We consider the minimum makespan problem for n tasks and two unrelated
parallel selfish machines. Let Rnβ be the best approximation ratio of
randomized monotone scale-free algorithms. This class contains the most
efficient algorithms known for truthful scheduling on two machines. We propose
a new MinβMax formulation for Rnβ, as well as upper and lower bounds on
Rnβ based on this formulation. For the lower bound, we exploit pointwise
approximations of cumulative distribution functions (CDFs). For the upper
bound, we construct randomized algorithms using distributions with piecewise
rational CDFs. Our method improves upon the existing bounds on Rnβ for small
n. In particular, we obtain almost tight bounds for n=2 showing that
β£R2ββ1.505996β£<10β6.Comment: 28 pages, 3 tables, 1 figure. Theory Comput Syst (2019