l22 Spreading Metrics for Vertex Ordering Problems

Moses Charikar, Mohammad Taghi Hajiaghayi, Howard J. Karloff, Satish Rao

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 $O(\sqrt{\log n}\log\log n)$ , $O(\sqrt{\log n}\log\log n)$ , and $O(\sqrt{\log T}\log\log T)$ , 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 “ℓ 2 2 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 (Proc. of 36th STOC, pp. 222–231, 2004) can be adapted to exploit the geometry of such ℓ 2 2 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 ℓ 2 2 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, ℓ 2 2 spreading metrics approximate permutation metrics on n points to a factor of $O(\sqrt{\log n}\log\log n)$ .