Almost Tight Approximation Hardness and Online Algorithms for Resource Scheduling

We study the precedence-constrained resource scheduling problem [SICOMP'75]. There are $n$ jobs where each job takes a certain time to finish and has a resource requirement throughout the execution time. There are precedence among the jobs. The problem asks that given a resource budget, schedule the jobs obeying the precedence constraints to minimize makespan (maximum completion time of a job) such that at any point in time, the total resource being used by all the jobs is at most the given resource budget. In the offline setting, an important open question is whether a polynomial-time $O(1)$-factor approximation algorithm can be found. We prove almost tight hardness of approximation: For some constant $\alpha > 0$, there is no $o((\log t_{\max})^{\alpha})$-factor ( or $o( ( \log n )^\alpha )$-factor ) approximation algorithm with $n$ jobs of maximum job length $t_{\max}$, unless P = NP ( or NP $\subset$ DTIME$(O( 2^{\text{polylog}(n)}))$ ). We further show a connection between this scheduling problem and a seemingly unrelated problem called the shortest common super-sequence (SCS) problem, which has wide application in Biology and Genomics. We prove that an $o(\log t_{\max})$-factor approximation of the scheduling problem would imply the existence of an $o(|\Sigma|)$-approximation algorithm for SCS with alphabet $\Sigma$. We then consider the online setting. We present $\Omega(\log n)$ and $\Omega(\log t_{\max})$ lower bounds of the competitive ratio of any randomized online algorithm. Moreover, we present a matching $O(\min\{\log n, \log t_{\max}\})$-competitive deterministic online algorithm.