# Bin Packing with Colocations

• Jean-Claude Bermond
• Nathann Cohen
• David Coudert
• Dimitrios Letsios
• Ioannis Milis
• Stéphane Pérennes
• Vassilis Zissimopoulos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10138)

## 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.

### References

1. 1.
Afrati, F.N., Dolev, S., Korach, E., Sharma, S., Ullman, J.D.: Assignment of different-sized inputs in mapreduce. In: Proceedings of the EDBT/ICDT Conference, Brussels, Belgium, pp. 28–37 (2015)Google Scholar
2. 2.
Afrati, F.N., Dolev, S., Korach, E., Sharma, S., Ullman, J.D.: Assignment problems of different-sized inputs in mapreduce. CoRR, abs/1507.04461 (2015)Google Scholar
3. 3.
Alon, N., Spencer, J.H.: The Probabilistic Method, 4th edn. Wiley, New York (2016)
4. 4.
Bermond, J.-C., Cohen, N., Coudert, D., Letsios, D., Milis, I., Pérennes, S., Zissimopoulos, V.: Bin packing with colocations. Technical report hal-01381333, Inria, I3S (2016). https://hal.inria.fr/hal-01381333
5. 5.
Bhaskara, A., Charikar, M., Chlamtac, E., Feige, U., Vijayaraghavan, A.: Detecting high log-densities: an $$O(n^{1/4})$$ approximation for densest k-subgraph. In: ACM Symposium on Theory of Computing (STOC), pp. 201–210 (2010)Google Scholar
6. 6.
Coffman Jr., 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)Google Scholar
7. 7.
Colbourn, C., Dinitz, J. (eds.): Handbook of Combinatorial Designs: Discrete Mathematics and its Applications, vol. 42, 2nd edn. Chapman and Hall/CRC, Boca Raton (2006)Google Scholar
8. 8.
Goldschmidt, O., Hochbaum, D.S., Hurkens, C.A.J., Yu, G.: Approximation algorithms for the k-clique covering problem. SIAM J. Discrete Math. 9(3), 492–509 (1996)
9. 9.
Hakimi, S.L.: On the degrees of the vertices of a directed graph. J. Frankl. Inst. 279, 290–308 (1965)
10. 10.
Mills, W., Mullin, R.: Coverings and packings. In: Colbourn, C., Dinitz, J. (eds.) Contemporary Design Theory: A Collection of Surveys, pp. 371–399. Wiley, New York (1992)Google Scholar
11. 11.
Rödl, V.: On a packing and covering problem. Eur. J. Comb. 6, 69–78 (1985)
12. 12.
Stinson, D.R.: Coverings. In: Colbourn, C., Dinitz, J. (eds.) The CRC Handbook of Combinatorial Designs. Discrete Mathematics and Its Applications, vol. 42, 2nd edn, pp. 260–265. CRC Press, Boca Raton (2006). chapter IV.8Google Scholar
13. 13.
Wilson, R.M.: Decomposition of complete graphs into subgraphs isomorphic to a given graph. Congres. Numer. 15, 647–659 (1976)

© Springer International Publishing AG 2017

## Authors and Affiliations

• Jean-Claude Bermond
• 1
• Nathann Cohen
• 2
• David Coudert
• 1
• Dimitrios Letsios
• 1
• Ioannis Milis
• 3
• Stéphane Pérennes
• 1
• Vassilis Zissimopoulos
• 4
1. 1.Université Côte d’Azur, Inria, CNRS, I3SSophia AntipolisFrance
2. 2.CNRS and University of Paris SudOrsayFrance
3. 3.Department of InformaticsAthens University of Economics and BusinessAthensGreece
4. 4.Department of Informatics and TelecommunicationsNational and Kapodistrian University of AthensAthensGreece