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.

We study various models for the one-dimensional multi-stage facility location problems with transient agents, where a transient agent arrives in some stage and stays for a number of consecutive stages. In the problems, we need to serve each agent in one of their stages by determining the location of the facility at each stage. In the first model, we assume there is no cost for moving the facility across the stages. We focus on optimal algorithms to minimize both the social cost objective, defined as the total distance of all agents to the facility over all stages, and the maximum cost objective, defined as the max distance of any agent to the facility over all stages. For each objective, we give a slice-wise polynomial (XP) algorithm (i.e., solvable in m^f(k) for some fixed parameter k and computable function f, where m is the input size) and show that there is a polynomial-time algorithm when a natural first-come-first-serve (FCFS) order of agent serving is enforced. We then consider the mechanism design problem, where the agents' locations and arrival stages are private, and design a group strategy-proof mechanism that achieves good approximation ratios for both objectives and settings with and without FCFS ordering. In the second model, we consider the facility's moving cost between adjacent stages under the social cost objective, which accounts for the total moving distance of the facility. Correspondingly, we design XP (and polynomial time) algorithms and a group strategy-proof mechanism for settings with or without the FCFS ordering.

In this paper, we propose a fractional preference model for the facility location game with two facilities that serve the similar purpose on a line where each agent has his location information as well as fractional preference to indicate how well they prefer the facilities. The preference for each facility is in the range of [0, L] such that the sum of the preference for all facilities is equal to 1. The utility is measured by subtracting the sum of the cost of both facilities from the total length L where the cost of facilities is defined as the multiplication of the fractional preference and the distance between the agent and the facilities. We first show that the lower bound for the objective of minimizing total cost is at least Ω(n^1/3). Hence, we use the utility function to analyze the agents' satification. Our objective is to place two facilities on [0, L] to maximize the social utility or the minimum utility. For each objective function, we propose deterministic strategy-proof mechanisms. For the objective of maximizing the social utility, we present an optimal deterministic strategy-proof mechanism in the case where agents can only misreport their locations. In the case where agents can only misreport their preferences, we present a 2-approximation deterministic strategy-proof mechanism. Finally, we present a 4-approximation deterministic strategy-proof mechanism and a randomized strategy-proof mechanism with an approximation ratio of 2 where agents can misreport both the preference and location information. Moreover, we also give a lower-bound of 1.06. For the objective of maximizing the minimum utility, we give a lower-bound of 1.5 and present a 2-approximation deterministic strategy-proof mechanism where agents can misreport both the preference and location.

In this paper, we propose the optional preference model for the facility location game with two heterogeneous facilities on a line. Agents in this new model are allowed to have optional preference, which gives more flexibility for agents to report. Aiming at minimizing maximum cost or sum cost of agents, we propose different deterministic strategy-proof mechanisms without monetary transfers. Depending on which facility the agent with optional preference cares for, we consider two variants of the optional preference model: Min (caring for the closer one) and Max (caring for the further one). For the Min variant, we propose a 2-approximation mechanism for the maximum cost objective, as well as a lower bound of 4/3, and a (n/2+1)-approximation mechanism for the sum cost objective, as well as a lower bound of 2. For Max variant, we propose an optimal mechanism for the maximum cost objective and a 2-approximation mechanism for the sum cost objective.