Opt Versus Load In Dynamic Storage Allocation

Dynamic storage allocation is the problem of packing given axis-aligned rectangles into a horizontal strip of minimum height by sliding the rectangles vertically but not horizontally. Where L=LOAD is the maximum sum of heights of rectangles that intersect any vertical line and OPT is the minimum height of the enclosing strip, it is obvious that OPTgreater than or equal toLOAD; previous work showed that OPTless than or equal to3.LOAD. We continue the study of the relationship between OPT and LOAD, proving that OPT=L+O((h(max)/L)(1/7))L, where h(max) is the maximum job height. Conversely, we prove that for any epsilon>0, there exists a c>0 such that for all sufficiently large integers h(max), there is a dynamic storage allocation instance with maximum job height h(max), maximum load at most L, and OPTgreater than or equal toL+c(h(max)/L)(1/2+epsilon) L, for infinitely many integers L. En route, we construct several new polynomial-time approximation algorithms for dynamic storage allocation, including a (2+epsilon)-approximation algorithm for the general case and polynomial-time approximation schemes for several natural special cases.