Approximating the bandwidth via volume respecting embeddings (extended abstract)
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Divide-and-conquer approximation algorithms via spreading metrics
Journal of the ACM (JACM)
Semi-definite relaxations for minimum bandwidth and other vertex-ordering problems
Theoretical Computer Science - Selected papers in honor of Manuel Blum
Expander flows, geometric embeddings and graph partitioning
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
New Approximation Techniques for Some Linear Ordering Problems
SIAM Journal on Computing
O(√log n) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
l22 spreading metrics for vertex ordering problems
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Combinatorial algorithms for nearest neighbors, near-duplicates and small-world design
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Combinatorial Framework for Similarity Search
SISAP '09 Proceedings of the 2009 Second International Workshop on Similarity Search and Applications
d-dimensional arrangement revisited
Information Processing Letters
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We give an O(√log n)-approximation algorithm for d-dimensional arrangement - the problem of mapping a graph to a d-dimensional grid (for constant d ≥ 2) to minimize the sum of edge lengths. This improves the previous best O(log n log log n) approximation of Even, Naor, Rao and Schieber. The d = 1 case is the well studied Minimum Linear Arrangement problem. The problem is equivalent to the question of mapping a graph to integer points on a line so as to minimize the sum of edge costs, where edge costs are measured by edge lengths raised to the exponent α = 1/d. We give a simple recursive partitioning algorithm for this variant of linear arrangement for any exponent α ε (0, 1). Our analysis also applies to a directed version of the problem: given a directed graph, the goal is to map vertices to the line so as to minimize the sum of costs of forward edges. As before, edge costs are edge lengths raised to the exponent α. The α = 0 case is the well known Minimum Feedback Arc Set problem, and the α = 1 case is essentially the Minimum Storage-Time Product problem. We analyze an extremely simple divide and conquer algorithm that uses a balanced cut subroutine with approximation ratio β to recursively partition the graph. Our analysis shows that this approach gives an approximation ratio of O(β) for the minimum linear arrangement problem with exponent α for any fixed α ε (0, 1).