Polynomial-time approximation schemes for scheduling problems with time lags

We identify two classes of machine scheduling problems with time lags that possess Polynomial-Time Approximation Schemes (PTASs). These classes together, one for minimizing makespan and one for minimizing total completion time, include many well-studied time lag scheduling problems. The running times of these approximation schemes are polynomial in the number of jobs, but exponential in the number of machines and the ratio between the largest time lag and the smallest positive operation time. These classes constitute the first PTAS results for scheduling problems with time lags

Parallel Machine Scheduling with Nested Processing Set Restrictions and Job Delivery Times

The problem of scheduling jobs with delivery times on parallel machines is studied, where each job can only be processed on a specific subset of the machines called its processing set. Two distinct processing sets are either nested or disjoint; that is, they do not partially overlap. All jobs are available for processing at time 0. The goal is to minimize the time by which all jobs are delivered, which is equivalent to minimizing the maximum lateness from the optimization viewpoint. A list scheduling approach is analyzed and its approximation ratio of 2 is established. In addition, a polynomial time approximation scheme is derived

The Open Shop Scheduling Problem

We discuss the computational complexity, the approximability, the algorithmics and the combinatorics of the open shop scheduling problem. We summarize the most important results from the literature and explain their main ideas, we sketch the most beautiful proofs, and we also list a number of open problems

Scheduling parallel machines with inclusive processing set restrictions and job release times

2009-2010 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe

Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints

We consider the ILP Feasibility problem: given an integer linear program $\{Ax = b, x\geq 0\}$, where $A$ is an integer matrix with $k$ rows and $\ell$ columns and $b$ is a vector of $k$ integers, we ask whether there exists $x\in\mathbb{N}^\ell$ that satisfies $Ax = b$. Our goal is to study the complexity of ILP Feasibility when both $k$, the number of constraints (rows of $A$), and $\|A\|_\infty$, the largest absolute value in $A$, are small. Papadimitriou [J. ACM, 1981] was the first to give a fixed-parameter algorithm for ILP Feasibility in this setting, with running time $\left((\|A\mid b\|_\infty) \cdot k\right)^{O(k^2)}$. This was very recently improved by Eisenbrand and Weismantel [SODA 2018], who used the Steinitz lemma to design an algorithm with running time $(k\|A\|_\infty)^{O(k)}\cdot \|b\|_\infty^2$, and subsequently by Jansen and Rohwedder [2018] to $O(k\|A\|_\infty)^{k}\cdot \log \|b\|_\infty$. We prove that for $\{0,1\}$-matrices $A$, the dependency on $k$ is probably optimal: an algorithm with running time $2^{o(k\log k)}\cdot (\ell+\|b\|_\infty)^{o(k)}$ would contradict ETH. This improves previous non-tight lower bounds of Fomin et al. [ESA 2018]. We then consider ILPs with many constraints, but structured in a shallow way. Precisely, we consider the dual treedepth of the matrix $A$, which is the treedepth of the graph over the rows of $A$, with two rows adjacent if in some column they both contain a non-zero entry. It was recently shown by Kouteck\'{y} et al. [ICALP 2018] that ILP Feasibility can be solved in time $\|A\|_\infty^{2^{O(td(A))}}\cdot (k+\ell+\log \|b\|_\infty)^{O(1)}$. We present a streamlined proof of this fact and prove optimality: even assuming that all entries of $A$ and $b$ are in $\{-1,0,1\}$, the existence of an algorithm with running time $2^{2^{o(td(A))}}\cdot (k+\ell)^{O(1)}$ would contradict ETH.Comment: Added Corollary 2, extended Conclusion

Fast approximation schemes for Boolean programming and scheduling problems related to positive convex Half-Product

We address a version of the Half-Product Problem and its restricted variant with a linear knapsack constraint. For these minimization problems of Boolean programming, we focus on the development of fully polynomial-time approximation schemes with running times that depend quadratically on the number of variables. Applications to various single machine scheduling problems are reported: minimizing the total weighted flow time with controllable processing times, minimizing the makespan with controllable release dates, minimizing the total weighted flow time for two models of scheduling with rejection

Empowering the Configuration-IP - New PTAS Results for Scheduling with Setups Times

Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems, where a set of items has to be placed in multiple target locations. Herein a configuration describes a possible placement on one of the target locations, and the IP is used to chose suitable configurations covering the items. We give an augmented IP formulation, which we call the module configuration IP. It can be described within the framework of n-fold integer programming and therefore be solved efficiently. As an application, we consider scheduling problems with setup times, in which a set of jobs has to be scheduled on a set of identical machines, with the objective of minimizing the makespan. For instance, we investigate the case that jobs can be split and scheduled on multiple machines. However, before a part of a job can be processed an uninterrupted setup depending on the job has to be paid. For both of the variants that jobs can be executed in parallel or not, we obtain an efficient polynomial time approximation scheme (EPTAS) of running time f(1/epsilon) x poly(|I|) with a single exponential term in f for the first and a double exponential one for the second case. Previously, only constant factor approximations of 5/3 and 4/3 + epsilon respectively were known. Furthermore, we present an EPTAS for a problem where classes of (non-splittable) jobs are given, and a setup has to be paid for each class of jobs being executed on one machine

Variable Parameter Analysis for Scheduling One Machine

In contrast to the fixed parameter analysis (FPA), in the variable parameter analysis (VPA) the value of the target problem parameter is not fixed, it rather depends on the structure of a given problem instance and tends to have a favorable asymptotic behavior when the size of the input increases. While applying the VPA to an intractable optimization problem with $n$ objects, the exponential-time dependence in enumeration of the feasible solution set is attributed solely to the variable parameter $\nu$, $\nu<<n$. As opposed to the FPA, the VPA does not imply any restriction on some problem parameters, it rather takes an advantage of a favorable nature of the problem, which permits to reduce the cost of enumeration of the solution space. Our main technical contribution is a variable parameter algorithm for a strongly $\mathsf{NP}$-hard single-machine scheduling problem to minimize maximum job lateness. The target variable parameter $\nu$ is the number of jobs with some specific characteristics, the ``emerging'' ones. The solution process is separated in two phases. At phase 1 a partial solution including $n-\nu$ non-emerging jobs is constructed in a low degree polynomial time. At phase 2 less than $\nu!$ permutations of the $\nu$ emerging jobs are considered. Each of them are incorporated into the partial schedule of phase 1. Doe to the results of an earlier conducted experimental study, $\nu/n$ varied from $1/4$ for small problem instances to $1/10$ for the largest tested problem instances, so that that the ratio becomes closer to 0 for large $n$s.Comment: arXiv admin note: substantial text overlap with arXiv:2103.0990