New Approximation Bounds for Lpt Scheduling

Abstract

We provide new bounds for the worst case approximation ratio of the classic Longest Processing Time (Lpt) heuristic for related machine scheduling (Q||C _max). For different machine speeds, Lpt was first considered by Gonzalez et al. (SIAM J. Comput. 6(1):155–166, 1977). The best previously known bounds originate from more than 20 years back: Dobson (SIAM J. Comput. 13(4):705–716, 1984), and independently Friesen (SIAM J. Comput. 16(3):554–560, 1987) showed that the worst case ratio of Lpt is in the interval (1.512,1.583), and in (1.52,1.67), respectively. We tighten the upper bound to $1+\sqrt{3}/3\approx1.5773$ , and the lower bound to 1.54. Although this improvement might seem minor, we consider the structure of potential lower bound instances more systematically than former works. We present a scheme for a job-exchanging process, which, repeated any number of times, gradually increases the lower bound. For the new upper bound, this systematic method together with a new idea of introducing fractional jobs, facilitated a proof that is surprisingly simple, relative to the result. We present the upper-bound proof in parameterized terms, which leaves room for further improvements.