Makespan scheduling on identical machines is one of the most basic and fundamental packing problems studied in the discrete optimization literature. It asks for an assignment of n jobs to a set of m identical machines that minimizes the makespan. The problem is strongly NP-hard, and thus we do not expect a ([Formula: see text])-approximation algorithm with a running time that depends polynomially on [Formula: see text]. It has been recently shown that a subexponential running time on [Formula: see text] would imply that the Exponential Time Hypothesis (ETH) fails. A long sequence of algorithms have been developed that try to obtain low dependencies on [Formula: see text], the better of which achieves a quadratic running time on the exponent. In this paper we obtain an algorithm with an almost-linear dependency on [Formula: see text] in the exponent, which is tight under ETH up to logarithmic factors. Our main technical contribution is a new structural result on the configuration-IP integer linear program. More precisely, we show the existence of a highly symmetric and sparse optimal solution, in which all but a constant number of machines are assigned a configuration with small support. This structure can then be exploited by integer programming techniques and enumeration. We believe that our structural result is of independent interest and should find applications to other settings. We exemplify this by applying our structural results to the minimum makespan problem on related machines and to a larger class of objective functions on parallel machines. For all these cases, we obtain an efficient PTAS with running time with an almost-linear dependency on [Formula: see text] and polynomial in n.

We study scheduling problems on a machine with varying speed. Assuming a known speed function we ask for a cost-efficient scheduling solution. Our main result is a PTAS for minimizing the total weighted completion time in this setting. This also implies a PTAS for the closely related problem of scheduling to minimize generalized global cost functions. The key to our results is a re-interpretation of the problem within the wellknown two-dimensional Gantt chart: instead of the standard approach of scheduling in the time-dimension, we construct scheduling solutions in the weight-dimension.We also consider a dynamic problem variant in which deciding upon the speed is part of the scheduling problem and we are interested in the tradeoff between scheduling cost and speed-scaling cost, which is typically the energy consumption. We observe that the optimal order is independent of the energy consumption and that the problem can be reduced to the setting where the speed of the machine is fixed, and thus admits a PTAS. Furthermore, we provide an FPTAS for the NP-hard problem variant in which the machine can run only on a fixed number of discrete speeds. Finally, we show how our results can be used to obtain a (2+ε)-approximation for scheduling preemptive jobs with release dates on multiple identical parallel machines.

Abstract. We consider the online MST and TSP problems with recourse. The nodes of an unknown graph with metric edge cost appear one by one and must be connected in such a way that the resulting tree or tour has low cost. In the standard online setting, with irrevocable decisions, no algorithm can guarantee a constant competitive ratio. In our model we allow recourse actions by giving a limited budget of edge rearrangements per iteration. It has been an open question for more than 20 years if an online algorithm equipped with a constant (amortized) budget can guarantee constant-approximate solutions [7].As our main result, we answer this question affirmatively in an amortized setting. We introduce an algorithm that maintains a nearly optimal tree when given constant amortized budget. In the non-amortized setting, we specify a promising proof technique and conjecture a structural property of optimal solutions that would prove a constant competitive ratio with a single recourse action. It might seem rather optimistic that such a small budget should be sufficient for a significant cost improvement. However, we can prove such power of recourse in the offline setting in which the sequence of node arrivals is known. Even this problem prohibits constant approximations if there is no recourse allowed. Surprisingly, already a smallest recourse budget significantly improves the performance guarantee from non-constant to constant.Unlike in classical TSP variants, the standard double-tree and shortcutting approach does not give constant guarantees in the online setting. However, a non-trivial robust shortcutting technique allows to translate online MST results into TSP results at the loss of small factors.

Scheduling a set of n jobs on m identical parallel machines so as to minimize the makespan or maximize the minimum machine load are two of the most important and fundamental scheduling problems studied in the literature. We consider the general online scenario where jobs are consecutively added to and/or deleted from an instance. The goal is to maintain a near-optimal assignment of the current set of jobs to the m machines. This goal is essentially doomed to failure unless, upon arrival or departure of a job, we allow reassigning some other jobs. Considering that the reassignment of a job induces a cost proportional to its size, the total cost for reassigning jobs must preferably be bounded by a constant r times the total size of added or deleted jobs. The value r is called the reassignment factor of the solution and it is a measure of our willingness to adapt the solution over time. Our main result is that, for any ε > 0, it is possible to achieve (1 + ε)-competitive solutions with constant reassignment factor r(ε). For the minimum makespan problem this is the first improvement on the (2 + ε)-competitive algorithm by Andrews et al. (1999) [Andrews M, Goemans M, Zhang L (1999) Improved bounds for on-line load balancing. Algorithmica 23(4):278–301]. Crucial to our algorithm is a new insight into the structure of robust, almost optimal schedules.

A natural extension of the makespan minimization problem on unrelated machines is to allow jobs to be partially processed by different machines while incurring an arbitrary setup time. In this paper we present increasingly stronger LP-relaxations for this problem and their implications on the approximability of the problem. First we show that the straightforward LP, extending the approach for the original problem, has an integrality gap of 3 and yields an approximation algorithm of the same factor. By applying a lift-and-project procedure, we are able to improve both the integrality gap and the implied approximation factor to 1 + φ, where φ is the golden ratio. Since this bound remains tight for the seemingly stronger machine configuration LP, we propose a new job configuration LP that is based on an infinite continuum of fractional assignments of each job to the machines. We prove that this LP has a finite representation and can be solved in polynomial time up to any accuracy. Interestingly, we show that our problem cannot be approximated within a factor better than e e−1 ≈ 1.582 (unless P = N P), which is larger than the inapproximability bound of 1.5 for the original problem.Mathematics Subject Classification Primary 90B35 · 68W25; Secondary 68Q25 · 90C10