This work provides an analysis of checkpointing strategies for minimizing expected job execution times in an environment that is subject to processor failures. In the case of both sequential and parallel jobs, we give the optimal solution for exponentially distributed failure inter-arrival times, which, to the best of our knowledge, is the first rigorous proof that periodic checkpointing is optimal. For non-exponentially distributed failures, we develop a dynamic programming algorithm to maximize the amount of work completed before the next failure, which provides a good heuristic for minimizing the expected execution time. Our work considers various models of job parallelism and of parallel checkpointing overhead. We first perform extensive simulation experiments assuming that failures follow Exponential or Weibull distributions, the latter being more representative of real-world systems. The obtained results not only corroborate our theoretical findings, but also show that our dynamic programming algorithm significantly outperforms previously proposed solutions in the case of Weibull failures. We then discuss results from simulation experiments that use failure logs from production clusters. These results confirm that our dynamic programming algorithm significantly outperforms existing solutions for real-world clusters.

In this work we study min max robust scheduling problems assuming that the processing times can take any value in the budgeted uncertainty set introduced by Sim (2003,2004). We consider problems on a single machine that minimize the (weighted and unweighted) sum of completion times and problems that minimize the makespan on parallel and unrelated machines. We provide polynomial algorithms and approximation algorithms: constant factor, average non-constant factor, (fully or not) polynomial time approximation schemes. In addition, we prove that the robust version of minimizing the weighted completion time on a single machine is N P-hard in the strong sense.Keywords approximation algorithms, robust optimization, scheduling IntroductionScheduling is a very wide topic in combinatorial optimization with applications ranging from production and manufacturing systems to transportation and logistics systems. Stated generally, the objective of scheduling is to allocate optimally scarce resources to activities over time. The practical relevance and the difficulty of solving the general scheduling problem have motivated an intense research activity in a large variety of scheduling environments. Scheduling problems are usually defined in the following way: given a set of n jobs represented by J , a set of m machines represented by M, and processing times represented by the tuple p, we look for a schedule σ of the jobs on the machines that satisfies the side constraints, represented by the set S of feasible schedules, and minimize objective function f (σ, p). Formally, this amounts to solve optimization problem min σ∈S f (σ, p).Various sources of uncertainty affect real scheduling problems, among which machine breakdowns, working environment changes, worker performance instabilities, tool quality variations and unavailability. Ignoring these uncertainties usually yields schedules that perform poorly under real conditions. Hence, researchers have introduced frameworks where the uncertainty is directly taken into account either by considering random variables as input or in a worst-case approach where the uncertainty parameters are constrained in a set. These frameworks are respectively denoted by Stochastic Programming and Robust Optimization (RO). We disregard the former in this paper because of its requirement for a probabilistic distribution of the random inputs, which is very difficult to obtain in practice. We focus instead on Robust Scheduling, which models the uncertainty on the processing times by a finite set U ⊂ N n . 1 In the robust problem, the maximum value of f (σ, p) over all p ∈ U should be minimized. Formally, this amounts to solve optimization problem min σ∈S max p∈U f (σ, p), or equivalently, min σ∈S F (σ, U ) where F (σ, U ) = max p∈U f (σ, p) represents the robust objective function. We say that a schedule σ * ∈ S is robust if it solves the associated scheduling problem min σ∈S F (σ, U ).Robust schedules are desirable from a practical perspective because they hedge against adverse conditions of t...

International audiencen this paper we study the Multiple Strip Packing (MSP) problem, a generalization of the well-known Strip Packing problem. For a given set of rectangles, r 1,...,r n , with heights and widths ≤ 1, the goal is to find a non-overlapping orthogonal packing without rotations into k ∈ ℕ strips [0,1]×[0, ∞ ), minimizing the maximum of the heights. We present an approximation algorithm with absolute ratio 2, which is the best possible, unless P=NP , and an improvement of the previous best result with ratio 2 + ε. Furthermore we present simple shelf-based algorithms with short running-time and an AFPTAS for MSP. Since MSP is strongly NP -hard, an FPTAS is ruled out and an AFPTAS is also the best possible result in the sense of approximation theory

Motivated by the yield optimization problem in semiconductor manufacturing, we model the wafer to wafer integration problem as a special multi-dimensional assignment problem (called WWI-m), and study it from an approximation point of view. We give approximation algorithms achieving an approximation factor of 3 2 and 4 3 for WWI-3, and we show that extensions of these algorithms to the case of arbitrary m do not give constant factor approximations. We argue that a special case of the yield optimization problem can be solved in polynomial time.