Scheduling under Uncertainty: A Query-based Approach

Abstract

We consider a single machine, a set of unit-time jobs, and a set of unit-time errors. We assume that the time-slot at which each error will occur is not known in advance but, for every error, there exists an uncertainty area during which the error will take place. In order to find if the error occurs in a specific time-slot, it is necessary to issue a query to it. In this work, we study two problems: (i) the error-query scheduling problem, whose aim is to reveal enough error-free slots with the minimum number of queries, and (ii) the lexicographic error-query scheduling problem where we seek the earliest error-free slots with the minimum number of queries. We consider both the off-line and the on-line versions of the above problems. In the former, the whole instance and its characteristics are known in advance and we give a polynomial-time algorithm for the error-query scheduling problem. In the latter, the adversary has the power to decide, in an on-line way, the time-slot of appearance for each error. We propose then both lower bounds and algorithms whose competitive ratios asymptotically match these lower bounds.

Speed Scaling with Explorable Uncertainty

“…More close to our work are the works on scheduling under explorable uncertainty [4,7,14]. In [14], the authors consider the problem of scheduling jobs on a single machine when the cost of each query/test is unitary.…”

“…In [4], the authors extend the problem to non-uniform testing times and they present new competitive algorithms for different variants of the problem. In [7], a singlemachine scheduling problem is considered, where given a set of 𝑛 unit-time jobs, and a set of 𝑘 unit-time errors, the objective is to reveal 𝑛 error-free timeslots with the minimum number of queries. The authors present both lower bounds and asymptotically tight upper bounds for different variants of the problem.…”

Query-competitive sorting with uncertainty

“…Note that all the work cited so far deals with problems whose classical (offline) versions can be solved in polynomial time. Uncertainty versions with queries have been proposed for the knapsack problem [17] and the scheduling problem [2,9]. Since those problems are NP-hard, we might include the query cost into the solution cost and look for a competitive algorithm if we are looking for a polynomial-time algorithm.…”

“…In this paper, both algorithms are compared with an offline optimum query set, i.e., a minimum-cost set of queries that proves the obtained solution to be correct. 2 An algorithm (either adaptive or oblivious) is α-query-competitive if it performs a total query cost of at most α times the cost of an offline optimum query set.…”

Explorable Uncertainty Meets Decision-Making in Logistics

“…Other optimization problems studied in the context of explorable uncertainty include the shortest path problem (Feder et al 2007), the knapsack problem (Goerigk et al 2015), and scheduling problems (Albers and Eckl 2020;Arantes et al 2018;Dürr et al 2020).…”