Truthful interval covering

Autonomous Agents and Multi-Agent Systems https://doi.org/10.1007/s10458-024-09673-6 (2024) 38:41 Truthful interval covering Argyrios Deligkas1 · Aris Filos‑Ratsikas2 · Alexandros A. Voudouris3 Accepted: 9 August 2024 © The Author(s) 2024 Abstract We initiate the study of a novel problem in mechanism design without money, which we term Truthful Interval Covering (TIC). An instance of TIC consists of a set of agents each associated with an individual interval on a line, and the objective is to decide where to place a covering interval to minimize the total social or egalitarian cost of the agents, which is determined by the intersection of this interval with their individual ones. This fundamental problem can model situations of provisioning a public good, such as the use of power generators to prevent or mitigate load shedding in developing countries. In the strategic version of the problem, the agents wish to minimize their individual costs, and might misreport the position and/or length of their intervals to achieve that. Our goal is to design truthful mechanisms to prevent such strategic misreports and achieve good approximations to the best possible social or egalitarian cost. We consider the fundamental setting of known intervals with equal lengths and provide tight bounds on the approximation ratios achieved by truthful deterministic mechanisms. For the social cost, we also design a randomized truthful mechanism that outperforms all possible deterministic ones. Finally, we highlight a plethora of natural extensions of our model for future work, as well as some natural limitations of those settings. Keywords Mechanism design · Approximation · Interval covering 1 Introduction We introduce the Truthful Interval Covering (TIC) problem, a novel problem in the field of mechanism design without money Procaccia and Tennenholtz [16]. In this problem, there is a set N of n agents, each of whom is associated with an interval Ii on the line of real A preliminary version of this paper appears in the Proceedings of the 33rd International Joint Conference on Artificial Intelligence (IJCAI), 2024. This full version contains extended discussion of the model and the full proofs of all statements. * Alexandros A. Voudouris 1 Royal Holloway University of London, London, UK 2 University of Edinburgh, Edinburgh, UK 3 University of Essex, Colchester, UK Vol.:(0123456789) 41 Page 2 of 25 Autonomous Agents and Multi-Agent Systems (2024) 38:41 numbers. There is also a covering interval C, which should be placed somewhere on the line. The cost of agent i ∈ N is a function of the portion of Ii that C covers; in the simplest version of the problem, the cost is just the part of Ii that is not covered by C. The goal is to place the interval so as to minimize the social cost (total cost of the agents) or the max cost (maximum individual agent cost), while taking the incentives of the agents into account. Indeed, agents might misreport information about their intervals (e.g., their position or length) if that would lead to an outcome that is preferable for them. The TIC problem captures applications in which a public good is provisioned and shared among a set of participants. We provide a few indicative of many examples below. • The covering interval could represent the time interval during which a power generator can be operated, and the individual intervals capture the times during which each citizen would like to have access to electricity. The minimum-social cost solution is one that covers as much demand for electricity as possible. This is particularly relevant in developing countries where electricity might be a scarce resource, and can be used to prevent or mitigate the effects of load shedding.1 • The covering interval could correspond to the range of a public WiFi hotspot to be placed in an area with low broadband connectivity, when the agents’ intervals are the signal ranges of their devices. • The covering interval could capture the time in which to schedule a university openday or a job fair, given the preferences of the potential attendees over the different time intervals in the day. • The covering interval could be an express public transportation line connecting parts of a city or intercity network, and the agents express which parts of the route they would like this service to cover. Despite its fundamental nature, and its resemblance to other classic algorithmic problems like the interval scheduling problem and its variants Kolen et al., [12], the interval covering problem has seemingly not been studied systematically from a purely algorithmic point of view. This can likely be attributed to the fact that the optimal covering can be found in polynomial time via a rather simple algorithm (see Theorem 2.3 in Sect. 2). Once we move to a mechanism design regime however, where the incentives of the agents for misresorting come into effect, the problem becomes much more challenging. Truthful mechanisms, which eliminate those incentives, are necessarily suboptimal, and resort to approximations. Our goal is to design truthful mechanisms that achieve approximations that are as small as possible, and identify the limitations of such mechanisms via appropriate inapproximability results. 1.1 Our contribution In this paper, we introduce the Truthful Interval Covering (TIC) problem as a novel and interesting problem in mechanism design without money. Our technical contribution is as follows; see also Table 1 for an overview. 1 E.g., see https://en.wikipedia.org/wiki/South_African_energy_crisis. Autonomous Agents and Multi-Agent Systems Page 3 of 25 (2024) 38:41 41 Table 1  An overview of our bounds on the approximation ratio of truthful mechanisms for the different settings and social objectives that we consider Social cost Max cost Deterministic Randomized Deterministic Randomized Known, equal 2 Known, unequal Unknown n Unbounded [3∕2⋆ , 5∕3] 2 2⋆ Unbounded 2 Unbounded Unbounded Here, n is the number of agents, and unbounded means that there is no truthful mechanism with finite approximation ratio. The results marked with ⋆ hold for randomized truthful mechanisms that are convex combinations of statistic mechanisms • We provide upper and lower bounds on the approximation ratio of truthful mechanisms for the most fundamental version of the problem, where all of the interval lengths are known and equal, which already turns out to be quite challenging.2 We start with the social cost objective and deterministic truthful mechanisms, for which, in Sect. 3, we prove a tight bound of 2 − 2∕n on the approximation ratio. In Sect. 4, we present a simple randomized, universally truthful mechanism that achieves an approximation ratio of 5/3, thus outperforming all deterministic ones. In Sect. 5, we turn our attention to the max cost objective, for which we show a tight approximation ratio of 2 for deterministic mechanisms, and a lower bound of 2 for a natural (...truncated)