, and Programming (ICALP), pp. 72:1–72:13 (2016)
Jansen, K., Öhring, S.R.: Approximation algorithms for time constrained scheduling. Inf. Comput. 132(2), 85–108 (1997)
Karmarkar, N., Karp, R.M.: An efficient approximation scheme for the one-dimensional bin-packing problem. In: 23rd Annual Symposium on Foundations of Computer Science, pp. 312–320. IEEE (1982)
Leung, J.Y.: Bin packing with restricted piece sizes. Inf. Process. Lett. 31(3), 145–149 (1989)
McCloskey, B., Shankar, A.: Approaches to bin packing with clique-graph conflicts. Computer Science Division, University of California (2005)
Oh, Y., Son, S.: On a constrained bin-packing problem. Technical report CS-95-14 (1995)
Rothvoß, T.: Approximating bin packing within O(log OPT * log log OPT) bins. In: 54th Annual IEEE Symposium on Foundations of Computer Science, pp. 20–29. IEEE Computer Society (2013)
Simchi-Levi, D.: New worst-case results for the bin-packing problem. Naval Res. Logist. (NRL) 41(4), 579–585 (1994)
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3(1), 103–128 (2007)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Doron-Arad, I., Kulik, A., Shachnai, H. (2021). An APTAS for Bin Packing with Clique-Graph Conflicts. In: Lubiw, A., Salavatipour, M., He, M. (eds) Algorithms and Data Structures. WADS 2021. Lecture Notes in Computer Science(), vol 12808. Springer, Cham. https://doi.org/10.1007/978-3-030-83508-8_21
Download citation
DOI: https://doi.org/10.1007/978-3-030-83508-8_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-83507-1
Online ISBN: 978-3-030-83508-8
eBook Packages: Computer ScienceComputer Science (R0)