Multistage online maxmin allocation of indivisible entities

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