Max-min Allocation Problem Research Articles

Multistage online maxmin allocation of indivisible entities

We consider an online allocation problem that involves a set P of n players and a set E of m indivisible entities over discrete time steps 1,2,…,τ. At each time step t∈[1,τ], for every entity e∈E, there is a restriction list Lt(e) that prescribes the subset of players to whom e can be assigned and a non-negative value vt(e,p) of e to every player p∈P. The sets P and E are fixed beforehand. The sets Lt(⋅) and values vt(⋅,⋅) are given in an online fashion. An allocation is a distribution of E among P, and we are interested in the minimum total value of the entities received by a player according to the allocation. In the static case, it is NP-hard to find an optimal allocation the maximizes this minimum value. On the other hand, ρ-approximation algorithms have been developed for certain values of ρ∈(0,1]. We propose a w-lookahead algorithm for the multistage online maxmin allocation problem for any fixed w⩾1 in which the restriction lists and values of entities may change between time steps, and there is a fixed stability reward for an entity to be assigned to the same player from one time step to the next. The objective is to maximize the sum of the minimum values and stability rewards over the time steps 1,2,…,τ. Our algorithm achieves a competitive ratio of (1−c)ρ, where c is the positive root of the equation wc2=ρ(w+1)(1−c). When w=1, it is greater than ρ4ρ+2+ρ10, which improves upon the previous ratio of ρ4ρ+2−21−τ(2ρ+1) obtained for the case of 1-lookahead.

POLYNOMIAL APPROXIMATION SCHEMES FOR THE MAX-MIN ALLOCATION PROBLEM UNDER A GRADE OF SERVICE PROVISION

The max-min allocation problem under a grade of service provision is defined in the following model: given a set [Formula: see text] of m parallel machines and a set [Formula: see text] of n jobs, where machines and jobs are all entitled to different levels of grade of service (GoS), each job [Formula: see text] has its processing time pj and it can only be allocated to a machine Mi whose GoS level is no more than the GoS level the job Jj has. The goal is to allocate all jobs to m machines to maximize the minimum machine load among these m machines, where the machine load of Mi is the sum of the processing time of jobs executed on Mi. The best known approximation algorithm [4] to solve this problem produces an allocation in which the minimum machine completion time is at least Ω ( log log log m/ log log m) of the optimal value. In this paper, we respectively present four approximation schemes to solve this problem and its two special versions: (1) a polynomial time approximation scheme (PTAS) with a running time [Formula: see text] for the general version, where ϵ > 0; (2) a PTAS and a fully polynomial time approximation scheme (FPTAS) with the running time O(n) for the version where the number m of machines is fixed; (3) a PTAS with the running time O(n) for the version where the number of GoS levels is bounded by k.

A Max-Min Allocation Problem: Its Solutions and Applications

This paper describes a max-min allocation problem that has M constraints and N nonnegative integer variables. By exploring the structure of the optimal solution to this max-min allocation problem, we develop a nonsimplex-based algorithm that finds an optimal solution within O(MN2) operations. In addition, we show that a space allocation problem for a flexible component insertion machine can be formulated as a special case of this max-min allocation problem with one constraint. Furthermore, we present some manufacturing and production problems that can he formulated as max-min allocation problems, and hence, can be solved efficiently.