Integrality gaps for sparsest cut and minimum linear arrangement problems
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
New theoretical results on quadratic placement
Integration, the VLSI Journal
Interval completion with few edges
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
An improved approximation ratio for the minimum linear arrangement problem
Information Processing Letters
A divide and conquer algorithm for d-dimensional arrangement
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
On compressing social networks
Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining
Decorous Lower Bounds for Minimum Linear Arrangement
INFORMS Journal on Computing
Revised GRASP with path-relinking for the linear ordering problem
Journal of Combinatorial Optimization
Inapproximability Results for Maximum Edge Biclique, Minimum Linear Arrangement, and Sparsest Cut
SIAM Journal on Computing
d-dimensional arrangement revisited
Information Processing Letters
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We describe logarithmic approximation algorithms for the NP-hard graph optimization problems of minimum linear arrangement, minimum containing interval graph, and minimum storage--time product. This improves upon the best previous approximation bounds of Even, Naor, Rao, and Schieber [J. ACM, 47 (2000), pp. 585--616] for these problems by a factor of $\Omega$(log log n). We use the lower bound provided by the volume W of a spreading metric for each of the ordering problems above (as defined by Even et al.) in order to find a solution with cost at most a logarithmic factor times W for these problems. We develop a divide-and-conquer strategy where the cost of a solution to a problem at a recursive level is $C$ plus the cost of a solution to the subproblems at this level, and where the spreading metric volume on the subproblems is less than the original volume by $\Omega$(C log n), ensuring that the resulting solution has cost O(log n) times the original spreading metric volume. We note that this is an existentially tight bound on the relationship between the spreading metric volume and the true optimal values for these problems. For planar graphs, we combine a structural theorem of Klein, Plotkin, and Rao [Proceedings of the 25th ACM Symposium on Theory of Computing, 1993, pp. 682--690] with our new recursion technique to show that the spreading metric cost volumes are within an O(log log n) factor of the cost of an optimal solution for the minimum linear arrangement, and the minimum containing interval graph problems.