Close the approximation gaps for the min-variant cost

Determine whether the approximation gaps for deterministic strategyproof mechanisms with min-variant individual costs can be closed under both the social cost objective and the maximum cost objective: specifically, improve the social-cost ratio from 3 toward the lower bound of 2 and the maximum-cost ratio from 3 toward the lower bound of 5/3.

Background

The paper studies deterministic strategyproof mechanisms for locating a bridge between two parallel regions containing prelocated facilities, when agents have private locations and private optional preferences over the facilities. Under the min-variant individual cost, an agent pays the minimum distance to the facilities in which she is interested.

For social-cost minimization, the paper provides a strategyproof 3-approximation mechanism and proves a lower bound of 2. For maximum-cost minimization, it provides a strategyproof 3-approximation mechanism and proves a lower bound of 5/3. Thus, the optimal deterministic strategyproof approximation ratios remain unresolved for both objectives.