We contribute the first randomized algorithm that is an integration of
arbitrarily many deterministic algorithms for the fully online multiprocessor
scheduling with testing problem. When there are two machines, we show that with
two component algorithms its expected competitive ratio is already strictly
smaller than the best proven deterministic competitive ratio lower bound. Such
algorithmic results are rarely seen in the literature. Multiprocessor
scheduling is one of the first combinatorial optimization problems that have
received numerous studies. Recently, several research groups examined its
testing variant, in which each job Jjβ arrives with an upper bound ujβ on
the processing time and a testing operation of length tjβ; one can choose to
execute Jjβ for ujβ time, or to test Jjβ for tjβ time to obtain the
exact processing time pjβ followed by immediately executing the job for pjβ
time. Our target problem is the fully online version, in which the jobs arrive
in sequence so that the testing decision needs to be made at the job arrival as
well as the designated machine. We propose an expected (Ο+3β+1)(β3.1490)-competitive randomized algorithm as a non-uniform
probability distribution over arbitrarily many deterministic algorithms, where
Ο=25β+1β is the Golden ratio. When there are two
machines, we show that our randomized algorithm based on two deterministic
algorithms is already expected 43Ο+313β7Οββ(β2.1839)-competitive. Besides, we use Yao's principle to prove lower
bounds of 1.6682 and 1.6522 on the expected competitive ratio for any
randomized algorithm at the presence of at least three machines and only two
machines, respectively, and prove a lower bound of 2.2117 on the competitive
ratio for any deterministic algorithm when there are only two machines.Comment: 21 pages with 1 plot; an extended abstract to be submitte