Max-min Online Allocations with a Reordering Buffer

Abstract

Abstract. We consider online scheduling so as to maximize the minimum load, using a reordering buffer which can store some of the jobs before they are assigned irrevocably to machines. For m identical machines, we show an upper bound of Hm−1 + 1 for a buffer of size m − 1. A competitive ratio below Hm is not possible with any finite buffer size, and it requires a buffer of sizeΩ(m) to get a ratio of O(log m). For uniformly related machines, we show that a buffer of size m + 1 is sufficient to get an approximation ratio of m, which is best possible for any finite sized buffer. Finally, for the restricted assignment model, we show lower bounds identical to those of uniformly related machines, but using different constructions. In addition, we design an algorithm of approximation ratio O(m) which uses a finite sized buffer. We give tight bounds for two machines in all the three models. These results sharply contrast to the (previously known) results which can be achieved without the usage of a reordering buffer, where it is not possible to get a ratio below an approximation ratio of m already for identical machines, and it is impossible to obtain an algorithm of finite approximation ratio in the other two models, even for m = 2. Our results strengthen the previous conclusion that a reordering buffer is a powerful tool and it allows a significant decrease in the competitive ratio of online algorithms for scheduling problems. Another interesting aspect of our results is that our algorithm for identical machines imitates the behavior of the greedy algorithm on (a specific set of) related machines, whereas our algorithm for related machines completely ignores the speeds until the end, and then only uses the relative order of the speeds.

The cost of selfishness for maximizing the minimum load on uniformly related machines

“…Additionally, we showed close bounds on the poa for the complete set of games, namely, that it is at least 1.691 and at most 1.7. For small numbers of machines we showed that poa(2) = poa(3) = 3 2 and poa(4) = 13 8 . In contrast to these results, we can show that for uniformly related machines even the pos for the complete set of instances is unbounded.…”

“…[4,2,6,8]). Various game-theoretic aspects of max-min fairness in resource allocation games were considered, but unlike the makespan minimization problem for which the poa and the pos were extensively studied (see [17,3,18]), these measures were not previously considered for the uncoordinated machine covering problem in the setting of selfish jobs and uniformly related machines.…”

An overview of some results for reordering buffers

“…Another example of an objective function is to maximize the minimum load on any machine. Epstein, Levin, and van Stee [16] give a comprehensive study of the use of reordering buffers for this max-min objective function.…”

Improved approaches to the exact solution of the machine covering problem

“…Further publications are concerned with a polynomial-time approximation scheme (PTAS) (cf. Woeginger 1997) and on-line as well as semi-on-line versions of P||C min (cf., e.g., Azar and Epstein 1998;He and Tan 2002;Luo and Sun 2005;Ebenlendr et al 2006;Cai 2007;Tan and Wu 2007;Epstein et al 2011). The sole publication devoted to exact solution procedures is due to Haouari and Jemmali (2008).…”