Restricted Max-Min Allocation: Integrality Gap and Approximation Algorithm

Given a set of players P, indivisible resources R, and non-negative values $$\{v_{pr}\}_{p\in r\in R}$$ , an allocation is partition R into disjoint subsets $$\{C_p\}_{p \in P}$$ so that each player p assigned the in $$C_p$$ . The max-min fair problem to determine maximizes $$\min _p \sum _{r\in C_p}v_{pr}$$ In restricted case this problem, resource r has intrinsic value $$v_r$$ $$v_{pr} = v_r$$ for every who desires 0$$ does not. We study paper. For configuration LP played important role estimating approximating optimal solution. Our first result upper bound $$3\frac{21}{26}$$ on integrality gap, which currently best. It obtained by tighter analysis local search Asadpour et al. [TALG’12]. remains unknown whether runs polynomial time or second polynomial-time algorithm achieves approximation ratio $$4 + \delta $$ any constant $$\delta (0,1)$$ can be seen as generalization aforementioned search.