We study the k-facility location games with optional preferences on the line. In the games, each strategic agent has a public location preference on the k facility locations and a private optional preference on the preferred/acceptable set of facilities out of the k facilities. Our goal is to design strategyproof mechanisms to elicit agents’ optional preferences and locate k facilities to minimize the social or maximum cost of agents based on their facility preferences and public agent locations. We consider two variants of the facility location games with optional preferences: the Min variant and the Max variant where the agent’s cost is defined as their distance to the closest acceptable facility and the farthest acceptable facility, respectively. For the Min variant, we present two deterministic strategyproof mechanisms to minimize the maximum cost and social cost with k ≥ 3 facilities, achieving approximation ratios of 3 and 2n+1 respectively. We complement the results by establishing lower bounds of 3/2 and n/4 for the approximation ratios achievable by any deterministic strategyproof mechanisms for the maximum cost and social cost, respectively. We then improve our results in a special setting of the Min variant where there are exactly three facilities and present two deterministic strategyproof mechanisms to minimize the maximum cost and social cost. For the Max variant, we present an optimal deterministic strategyproof mechanism for the maximum cost and a k-approximation deterministic strategyproof mechanism for the social cost.

We consider the k-facility location problems with capacity constraints in bounded location space from the mechanism design perspective. In this problem, we seek to locate k capacity constrained facilities in a bounded interval (i.e., B=[bl,br]) to serve agents, who have preferences on the ideal locations of the facilities in the interval. Our goal is to design strategyproof mechanisms to elicit agents’ true ideal locations and locate facilities that minimize the social cost and maximum cost, which are defined to be the sum and the maximum of the agents’ costs (i.e., agents’ distances to their facilities), respectively. For the equal capacity setting without spare capacity (i.e., all the agents can be served exactly), we provide a deterministic strategyproof mechanism. For any bounded interval (i.e., bl, br∈R), our mechanism has approximation ratios of n-1 for the social cost and 4 for the maximum cost with k≥3 facilities and n≥3 agents. We also establish lower bounds of n/2 for the social cost by a common class of deterministic mechanisms that order agents from left to right, and 2 for the maximum cost by any deterministic mechanism. Our mechanism also achieves tight bounds for both costs with k<3 facilities. We then consider the equal capacity setting with spare capacity and the arbitrary capacity setting without spare capacity. For these two settings and any bounded interval, we provide randomized strategyproof mechanisms with approximation ratios of n/2 for the social cost and 2 for the maximum cost with any number of facilities. We complement this result by establishing lower bounds of 5/3 for the social cost and 3/2 for the maximum cost.