Scheduling UET-tasks on a star network: complexity and approximation

In this article we investigate complexity and approximation on a processor network where the communication delay depends on the distance between the processors performing tasks. We then prove that there is no polynomial-time heuristic with a performance guarantee smaller than ${\frac{6}{5}}$ (respectively ${\frac{14}{13}}$ ) for minimization of the makespan (respectively the total job completion time) on a processor network represented by a star. Moreover, we design an efficient polynomial-time approximation algorithm with a worst-case ratio of four. We also prove that a polynomial-time algorithm exists for a schedule with a length of at most four. Lastly, we generalize all previous results on complexity and approximation.     

Scheduling subject to resource constraints: classification and complexity

In deterministic sequencing and scheduling problems, jobs are to be processed on machines of limited capacity. We consider an extension of this class of problems, in which the jobs require the use of additional scarce resources during their execution. A classification scheme for resource constraints is proposed and the computational complexity of the extended problem class is investigated in terms of this classification. Models involving parallel machines, unit-time jobs and the maximum completion time criterion are studied in detail; other models are briefly discussed.     

The routing open-shop problem on a network: Complexity and approximation

In the routing open-shop problem, jobs are located at nodes of an undirected transportation network, and the machines travel on the network to execute jobs in the open-shop environment. The machines are initially located at the same node (depot) and must return to the depot after completing all the jobs. It is required to find a nonpreemptive schedule that minimizes the makespan. We prove that the problem is NP-hard even on a two-node network with two machines, and even on a two-node network with two jobs and m machines. We develop polynomial-time approximation heuristics and obtain bounds on their approximation performance.     

Scheduling transmissions in a network

Consider k stations wishing to transmit messages over a network of channels to a common receiver. The capacity of a channel is the maximum amount of data which can be transmitted in a time unit. In addition to the transmission stations, the network contains switching nodes. Given that the jth station has σ_j messages (j = 1,…, k) to transmit, it is desired to find a schedule with minimum completion time T. The amount of data sent over a channel may vary in time. A schedule is stationary if the amount of data sent in a time unit is constant throughout the schedule. It is first shown that for every schedule there exists a stationary schedule with the same completion time. Thus, the search for an optimum schedule is confined to stationary schedules. The problem of finding an optimum stationary schedule is formulated as a multisource single-sink network flow problem, in which the net amount of outgoing flow from each source (transmission station) within one time unit is $\text{σ}{}_{\mathrm{j}}\text{T}$. An O(k|E||V|²) time algorithm to find the minimum T similar to Dinic's flow algorithm is suggested. Using Sleator and Tarjan's techniques an O(k²|E||V|log|V|) algorithm is derived. The running time of both algorithms is independent of the σ_j's and the capacities.     

Erratum: Scheduling for a Known but Irregular Batch-Wise Demand

Journal of the Operational Research Society -     

Polynomial time approximation schemes and parameterized complexity

In this paper, we study the relationship between the approximability and the parameterized complexity of NP optimization problems. We introduce a notion of polynomial fixed-parameter tractability and prove that, under a very general constraint, an NP optimization problem has a fully polynomial time approximation scheme if and only if the problem is polynomial fixed-parameter tractable. By enforcing a constraint of planarity on the W-hierarchy studied in parameterized complexity theory, we obtain a class of NP optimization problems, the planar W-hierarchy, and prove that all problems in this class have efficient polynomial time approximation schemes (EPTAS). The planar W-hierarchy seems to contain most of the known EPTAS problems, and is significantly different from the class introduced by Khanna and Motwani in their efforts in characterizing optimization problems with polynomial time approximation schemes.     

Erratum: Scheduling jobs with position-dependent processing times

Bachman and Janiak provided a sketch of the proof that the problem 1∣r _i ,p _i (v)=a _i /v∣C_max is NP-hard in the strong sense. However, they did not show how to avoid using harmonic numbers whose encoding is not pseudo-polynomial, which makes the proof incomplete. In this corrigendum, we provide a new complete proof.     

A dynamic edge covering and scheduling problem: complexity results and approximation algorithms

We consider a new dynamic edge covering and scheduling problem that focuses on assigning resources to nodes in a network to minimize the amount of time required to process all edges in it. Resources need to be co-located at the endpoints of an edge for it to be processed and, therefore, this problem contains both edge covering and scheduling decisions. These new problems have motivating applications in traffic systems and military intelligence operations. We provide complexity results for the dynamic edge covering and scheduling problem over different types of networks. We then show that existing approximation algorithms for parallel machine scheduling problems can be leveraged to provide approximation algorithms for this new class of problems over certain types of networks.     

Fluid approximation of a controlled multiclass tandem network

A two-station, four-class queueing network with dynamic scheduling of servers is analyzed. It is shown that the corresponding Markov decision problem converges under fluid scaling to a fluid optimal control model. The structure of the optimal policy for the fluid network, and of an asymptotically optimal policy for the queueing network are derived in an explicit form. They concur with the tandem μ-rule, if this policy gives priority to the same flow of customers in both stations. In general, they are monotone with a linear switching surface. This revised version was published online in June 2006 with corrections to the Cover Date.     

Acyclic graph coloring and the complexity of the star chromatic number

Star chromatic number, introduced by A. Vince, is a natural generalization of chromatic number. We consider the question, “When is χ* < χ?” We show that χ* < χ if and only if a particular digraph is acyclic and that the decisioin problem associated with this question is probably not in NP though it is both NP-hard and NP-easy. © 1993 John Wiley & Sons, Inc.     

Partition into cliques for cubic graphs: Planar case, complexity and approximation

Given a graph G=(V,E) and a positive integer k, the partition into cliques (pic) decision problem consists of deciding whether there exists a partition of V into k disjoint subsets V₁,V₂,…,V_k such that the subgraph induced by each part V_i is a complete subgraph (clique) of G. In this paper, we establish both the NP-completeness of pic for planar cubic graphs and the Max SNP-hardness of pic for cubic graphs. We present a deterministic polynomial time -approximation algorithm for finding clique partitions in maximum degree three graphs.     

A note on the complexity of an algorithm for chebyshev approximation

Some remarks are given concerning the complexity of an exchange algorithm for Chebyshev Approximation.Dedicated to Germund Dahlquist on the occasion of his 60th birthday.Supported in part by NIH Grant RR01243 at the University of Washington. — The paper was revised at the Naval Postgraduate School, Monterey, California.