Every Deterministic Nonclairvoyant Scheduler has a Suboptimal Load Threshold

Abstract

The goal is to prove a surprising lower bound for resource augmented nonclairvoyant algorithms for scheduling jobs with sublinear nondecreasing speed-up curves on multiple processors with the objective of average response time.  Edmonds and Pruhs in SODA09 prove that for every $\e > 0$, there is an algorithm $\alg_{\e}$ that is $(1\!+\!\epsilon)$-speed $O({1 \over
\e2})$-competitive.  A problem, however, is that this algorithm
$\alg_{\e}$ depends on $\e$.  The goal is to prove that every fixed
deterministic nonclairvoyant algorithm has a suboptimal speed
threshold, namely for every (graceful) algorithm $\alg$, there is a
threshold $1\!+\!\beta_{\alg}$ that is $\beta_{\alg} > 0$ away from
being optimal such that the algorithm is $\Omega({1 \over \e
\beta_{\alg}})$ competitive with speed $(1 \!+\!  \beta_{\alg}) \!+\!
\e$ and is $\omega(1)$ competitive with speed $1 \!+\! \beta_{\alg}$.
I have worked very hard on it and have felt that I was close.  The
proof technique is to use Brouwer's fixed point theorem to break the
cycle of needing to know which input will be given before one can know
what the algorithm will do and needing to know what the algorithm will
do before one can know which input to give. Every thing I have can be
found at