NDLI: On Bin Packing with Conflicts

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Epstein, Leah Levin, Asaf
Copyright Year	2008
Abstract	We consider the offline and online versions of a bin packing problem called bin packing with conflicts. Given a set of items $V=\{ 1,2,\dots,n\}$ with sizes $s_1,s_2,\dots,s_n\in[0,1]$ 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 items in each independent set is at most 1 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 $E=\emptyset$ and of the graph coloring problem where $s_i=0$ for all $i=1,2,\dots,n$. 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 $\frac52=2.5$-approximation algorithm for perfect graphs; a $\frac73\approx2.33333$-approximation algorithm for a subclass of perfect graphs, which contains interval graphs and chordal graphs; and a $\frac74=1.75$-approximation for algorithm bipartite graphs. For the online problem on interval graphs, we design a 4.7-competitive algorithm and show a lower bound of $\frac{155}{36}\approx4.30556$ 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 $\frac{47}{36}\approx1.30556$.
Starting Page	1270
Ending Page	1298
Page Count	29
File Format	PDF
ISSN	10526234
DOI	10.1137/060666329
e-ISSN	10957189
Journal	SIAM Journal on Optimization (SJOPE8)
Issue Number	3
Volume Number	19
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2008-11-19
Access Restriction	Subscribed
Subject Keyword	Analysis of algorithms and problem complexity Approximation algorithms approximation algorithms Analysis of algorithms online algorithms bin packing
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Software

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