L22 spreading metrics for vertex ordering problems

Abstract

We design approximation algorithms for the vertex ordering problems MINIMUM LINEAR ARRANGEMENT, MINIMUM CONTAINING INTERVAL GRAPH, and MINIMUM STORAGE-TIME PRODUCT, achieving approximation factors of O√log n log log n), O√log n log log n), and O√log T log log T), respectively, the last running in time polynomial in T (T being the sum of execution times). The technical contribution of our paper is to introduce l22 spreading metrics" (that can be computed by semidefinite programming) as relaxations for both undirected and directed "permutation metrics," which are induced by permutations of {1, 2, . . ., n}. The techniques introduced in the recent work of Arora, Rao and Vazirani can be adapted to exploit the geometry of such l22 spreading metrics, giving a powerful tool for the design of divide-and-conquer algorithms. In addition to their applications to approximation algorithms, the study of such l22 spreading metrics as relaxations of permutation metrics is interesting in its own right. We show how our results imply that, in a certain sense we make precise, l22 spreading metrics approximate permutation metrics on n points to a factor of O√log n log log n).