Exact Algorithms and Lower Bounds for Forming Coalitions of Constrained Maximum Size

Imagine we want to split a group of agents into teams in the most \emph{efficient} way, considering that each agent has their own preferences about their teammates. This scenario is modeled by the extensively studied \textsc{Coalition Formation} problem. Here, we study a version of this problem where each team must additionally be of bounded size. We conduct a systematic algorithmic study, providing several intractability results as well as multiple exact algorithms that scale well as the input grows (FPT), which could prove useful in practice. Our main contribution is an algorithm that deals efficiently with tree-like structures (bounded \emph{treewidth}) for ``small'' teams. We complement this result by proving that our algorithm is asymptotically optimal. Particularly, there can be no algorithm that vastly outperforms the one we present, under reasonable theoretical assumptions, even when considering star-like structures (bounded \emph{vertex cover number}).

Solving Multiagent Path Finding on Highly Centralized Networks

The Mutliagent Path Finding (MAPF) problem consists of identifying the trajectories that a set of agents should follow inside a given network in order to reach their desired destinations as soon as possible, but without colliding with each other. We aim to minimize the maximum time any agent takes to reach their goal, ensuring optimal path length. In this work, we complement a recent thread of results that aim to systematically study the algorithmic behavior of this problem, through the parameterized complexity point of view. First, we show that MAPF is NP-hard when the given network has a star-like topology (bounded vertex cover number) or is a tree with $11$ leaves. Both of these results fill important gaps in our understanding of the tractability of this problem that were left untreated in the recent work of [Fioravantes et al. Exact Algorithms and Lowerbounds for Multiagent Path Finding: Power of Treelike Topology. AAAI'24]. Nevertheless, our main contribution is an exact algorithm that scales well as the input grows (FPT) when the topology of the given network is highly centralized (bounded distance to clique). This parameter is significant as it mirrors real-world networks. In such environments, a bunch of central hubs (e.g., processing areas) are connected to only few peripheral nodes.

Exact Algorithms for Multiagent Path Finding with Communication Constraints on Tree-Like Structures

Consider the scenario where multiple agents have to move in an optimal way through a network, each one towards their ending position while avoiding collisions. By optimal, we mean as fast as possible, which is evaluated by a measure known as the makespan of the proposed solution. This is the setting studied in the Multiagent Path Finding problem. In this work, we additionally provide the agents with a way to communicate with each other. Due to size constraints, it is reasonable to assume that the range of communication of each agent will be limited. What should be the trajectories of the agents to, additionally, maintain a backbone of communication? In this work, we study the Multiagent Path Finding with Communication Constraint problem under the parameterized complexity framework. Our main contribution is three exact algorithms that are efficient when considering particular structures for the input network. We provide such algorithms for the case when the communication range and the number of agents (the makespan resp.) are provided in the input and the network has a tree topology, or bounded maximum degree (has a tree-like topology, i.e., bounded treewidth resp.). We complement these results by showing that it is highly unlikely to construct efficient algorithms when considering the number of agents as part of the input, even if the makespan is $3$ and the communication range is $1$.

Exact Algorithms and Lowerbounds for Multiagent Pathfinding: Power of Treelike Topology

In the Multiagent Path Finding problem (MAPF for short), we focus on efficiently finding non-colliding paths for a set of $k$ agents on a given graph $G$, where each agent seeks a path from its source vertex to a target. An important measure of the quality of the solution is the length of the proposed schedule $\ell$, that is, the length of a longest path (including the waiting time). In this work, we propose a systematic study under the parameterized complexity framework. The hardness results we provide align with many heuristics used for this problem, whose running time could potentially be improved based on our fixed-parameter tractability results. We show that MAPF is W[1]-hard with respect to $k$ (even if $k$ is combined with the maximum degree of the input graph). The problem remains NP-hard in planar graphs even if the maximum degree and the makespan$\ell$ are fixed constants. On the positive side, we show an FPT algorithm for $k+\ell$. As we delve further, the structure of~$G$ comes into play. We give an FPT algorithm for parameter $k$ plus the diameter of the graph~$G$. The MAPF problem is W[1]-hard for cliquewidth of $G$ plus $\ell$ while it is FPT for treewidth of $G$ plus $\ell$.