We consider the strategyproof facility location problem on a circle. We focus on the case of 5 agents, and find a tight bound for the PCD strategyproof mechanism, which selects the reported location of an agent in proportion to the length of the arc in front of it. We methodically "reduce" the size of the instance space and then use standard optimization techniques to find and prove the bound is tight. Moreover we hypothesize the approximation ratio of PCD for general odd $n$.

We receive a prediction for each agent's location, and these predictions are crucially allowed to be only

Facility location is fundamental in operations research, mechanism design, and algorithmic game theory, with applications ranging from urban infrastructure planning to distributed systems. Recent research in this area has focused on augmenting classic strategyproof mechanisms with predictions to achieve an improved performance guarantee against the uncertainty under the strategic environment. Previous work has been devoted to address the trade-off obstacle of balancing the consistency (near-optimality under accurate predictions) and robustness (bounded inefficiency under poor predictions) primarily in the unweighted setting, assuming that all agents have the same importance. However, this assumption may not be true in some practical scenarios, leading to research of weighted facility location problems. The major contribution of the current work is to provide a prediction augmented algorithmic framework for balancing the consistency and robustness over strategic agents with non-uniform weights. In particular, through a reduction technique that identifies a subset of representative instances and maps the other given locations to the representative ones, we prove that there exists a strategyproof mechanism achieving a bounded consistency guarantee of $\frac{\sqrt{(1+c)^2W^2_{\min}+(1-c)^2W^2_{\max}}}{(1+c)W_{\min}}$ and a bounded robustness guarantee of $\frac{\sqrt{(1-c)^2W^2_{\min}+(1+c)^2W^2_{\max}}}{(1-c)W_{\min}}$ in weighted settings, where $c$ can be viewed as a parameter to make a trade-off between the consistency and robustness and $W_{\min}$ and $W_{\max}$ denote the minimum and maximum agents' weight. We also prove that there is no strategyproof deterministic mechanism that reach $1$-consistency and $O\left( n \cdot \frac{W_{\max}}{W_{\min}} \right)$-robustness in weighted FLP, even with fully predictions of all agents.

In the one-dimensional facility location problem, agents are located on the real line, and a planner's goal is to build one or more facilities on the line to serve the agents. The cost of an agent is her distance to the nearest facility. The problem asks to locate facilities to minimize the total cost of all agents (the utilitarian objective) or the maximum cost among all agents (the egalitarian objective). It is well-known that both optimization problems can be solved in polynomial time [19]. Over the past decade [1, 3, 7, 21, 23, 27], these problems have undergone intensive study from the perspectives of mechanism design and game theory. A key conversion in the models is that now each agent becomes strategic and may misreport her position to decrease her cost. These new problems are called the facility location games, which require to design mechanisms that elicit the true positions of agents and output facility locations to (approximately) minimize the total or maximum cost. Moulin [22] provides a complete characterization of strategyproof mechanisms for the facility location game on line where agents have single-peaked preferences, with no consideration for the optimization of the objective function.

Algorithms with predictions have attracted much attention in the last years across various domains, including variants of facility location, as a way to surpass traditional worst-case analyses. We study the $k$-facility location mechanism design problem, where the $n$ agents are strategic and might misreport their location. Unlike previous models, where predictions are for the $k$ optimal facility locations, we receive $n$ predictions for the locations of each of the agents. However, these predictions are only "mostly" and "approximately" correct (or MAC for short) -- i.e., some $\delta$-fraction of the predicted locations are allowed to be arbitrarily incorrect, and the remainder of the predictions are allowed to be correct up to an $\varepsilon$-error. We make no assumption on the independence of the errors. Can such predictions allow us to beat the current best bounds for strategyproof facility location? We show that the $1$-median (geometric median) of a set of points is naturally robust under corruptions, which leads to an algorithm for single-facility location with MAC predictions. We extend the robustness result to a "balanced" variant of the $k$ facilities case. Without balancedness, we show that robustness completely breaks down, even for the setting of $k=2$ facilities on a line. For this "unbalanced" setting, we devise a truthful random mechanism that outperforms the best known result of Lu et al. [2010], which does not use predictions. En route, we introduce the problem of "second" facility location (when the first facility's location is already fixed). Our findings on the robustness of the $1$-median and more generally $k$-medians may be of independent interest, as quantitative versions of classic breakdown-point results in robust statistics.

Multi-Agent Path Finding (MAPF) involves determining paths for multiple agents to travel simultaneously through a shared area toward particular goal locations. This problem is computationally complex, especially when dealing with large numbers of agents, as is common in realistic applications like autonomous vehicle coordination. Finding an optimal solution is often computationally infeasible, making the use of approximate algorithms essential. Adding to the complexity, agents might act in a self-interested and strategic way, possibly misrepresenting their goals to the MAPF algorithm if it benefits them. Although the field of mechanism design offers tools to align incentives, using these tools without careful consideration can fail when only having access to approximately optimal outcomes. Since approximations are crucial for scalable MAPF algorithms, this poses a significant challenge. In this work, we introduce the problem of scalable mechanism design for MAPF and propose three strategyproof mechanisms, two of which even use approximate MAPF algorithms. We test our mechanisms on realistic MAPF domains with problem sizes ranging from dozens to hundreds of agents. Our findings indicate that they improve welfare beyond a simple baseline.