New approximation techniques for some ordering problems

Abstract : We describe O(log n) times optimal approximation algorithms for the NP-hard graph optimization problems of minimum linear arrangement, minimum containing interval graph and minimum storage-time product. This improves on the O(log n log log n) approximation bounds provided in a previous paper by Even, Naor, Scheiber and Rao. Our techniques are based on using spreading metrics (as in Even Naor Rao, and Scheiber) to define a cost estimate for a problem. In this paper, we develop a recursion where at each level we identify cost which, if incurred, yields a reduction in the spreading metric cost estimates for the resulting subproblems. Specifically, we give a strategy where the cost of solving a problem at a recursive level is C plus the cost of solving the subproblems, and where the spreading metric cost estimate on the subproblem(s) is reduced by at least omego (C/log n). This ensures that the resulting solution has cost at most O(log n) times the original spreading metric cost estimate. We note that this is an existentially tight bound on the relationship between the spreading metric cost estimates and the true optimal values for these problems. For planar graphs, we continue a structural theorem of Klein, Plotkin and Rao with our new recursion and standard divide-and-conquer techniques to show that the spreading metric cost estimates are within an O(log log n) factor of the cost of the optimal solution for the minimum linear arrangement and the minimum containing interval graph problems.