Optimal Algorithms for Scheduling under Time-of-Use Tariffs

We consider a natural generalization of classical scheduling problems in which using a time unit for processing a job causes some time-dependent cost which must be paid in addition to the standard scheduling cost. We study the scheduling objectives of minimizing the makespan and the sum of (weighted) completion times. It is not difficult to derive a polynomial-time algorithm for preemptive scheduling to minimize the makespan on unrelated machines. The problem of minimizing the total (weighted) completion time is considerably harder, even on a single machine. We present a polynomial-time algorithm that computes for any given sequence of jobs an optimal schedule, i.e., the optimal set of time-slots to be used for scheduling jobs according to the given sequence. This result is based on dynamic programming using a subtle analysis of the structure of optimal solutions and a potential function argument. With this algorithm, we solve the unweighted problem optimally in polynomial time. For the more general problem, in which jobs may have individual weights, we develop a polynomial-time approximation scheme (PTAS) based on a dual scheduling approach introduced for scheduling on a machine of varying speed. As the weighted problem is strongly NP-hard, our PTAS is the best possible approximation we can hope for.Comment: 17 pages; A preliminary version of this paper with a subset of results appeared in the Proceedings of MFCS 201

Optimal Algorithms for Scheduling under Time-of-Use Tariffs

We consider a natural generalization of classical scheduling problems in which using a time unit for processing a job causes some time-dependent cost which must be paid in addition to the standard scheduling cost. We study the scheduling objectives of minimizing the makespan and the sum of (weighted) completion times. It is not difficult to derive a polynomial-time algorithm for preemptive scheduling to minimize the makespan on unrelated machines. The problem of minimizing the total (weighted) completion time is considerably harder, even on a single machine. We present a polynomial-time algorithm that computes for any given sequence of jobs an optimal schedule, i.e., the optimal set of time-slots to be used for scheduling jobs according to the given sequence. This result is based on dynamic programming using a subtle analysis of the structure of optimal solutions and a potential function argument. With this algorithm, we solve the unweighted problem optimally in polynomial time. For the more general problem, in which jobs may have individual weights, we develop a polynomial-time approximation scheme (PTAS) based on a dual scheduling approach introduced for scheduling on a machine of varying speed. As the weighted problem is strongly NP-hard, our PTAS is the best possible approximation we can hope for

Min-Sum Scheduling Under Precedence Constraints

In many scheduling situations, it is important to consider non-linear functions of job completions times in the objective. This was already recognized by Smith (1956). Recently, the theory community has begun a thorough study of the resulting problems, mostly on single-machine instances for which all permutations of jobs are feasible. However, a typical feature of many scheduling problems is that some jobs can only be processed after others. In this paper, we give the first approximation algorithms for min-sum scheduling with (nonnegative, non-decreasing) non-linear functions and general precedence constraints. In particular, for 1|prec|sum w_j f(C_j), we propose a polynomial-time universal algorithm that performs well for all functions f simultaneously. Its approximation guarantee is 2 for all concave functions, at worst. We also provide a (non-universal) polynomial-time algorithm for the more general case 1|prec|sum f_j(C_j). The performance guarantee is no worse than 2+epsilon for all concave functions. Our results match the best bounds known for the case of linear functions, a widely studied problem, and considerably extend the results for minimizing sum w_jf(C_j) without precedence constraints

Optimal algorithms for scheduling under time-of-use tariffs

We consider a natural generalization of classical scheduling problems to a setting in which using a time unit for processing a job causes some time-dependent cost, the time-of-use tariff, which must be paid in addition to the standard scheduling cost. We focus on preemptive single-machine scheduling and two classical scheduling cost functions, the sum of (weighted) completion times and the maximum completion time, that is, the makespan. While these problems are easy to solve in the classical scheduling setting, they are considerably more complex when time-of-use tariffs must be considered. We contribute optimal polynomial-time algorithms and best possible approximation algorithms. For the problem of minimizing the total (weighted) completion time on a single machine, we present a polynomial-time algorithm that computes for any given sequence of jobs an optimal schedule, i.e., the optimal set of time slots to be used for preemptively scheduling jobs according to the given sequence. This result is based on dynamic programming using a subtle analysis of the structure of optimal solutions and a potential function argument. With this algorithm, we solve the unweighted problem optimally in polynomial time. For the more general problem, in which jobs may have individual weights, we develop a polynomial-time approximation scheme (PTAS) based on a dual scheduling approach introduced for scheduling on a machine of varying speed. As the weighted problem is strongly NP-hard, our PTAS is the best possible approximation we can hope for. For preemptive scheduling to minimize the makespan, we show that there is a comparably simple optimal algorithm with polynomial running time. This is true even in a certain generalized model with unrelated machines

How Unsplittable-Flow-Covering helps Scheduling with Job-Dependent Cost Functions

Generalizing many well-known and natural scheduling problems, scheduling with job-specific cost functions has gained a lot of attention recently. In this setting, each job incurs a cost depending on its completion time, given by a private cost function, and one seeks to schedule the jobs to minimize the total sum of these costs. The framework captures many important scheduling objectives such as weighted flow time or weighted tardiness. Still, the general case as well as the mentioned special cases are far from being very well understood yet, even for only one machine. Aiming for better general understanding of this problem, in this paper we focus on the case of uniform job release dates on one machine for which the state of the art is a 4-approximation algorithm. This is true even for a special case that is equivalent to the covering version of the well-studied and prominent unsplittable flow on a path problem, which is interesting in its own right. For that covering problem, we present a quasi-polynomial time $(1+\epsilon)$-approximation algorithm that yields an $(e+\epsilon)$-approximation for the above scheduling problem. Moreover, for the latter we devise the best possible resource augmentation result regarding speed: a polynomial time algorithm which computes a solution with \emph{optimal }cost at $1+\epsilon$ speedup. Finally, we present an elegant QPTAS for the special case where the cost functions of the jobs fall into at most $\log n$ many classes. This algorithm allows the jobs even to have up to $\log n$ many distinct release dates.Comment: 2 pages, 1 figur

Geometry of Scheduling on Multiple Machines

We consider the following general scheduling problem: there are m identical machines and n jobs all released at time 0. Each job j has a processing time pj, and an arbitrary non-decreasing function fj that specifies the cost incurred for j, for each possible completion time. The goal is to find a preemptive migratory schedule of minimum cost. This models several natural objectives such as weighted norm of completion time, weighted tardiness and much more. We give the first O(1) approximation algorithm for this problem, improving upon the O(loglognP) bound due to Moseley (2019). To do this, we first view the job-cover inequalities of Moseley geometrically, to reduce the problem to that of covering demands on a line by rectangular and triangular capacity profiles. Due to the non-uniform capacities of triangles, directly using quasi-uniform sampling loses a O(loglogP) factor, so a second idea is to adapt it to our setting to only lose an O(1) factor. Our ideas for covering points with non-uniform capacity profiles (which have not been studied before) may be of independent int

Non-uniform Geometric Set Cover and Scheduling on Multiple Machines

We consider the following general scheduling problem studied recently by Moseley. There are $n$ jobs, all released at time $0$, where job $j$ has size $p_j$ and an associated arbitrary non-decreasing cost function $f_j$ of its completion time. The goal is to find a schedule on $m$ machines with minimum total cost. We give an $O(1)$ approximation for the problem, improving upon the previous $O(\log \log nP)$ bound ($P$ is the maximum to minimum size ratio), and resolving the open question of Moseley. We first note that the scheduling problem can be reduced to a clean geometric set cover problem where points on a line with arbitrary demands, must be covered by a minimum cost collection of given intervals with non-uniform capacity profiles. Unfortunately, current techniques for such problems based on knapsack cover inequalities and low union complexity, completely lose the geometric structure in the non-uniform capacity profiles and incur at least an $\Omega(\log\log P)$ loss. To this end, we consider general covering problems with non-uniform capacities, and give a new method to handle capacities in a way that completely preserves their geometric structure. This allows us to use sophisticated geometric ideas in a black-box way to avoid the $\Omega(\log \log P)$ loss in previous approaches. In addition to the scheduling problem above, we use this approach to obtain $O(1)$ or inverse Ackermann type bounds for several basic capacitated covering problems

Small Extended Formulation for Knapsack Cover Inequalities from Monotone Circuits

Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield linear programming (LP) relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. In this paper we address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. For the min-knapsack cover problem, our main result can be stated formally as follows: for any $\varepsilon >0$, there is a $(1/\varepsilon)^{O(1)}n^{O(\log n)}$-size LP relaxation with an integrality gap of at most $2+\varepsilon$, where $n$ is the number of items. Prior to this work, there was no known relaxation of subexponential size with a constant upper bound on the integrality gap. Our construction is inspired by a connection between extended formulations and monotone circuit complexity via Karchmer-Wigderson games. In particular, our LP is based on $O(\log^2 n)$-depth monotone circuits with fan-in~$2$ for evaluating weighted threshold functions with $n$ inputs, as constructed by Beimel and Weinreb. We believe that a further understanding of this connection may lead to more positive results complementing the numerous lower bounds recently proved for extended formulations.Comment: 21 page