Abstract
We consider the offline and online versions of a bin packing problem called bin packing with conflicts. Given a set of items V={ 1,2, ...,n} with sizes s 1,s 2 ...,s n ∈[0,1] and a conflict graph G=(V,E), the goal is to find a partition of the items into independent sets of G, where the total size of each independent set is at most one, so that the number of independent sets in the partition is minimized. This problem is clearly a generalization of both the classical (one-dimensional) bin packing problem where E=∅ and of the graph coloring problem where s i =0 for all i=1,2, ...,n. Since coloring problems on general graphs are hard to approximate, following previous work, we study the problem on specific graph classes. For the offline version we design improved approximation algorithms for perfect graphs and other special classes of graphs, these are a \(\frac 52=2.5\)-approximation algorithm for perfect graphs, a \(\frac 73\approx 2.33333\)-approximation for a sub-class of perfect graphs, which contains interval graphs, and a \(\frac 74=1.75\)-approximation for bipartite graphs. For the online problem on interval graphs, we design a 4.7-competitive algorithm and show a lower bound of \(\frac {155}{36}\approx 4.30556\) on the competitive ratio of any algorithm. To derive the last lower bound, we introduce the first lower bound on the asymptotic competitive ratio of any online bin packing algorithm with known optimal value, which is \(\frac {47}{36}\approx 1.30556\).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Arkin, E., Hassin, R.: On local search for weighted packing problems. Mathematics of Operations Research 23, 640–648 (1998)
Baker, B.S., Coffman Jr., E.G.: A tight asymptotic bound for next-fit-decreasing bin-packing. SIAM J. on Algebraic and Discrete Methods 2(2), 147–152 (1981)
Coffman Jr., E.G., Csirik, J., Leung, J.: Variants of classical bin packing. In: Gonzalez, T.F. (ed.) Approximation algorithms and metaheuristics. Chapman and Hall/CRC (to appear)
Crescenzi, P., Kann, V., Halldórsson, M.M., Karpinski, M., Woeginger, G.J.: A compendium of NP optimization problems, http://www.nada.kth.se/viggo/problemlist/compendium.html
de Werra, D.: An introduction to timetabling. European Journal of Operational Research 19, 151–162 (1985)
Galambos, G., Woeginger, G.J.: Repacking helps in bounded space online bin packing. Computing 49, 329–338 (1993)
Garey, M.R., Graham, R.L., Johnson, D.S., Yao, A.C.C.: Resource constrained scheduling as generalized bin packing. Journal of Combinatorial Theory (Series A) 21, 257–298 (1976)
Garey, M.R., Johnson, D.S.: Computers and intractability. W. H. Freeman and Company, New York (1979)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, London (1980)
Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM Journal on Applied Mathematics 17, 263–269 (1969)
Gyárfás, A., Lehel, J.: On-line and first-fit colorings of graphs. Journal of Graph Theory 12, 217–227 (1988)
Hujter, M., Tuza, Z.: Precoloring extension, III: Classes of perfect graphs. Combinatorics, Probability and Computing 5, 35–56 (1996)
Irani, S., Leung, V.J.: Scheduling with conflicts, and applications to traffic signal control. In: Proc. of 7th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1996), pp. 85–94 (1996)
Jansen, K.: An approximation scheme for bin packing with conflicts. Journal of Combinatorial Optimization 3(4), 363–377 (1999)
Jansen, K., Öhring, S.: Approximation algorithms for time constrained scheduling. Information and Computation 132, 85–108 (1997)
Jensen, T.R., Toft, B.: Graph coloring problems. Wiley, Chichester (1995)
Johnson, D.S., Demers, A., Ullman, J.D., Garey, M.R., Graham, R.L.: Worst-case performance bounds for simple one-dimensional packing algorithms. SIAM Journal on Computing 3, 256–278 (1974)
Kierstead, H.A., Trotter, W.T.: An extremal problem in recursive combinatorics. Congressus Numerantium 33, 143–153 (1981)
Lee, C.C., Lee, D.T.: A simple online bin packing algorithm. Journal of the ACM 32(3), 562–572 (1985)
Lovász, L., Saks, M., Trotter, W.T.: An on-line graph coloring algorithm with sublinear performance ratio. Discrete Math. 75, 319–325 (1989)
Marx, D.: Precoloring extension, http://www.cs.bme.hu/dmarx/prext.html
Marx, D.: Precoloring extension on chordal graphs (2004) (manuscript)
McCloskey, B., Shankar, A.: Approaches to bin packing with clique-graph conflicts. Technical Report UCB/CSD-05-1378, EECS Department, University of California, Berkeley (2005)
Oh, Y., Son, S.H.: On a constrained bin-packing problem. Technical Report CS-95-14, Department of Computer Science, University of Virginia (1995)
Schrijver, A.: Combinatorial optimization polyhedra and efficiency. Springer, Heidelberg (2003)
Seiden, S.S.: On the online bin packing problem. Journal of the ACM 49(5), 640–671 (2002)
Simchi-Levi, D.: New worst-case results for the bin-packing problem. Naval Res. Logist. 41(4), 579–585 (1994)
Ullman, J.D.: The performance of a memory allocation algorithm. Technical Report 100, Princeton University, Princeton, NJ (1971)
van Vliet, A.: An improved lower bound for online bin packing algorithms. Information Processing Letters 43(5), 277–284 (1992)
Yao, A.C.C.: New algorithms for bin packing. Journal of the ACM 27, 207–227 (1980)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Epstein, L., Levin, A. (2007). On Bin Packing with Conflicts. In: Erlebach, T., Kaklamanis, C. (eds) Approximation and Online Algorithms. WAOA 2006. Lecture Notes in Computer Science, vol 4368. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11970125_13
Download citation
DOI: https://doi.org/10.1007/11970125_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69513-4
Online ISBN: 978-3-540-69514-1
eBook Packages: Computer ScienceComputer Science (R0)