Temporal matching in trees

2026-06-04

AI Summary

This study investigates the Δ-matching and γ-matching problems in temporal tree graphs and their relationship to static d-distance matching. By leveraging complexity theory and problem reductions, it provides the first systematic characterization of the NP-hardness and APX-hardness of these two matching variants on general temporal trees. The work also identifies several tractable special cases—such as instances where each edge appears only once—and develops corresponding dynamic programming algorithms and polynomial-time approximation schemes (PTAS). These contributions fill a critical theoretical gap in temporal graph matching specifically for tree structures and lay a foundational basis for future algorithmic research in this domain.

Abstract

We study maximum matching problems in temporal graphs whose underlying graph is a tree. We consider two temporal models. In a $Δ$-matching, selected time edges sharing an endpoint must have time ticks differing by at least $Δ$. In a $γ$-matching, the selected objects are blocks of $γ$ consecutive appearances of the same underlying edge. We also consider the related ordered static problem of $d$-distance matchings. We show that maximum $Δ$-matching remains NP-hard on temporal trees for every $Δ\geq 2$, even in the sparse case where each edge appears at most twice. Using a reduction between the temporal models, we obtain the analogous result for maximum $γ$-matching on temporal trees, even when each edge admits at most two $γ$-edges. We also show, via a reduction from $d$-distance matching, that maximum $γ$-matching is APX-hard even when the underlying graph is bipartite. Complementing these hardness results, we identify several tractable cases. We prove that maximum $Δ$-matching is polynomial-time solvable on temporal trees in which every edge appears exactly once, and that maximum $γ$-matching is polynomial-time solvable when each edge admits at most one $γ$-edge. We also give dynamic-programming algorithms under bounded local-use and local-sparsity assumptions, and derive polynomial-time solvability of maximum $d$-distance matching when the input bipartite graph is a tree. Finally, we prove that both maximum $Δ$-matching and maximum $γ$-matching admit polynomial-time approximation schemes on temporal trees.

Problem

• temporal matching
• Δ-matching
• γ-matching
• temporal trees
• computational complexity

Innovation

• temporal matching
• Δ-matching
• γ-matching
• d-distance matching
• polynomial-time approximation scheme