robotics fleet managers

Abstract: Consider a finite population of agents with decoupled Markov transition dynamics and empirical-density feedback, subject to the following constraints: with probability at least $1-δ_r$, at least a fraction $α_r$ of agents must reach a target region at some time $t^*$, while, at each time up to $t^*$, the unsafe population fraction must remain below $β_u$ with probability at least $1-δ_u$. However, standard mean-field methods enforce these constraints only in expectation, which fails to account for stochastic fluctuations at finite fleet size $N$. To address this control problem, we propagate the second-order moment (variance) of the empirical density alongside the mean-field trajectory via a discrete-time Lyapunov recursion, and apply the Cantelli inequality to convert chance constraints into tractable deterministic conditions on the moments of the empirical density. We then incorporate these moment-based surrogate constraints into a gradient-based sequential convex approximation procedure for density-feedback policy synthesis. We further introduce additional moment-error bounds to construct a rigorous finite-$N$ certificate. The method is evaluated on a gridworld environment and a power-system EV-charging aggregation problem and compared with a standard deterministic population-level LP baseline.

Abstract: In the online metric matching problem, we have $n$ servers with known locations in some metric space. Requests arrive one-by-one at certain locations, and upon arrival a request must be matched to a server that was not matched to a previous request. The goal is to minimize the matching cost. For randomized algorithms with an oblivious adversary, the best known competitive ratio is obtained by embedding the metric space into an HST, and then solving the problem in the setting where the metric space is defined by the HST. Bansal et al. (Algorithmica, 2014) introduced a framework for online metric matching where one develops an algorithm in a restricted reassignment model, and then transforms this into a true online algorithm. Using this framework, they obtained an expected competitive ratio of $O(\log n)$ for HSTs; this also gives the best known competitive ratio of $O(\log^2 n)$ for general metrics. In this paper, we revisit this framework with the aim of developing new algorithms. For HSTs where each node has at most $Δ$ children, we develop an algorithm via this framework with an expected competitive ratio of $O((\log\log Δ) \cdot \log Δ)$. In particular, this ratio is independent of $n$, the number of servers/requests. It is near-optimal, as the expected competitive ratio of any algorithm is $Ω(\log Δ)$.