Bi-Criteria Approximation Algorithms for Load Balancing on Unrelated Machines with Costs

We study a generalized version of the load balancing problem on unrelated machines with cost constraints: Given a set of m machines (of certain types) and a set of n jobs, each job j processed on machine i requires p_{i,j} time units and incurs a cost c_{i,j}, and the goal is to find a schedule of jobs to machines, which is defined as an ordered partition of n jobs into m disjoint subsets, in such a way that some objective function of the vector of the completion times of the machines is optimized, subject to the constraint that the total costs by the schedule must be within a given budget B. Motivated by recent results from the literature, our focus is on the case when the number of machine types is a fixed constant and we develop a bi-criteria approximation scheme for the studied problem. Our result generalizes several known results for certain special cases, such as the case with identical machines, or the case with a constant number of machines with cost constraints. Building on the elegant technique recently proposed by Jansen and Maack [K. Jansen and M. Maack, 2019], we construct a more general approach that can be used to derive approximation schemes to a wider class of load balancing problems with constraints

Approximation Schemes for Machine Scheduling

In the classical problem of makespan minimization on identical parallel machines, or machine scheduling for short, a set of jobs has to be assigned to a set of machines. The jobs have a processing time and the goal is to minimize the latest finishing time of the jobs. Machine scheduling is well known to be NP-hard and thus there is no polynomial time algorithm for this problem that is guaranteed to find an optimal solution unless P=NP. There is, however, a polynomial time approximation scheme (PTAS) for machine scheduling, that is, a family of approximation algorithms with ratios arbitrarily close to one. Whether a problem admits an approximation scheme or not is a fundamental question in approximation theory. In the present work, we consider this question for several variants of machine scheduling. We study the problem where the machines are partitioned into a constant number of types and the processing time of the jobs is also dependent on the machine type. We present so called efficient PTAS (EPTAS) results for this problem and variants thereof. We show that certain cases of machine scheduling with assignment restrictions do not admit a PTAS unless P=NP. Moreover, we introduce a graph framework based on the restrictions of the jobs and use it in the design of approximation schemes for other variants. We introduce an enhanced integer programming formulation for assignment problems, show that it can be efficiently solved, and use it in the EPTAS design for variants of machine scheduling with setup times. For one of the problems, we show that there is also a PTAS in the case with uniform machines, where machines have speeds influencing the processing times of the jobs. We consider cases in which each job requires a certain amount of a shared renewable resource and the processing time is depended on the amount of resource it receives or not. We present so called asymptotic fully polynomial time approximation schemes (AFPTAS) for the problems

Finding Optimal Diverse Feature Sets with Alternative Feature Selection

Feature selection is popular for obtaining small, interpretable, yet highly accurate prediction models. Conventional feature-selection methods typically yield one feature set only, which might not suffice in some scenarios. For example, users might be interested in finding alternative feature sets with similar prediction quality, offering different explanations of the data. In this article, we introduce alternative feature selection and formalize it as an optimization problem. In particular, we define alternatives via constraints and enable users to control the number and dissimilarity of alternatives. Next, we analyze the complexity of this optimization problem and show NP-hardness. Further, we discuss how to integrate conventional feature-selection methods as objectives. Finally, we evaluate alternative feature selection with 30 classification datasets. We observe that alternative feature sets may indeed have high prediction quality, and we analyze several factors influencing this outcome

Finding Optimal Diverse Feature Sets with Alternative Feature Selection

Feature selection is popular for obtaining small, interpretable, yet highly accurate prediction models. Conventional feature-selection methods typically yield one feature set only, which might not suffice in some scenarios. For example, users might be interested in finding alternative feature sets with similar prediction quality, offering different explanations of the data. In this article, we introduce alternative feature selection and formalize it as an optimization problem. In particular, we define alternatives via constraints and enable users to control the number and dissimilarity of alternatives. Next, we analyze the complexity of this optimization problem and show NP-hardness. Further, we discuss how to integrate conventional feature-selection methods as objectives. Finally, we evaluate alternative feature selection with 30 classification datasets. We observe that alternative feature sets may indeed have high prediction quality, and we analyze several factors influencing this outcome

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Computer science: the hardware software and heart of IT

1st edition, 201

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum