Approximation Algorithms for Generalized Path Scheduling

Abstract

Scheduling problems where the machines can be represented as the edges of a network and each job needs to be processed by a sequence of machines that form a path in this network have been the subject of many research articles (e.g. flow shop is the special case where the network as well as the sequence of machines for each job is a simple path). In this paper we consider one such problem, called Generalized Path Scheduling (GPS) problem, which can be defined as follows. Given a set of non-preemptive jobs J and identical machines M ( |J| = n and |M| = m ). The machines are ordered on a path. Each job j = {P_j = {l_j, r_j}, p_j} is defined by its processing time p_j and a sub-path P_j from machine with index l_j to r_j (l_j, r_j ? M, and l_j ? r_j) specifying the order of machines it must go through. We assume each machine has a queue of infinite size where jobs can sit in the queue to resolve conflicts. Two objective functions, makespan and total completion time, are considered. Machines can be identical or unrelated. In the latter case, this problem generalizes the classical Flow shop problem (in which all jobs have to go through all machines from 1 to m in that order). Generalized Path Scheduling has been studied (e.g. see [Ronald Koch et al., 2009; Zachary Friggstad et al., 2019]). In this paper, we present several improved approximation algorithms for both objectives. For the case of number of machines being sub-logarithmic in the number of jobs we present a PTAS for both makespan and total completion time. The PTAS holds even on unrelated machines setting and therefore, generalizes the result of Hall [Leslie A. Hall, 1998] for the classic problem of Flow shop. For the case of identical machines, we present an O((log m)/(log log m))-approximation algorithms for both objectives, which improve the previous best result of [Zachary Friggstad et al., 2019]. We also show that the GPS problem is NP-complete for both makespan and total completion time objectives