Dušan Knop

Dušan Knop, Šimon Schierreich, Ondřej Suchý

Inspired by the famous Target Set Selection problem, we propose a new discrete model to simultaneously spread two opinions within a social network and perform an initial study of its complexity. Here, we are given a social network, a seed-set of agents for each opinion, two thresholds for each agent, a budget, and a number of rounds. The first threshold represents the willingness of an agent to adopt an opinion if the agent has no opinion at all, while the second threshold states the willingness to acquire a second opinion if the agent already has one. The goal is to add at most budget-many agents to the initial seed-sets such that the process started with these extended seed-sets stabilizes within the given number of rounds, with each agent having either both opinions or none. That is, our goal is to ensure that the spread of opinions is balanced. We show that the problem is NP-hard, and thus we study the problem from the perspective of parameterized complexity. In particular, we show that the problem is FPT when parameterized by the number of rounds, the maximum threshold, and the treewidth combined. This algorithm also applies to the combined parameter, the treedepth and the maximum threshold. Finally, we show that the problem is FPT when parameterized by the vertex cover number, the $3$-path vertex cover number, or the vertex integrity of the input network alone. To complement our tractability results, we show that the problem is W[1]-hard with respect to a) the sizes of the initial seed-sets and the feedback-vertex set number combined, even if all thresholds are bounded by a constant, and b) the budget, the 4-path vertex cover number, and the feedback-vertex set number combined, even if every activation process stabilizes in at most 4 rounds.

Foivos Fioravantes, Dušan Knop, Jan Matyáš Křišťan et al.

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 $\ell$.

Foivos Fioravantes, Dušan Knop, Jan Matyáš Křišťan et al.

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.

Foivos Fioravantes, Dušan Knop, Jan Matyáš Křišťan et al.

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 $1$.

Foivos Fioravantes, Dušan Knop, Nikolaos Melissinos et al.

In Multiagent Path Finding (MAPF), the goal is to compute efficient, collision-free paths for multiple agents navigating a network from their sources to targets, minimizing the schedule's makespan-the total time until all agents reach their destinations. We introduce a new variant that formally models scenarios where some agents may experience delays due to malfunctions, posing significant challenges for maintaining optimal schedules. Recomputing an entirely new schedule from scratch after each malfunction is often computationally infeasible. To address this, we propose a framework for dynamic schedule adaptation that does not rely on full replanning. Instead, we develop protocols enabling agents to locally coordinate and adjust their paths on the fly. We prove that following our primary communication protocol, the increase in makespan after k malfunctions is bounded by k additional turns, effectively limiting the impact of malfunctions on overall efficiency. Moreover, recognizing that agents may have limited computational capabilities, we also present a secondary protocol that shifts the necessary computations onto the network's nodes, ensuring robustness without requiring enhanced agent processing power. Our results demonstrate that these protocols provide a practical, scalable approach to resilient multiagent navigation in the face of agent failures.

Kateřina Altmanová, Dušan Knop, Martin Koutecký

In recent years, algorithmic breakthroughs in stringology, computational social choice, scheduling, etc., were achieved by applying the theory of so-called $n$-fold integer programming. An $n$-fold integer program (IP) has a highly uniform block structured constraint matrix. Hemmecke, Onn, and Romanchuk [Math. Programming, 2013] showed an algorithm with runtime $a^{O(rst + r^2s)} n^3$, where $a$ is the largest coefficient, $r,s$, and $t$ are dimensions of blocks of the constraint matrix and $n$ is the total dimension of the IP; thus, an algorithm efficient if the blocks are of small size and with small coefficients. The algorithm works by iteratively improving a feasible solution with augmenting steps, and $n$-fold IPs have the special property that augmenting steps are guaranteed to exist in a not-too-large neighborhood. We have implemented the algorithm and learned the following along the way. The original algorithm is practically unusable, but we discover a series of improvements which make its evaluation possible. Crucially, we observe that a certain constant in the algorithm can be treated as a tuning parameter, which yields an efficient heuristic (essentially searching in a smaller-than-guaranteed neighborhood). Furthermore, the algorithm uses an overly expensive strategy to find a "best" step, while finding only an "approximatelly best" step is much cheaper, yet sufficient for quick convergence. Using this insight, we improve the asymptotic dependence on $n$ from $n^3$ to $n^2 \log n$. We show that decreasing the tuning parameter initially leads to an increased number of iterations needed for convergence and eventually to getting stuck in local optima, as expected. However, surprisingly small values of the parameter already exhibit good behavior. Second, our new strategy for finding "approximatelly best" steps wildly outperforms the original construction.