Pivot-and-Station Multi-Agent Path Finding: Solvability, Complexity, and Algorithms

Automated high-density storage systems (warehouses, robotic parking, plant logistics, etc.) require fleets of agents to move through scarce task-critical resources and then park without obstructing future operations. We introduce Pivot-and-Station Multi-Agent Path Finding (PS-MAPF), a MAPF variant in which a subset of tasked agents must each visit one of a set of interchangeable pivots (e.g., workstations) before the entire fleet terminates at anonymous stations, one agent per station. We characterize solvability completely: every instance on a 2-edge-connected graph is solvable, and, on arbitrary connected graphs, a structural effective-distance measure relative to the number of unoccupied vertices gives a necessary and sufficient condition. We prove that minimizing station-makespan or station-flowtime is NP-hard already with a single pivot. We present three algorithms, a complete baseline, a SAT-based optimal solver, and Pivot-Prioritized Planning (PPP), the last solving 74-89% of benchmark instances with makespan and flowtime orders of magnitude below the baseline.

Learning Partition Trees for Nearest Neighbor Search

We study nearest neighbor search from the perspective of data-driven algorithm design: given a dataset $P \subset \mathbb{R}^d$ of size $n$ and sample access to a query distribution over $\mathbb{R}^d$, the goal is to learn a data structure optimized for queries drawn from that specific distribution. We focus on the class of balanced halfspace trees, which naturally abstracts space-partitioning frameworks like locality-sensitive hashing. Assuming Gaussian-like marginal conditions on the dataset and query distribution, we give an efficient algorithm that learns a tree achieving $o(nd)$ query time, provided that a perfect tree exists. At the core of our algorithmic approach is the balanced halfspace cut problem, where we are given a distribution over $\mathbb{R}^d \times \mathbb{R}^d$ and must find a balanced halfspace that minimizes the fraction of cut pairs. We prove that without distributional assumptions, finding the optimal balanced halfspace is NP-hard. To circumvent this computational barrier, we design an efficient improper learning algorithm: if the optimal halfspace cuts an $α$ fraction of pairs, our algorithm outputs a balanced polynomial threshold function of degree $\tilde{O}(1/\varepsilon^2)$ that cuts at most an $O(\sqrt{α+\varepsilon})$ fraction.

On the Complexity of Entrywise Power Matrix Factorization

Given a nonnegative matrix $X$, a factorization rank $r$ and a real parameter $p$, entrywise power matrix factorization (EPMF) looks for a low-rank matrix $X_r$ such that $X = |X_r|^{\circ p}$ (exact case) or $X \approx |X_r|^{\circ p}$ (approximate case), where $(\cdot)^{\circ p}$ denotes the component-wise exponent. EPMF includes the modulus model ($p=1$) and component-wise square factorization ($p=2$) as special cases, the latter being closely related to the square root rank. We analyze the computational complexity of the exact decision problem and the Frobenius-norm approximation problem, and establish a complete complexity landscape. In the exact case, we show that EPMF is equivalent to the combinatorial problem of flipping the signs of the entries of a given matrix $X$ to obtain a rank-$r$ matrix, which we refer to as the signing problem. We first show that the signing problem, and hence exact EPMF, is strongly NP-hard, improving a weak NP-hardness result for the square-root-rank of Fawzi et al. (Math. Prog., 2015). We then show that the signing problem can be solved in polynomial-time when $r$ is fixed. Moreover, when the rank $r$ is part of the input, we show that for generic matrices the algorithm is fixed-parameter tractable (FPT) in the parameter $r$; in fact, the running time is linear in the input size $X$. In the approximate case using the Frobenius norm as an error measure, we show that EPMF is NP-hard, already when $r=2$, the smallest nontrivial case.