Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.18802

Routing Multiple Agents Below the Sum of Distances

Matthias Bentert·Eduard Eiben·Fedor V. Fomin·Petr A. Golovach
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study Transient Multiagent Pathfinding, a variant of the classical Multi-Agent Pathfinding problem in which a set of agents must be routed without collisions from designated start vertices to designated destination vertices in a graph. We analyze the problem within the above-and-below-guarantee paradigm of parameterized complexity. In particular, we consider the natural upper bound \(L\), given by the sum of the shortest-path distances between pairs of agents' terminals (corresponding to sequential routing of the agents). The parameterization is given by the gap \(ζ= L - λ\) between this bound and the target makespan \(λ\), together with the number \(k\) of agents. Our main result establishes fixed-parameter tractability for the combined parameter \(k + ζ\). Matching lower bounds show that parameterization by \(k\) alone is W[1]-hard, and that parameterization by \(ζ\) alone is W[1]-hard when terminals are not required to be distinct. On the positive side, if all terminals are distinct, the problem becomes fixed-parameter tractable when parameterized solely by \(ζ\). Finally, we show that Transient Multiagent Pathfinding is unlikely to admit a polynomial kernel when parameterized by \(k + ζ\). Together, our results provide an almost complete characterization of the parameterized complexity landscape of the problem for the considered parameters.

Explore related subjects

cs.DScs.DM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Matthias Bentert, Eduard Eiben, Fedor V. Fomin, Petr A. Golovach. 2026-09-16. Routing Multiple Agents Below the Sum of Distances. https://arxiv.org/abs/2609.18802

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Beyond Kruskal: Polynomial-Time Tensor Decomposition under the Lovitz-Petrov Condition

Identifiability criteria certify that a given tensor decomposition is a unique rank decomposition. Kruskal's classical condition is one of the best-known deterministic criteria for identifiability. However, no polynomial-time decomposition algorithm is known under the Kruskal condition, and verifying the condition itself is NP-hard. Lovitz and Petrov introduced a strictly more general identifiability condition which, in contrast, is polynomial-time verifiable, but no polynomial-time decomposition algorithm was previously known under this condition. We give a polynomial-time algorithm for tensor decomposition under the Lovitz--Petrov condition. Moreover, combining our algorithm with polynomial-time verification of the Lovitz--Petrov condition yields an efficient end-to-end certification procedure: after computing a decomposition, one can deterministically certify in polynomial time that it is unique and therefore of minimum rank. This contrasts with an arbitrary tensor decomposition, which certifies only an upper bound on the tensor rank, while determining tensor rank is NP-hard in general.

cs.DS↗
arXiv2026

Poisson Exchange Beyond Submodularity: Effective Approximation Algorithms for Offline and Online Subset Selection over Matroids

Over the past decade, a growing body of research has shown that $γ$-weak submodularity broadly arises in numerous subset selection tasks, including feature selection, neural network pruning, and video summarization. Despite its prevalence, maximizing a $γ$-weakly submodular function subject to a general matroid constraint remains challenging. To date, the only known approximation guarantee is the conservative $(1+1/γ)^{-2}$ factor established by \citet{chen2018weakly}. To improve upon this result, this paper proposes a novel algorithm called \MGPE, which repeatedly performs maximum-gain local exchanges through careful control of a non-homogeneous Poisson clock, and proves that this \MGPE\ can attain an approximation ratio arbitrarily close to $ρ_γ=1-\left(γ/(2-γ)\right)^{ \frac{γ^2}{2(1-γ)} }$. In sharp contrast to the previous guarantee, our obtained factor $ρ_γ$ not only strictly improves upon $(1+1/γ)^{-2}$ for every $γ\in(0,1]$, but also can asymptotically approach the optimal $(1-1/e)$-approximation for submodular maximization as $γ\to1$. Furthermore, we surprisingly find that when the matroid constraint reduces to a cardinality or the objective satisfies the stronger notion of $α$-weak DR-submodularity, \MGPE\ can automatically recover the tight approximation ratios of $1-e^{-γ}$ and $1-e^{-α}$, respectively. Here, $α\in(0,1]$ denotes the DR ratio.

cs.DS↗
arXiv2026

Linear-Query Deterministic Approximation for Non-monotone Submodular Maximization under a Knapsack Constraint

Submodular maximization under a knapsack constraint (SMK) is a fundamental combinatorial optimization problem with broad applications across machine learning and data mining. Motivated by large-scale applications where query efficiency is paramount, we study non-monotone SMK and focus on deterministic algorithms with linear query complexity. Prior deterministic linear-query algorithms achieve at best a $1/5-\varepsilon$ approximation, falling short of the $1/4-\varepsilon$ ratio attainable by randomized algorithms. We close this gap by presenting a deterministic $(1/4-\varepsilon)$-approximation with $O(n\log^2(1/\varepsilon)/\varepsilon^2)$ queries. Our approach partitions the analysis based on the cost of the largest optimal element $r$: when the cost of $r$ is moderate, we refine the threshold-twin-greedy framework via residual-budget enumeration to tighten the analysis; when the cost of $r$ is large, we reduce the problem to bicriteria submodular maximization. As a secondary contribution, we obtain a $(1/2-\varepsilon, O(1/\varepsilon))$-bicriteria approximation with $O(n\log(1/\varepsilon)/\varepsilon^2)$ queries, improving over the previous $O(n^2/\varepsilon)$ query bound.

cs.DS↗