A 3/2-Approximation Algorithm for the Graph Balancing Problem with Two Weights

In the pursuit of finding subclasses of the makespan minimization problem on unrelated parallel machines that have approximation algorithms with approximation ratio better than 2, the graph balancing problem has been of current interest. In the graph balancing problem each job can be non-preemptively scheduled on one of at most two machines with the same processing time on either machine. Recently, Ebenlendr, Krčál, and Sgall (Algorithmica 2014, 68, 62–80.) presented a 7 / 4 -approximation algorithm for the graph balancing problem. Let r , s ∈ Z + . In this paper we consider the graph balancing problem with two weights, where a job either takes r time units or s time units. We present a 3 / 2 -approximation algorithm for this problem. This is an improvement over the previously best-known approximation algorithm for the problem with approximation ratio 1.652 and it matches the best known inapproximability bound for it

A 3/2-Approximation Algorithm for the Graph Balancing Problem with Two Weights

In the pursuit of finding subclasses of the makespan minimization problem on unrelated parallel machines that have approximation algorithms with approximation ratio better than 2, the graph balancing problem has been of current interest. In the graph balancing problem each job can be non-preemptively scheduled on one of at most two machines with the same processing time on either machine. Recently, Ebenlendr, Krčál, and Sgall (Algorithmica 2014, 68, 62–80.) presented a 7 / 4 -approximation algorithm for the graph balancing problem. Let r , s ∈ Z + . In this paper we consider the graph balancing problem with two weights, where a job either takes r time units or s time units. We present a 3 / 2 -approximation algorithm for this problem. This is an improvement over the previously best-known approximation algorithm for the problem with approximation ratio 1.652 and it matches the best known inapproximability bound for it

Santa Claus, Machine Scheduling and Bipartite Hypergraphs

Here we discuss two related discrete optimization problems, a prominent problem in scheduling theory, makespan minimization on unrelated parallel machines, and the other a fair allocation problem, the Santa Claus problem. In each case the objective is to make the least well off participant as well off as possible, and in each case we have complexity results that bound how close we may estimate optimal values of worst-case instances in polytime. We explore some of the techniques that have been used in obtaining approximation algorithms or optimal value guarantees for these problems, as well as those involved in getting hardness results, emphasizing the relationships between the problems. A framework for decisional variants of approximation and optimal value estimation for optimization problems is introduced to clarify the discussion. Also discussed are bipartite hypergraphs, which correspond naturally to these problems, including a discussion of Haxell's Theorem for bipartite hypergraphs. Conditions for edge covering in bipartite hypergraphs are introduced and their implications investigated. The conditions are motivated by analogy to Haxell's Theorem and from generalizing conditions that arose from bipartite hypergraphs associated with machine scheduling