Additive Approximation Schemes for Load Balancing Problems

High Multiplicity Scheduling on Uniform Machines in FPT-Time

“…Let us consider a configuration, that is a vector c ∈ [O d ; ν] Z d representing an assignment of jobs to a specific machine type t ∈ [τ ]. If the load induced by c, that is p c, is larger than some threshold, we can show that (for our objectives) we can always split c into two configurations c (1) , c (2) where p c (1) takes on an a priori known value.…”

“…This requires a procedure to decide if for a fixed value α there is a schedule whose objective value is at least as good as α. We propose a general mechanism reduce the maximal number of jobs that can possibly be scheduled on each machine in such a decision procedure for so called "Load Balancing Problems"; those are problems where in the decision procedure mentioned above, the possible job assignments for a machine of type t are all assignments whose processing time are between (machine-type-specific) given lower and upper bounds [2].…”

“…We then define a fractional schedule σ based on the two areas (see Figure 4a). This schedule consists of three phases σ (1a) , σ (2) , and σ (1b) ; their naming follows the area involved. First, we try to put p max jobs of every type on every machine in area 1.…”

“…(i) σ (2) does not exceed area 2 on any machine. (ii) If there exists a job type j ∈ [d] such that σ (1b) • (j) ≥ 1, then area 2 is completely filled.…”

“…Jansen and Maack [14] gave an approximation scheme for R||{C min , C max } that runs in time O * (f (τ, ε)) (where τ is the number of different machine types) for some function f ; they also gave an approximation scheme for Q||C max that runs in time O * (g(ε)) for some function g. Bansal and Sviridenko [1] gave an O( log log m log log log m )-approximation algorithm for P |p ij ∈ {p j , ∞}|C min . For the objective C envy , there cannot be a (multiplicative) approximation algorithm as it is NP-hard to decide if there is a schedule for that assigns the same load to every machine, even for scheduling on identical machines [2].…”