Optimal Time-Backlog Tradeoffs for the Variable-Processor Cup Game

The \emph{$p$-processor cup game} is a classic and widely studied scheduling problem that captures the setting in which a $p$-processor machine must assign tasks to processors over time in order to ensure that no individual task ever falls too far behind. The problem is formalized as a multi-round game in which two players, a filler (who assigns work to tasks) and an emptier (who schedules tasks) compete. The emptier's goal is to minimize backlog, which is the maximum amount of outstanding work for any task. Recently, Kuszmaul and Westover (ITCS, 2021) proposed the \emph{variable-processor cup game}, which considers the same problem, except that the amount of resources available to the players (i.e., the number $p$ of processors) fluctuates between rounds of the game. They showed that this seemingly small modification fundamentally changes the dynamics of the game: whereas the optimal backlog in the fixed $p$-processor game is $\Theta(\log n)$, independent of $p$, the optimal backlog in the variable-processor game is $\Theta(n)$. The latter result was only known to apply to games with \emph{exponentially many} rounds, however, and it has remained an open question what the optimal tradeoff between time and backlog is for shorter games. This paper establishes a tight trade-off curve between time and backlog in the variable-processor cup game. Importantly, we prove that for a game consisting of $t$ rounds, the optimal backlog is $\Theta(n)$ if and only if $t \ge \Omega(n^3)$. Our techniques also allow for us to resolve several other open questions concerning how the variable-processor cup game behaves in beyond-worst-case-analysis settings.