How to Reduce Temporal Cliques to Find Sparse Spanners

TL;DR

The paper addresses the problem of finding linear-size temporal spanners in temporal cliques by proving a robust reduction to bi-cliques and establishing an equivalence between the existence of linear-size spanners in cliques and bi-cliques. It introduces the edge-pivot framework, along with new tools such as dismountable vertices, $c$-pivot-edges, and $e$-reverted edges, to obtain linear-size spanners for broad classes, including edge-pivot graphs, shifted matching graphs, and product graphs. The results substantially broaden the known families of temporal cliques that admit sparse spanners and provide a unified, streamlined approach that improves understanding of temporal sparsification. This advances the quest toward sparsifying temporal connectivity in broad dynamic networks and offers techniques that may generalize to other temporal graph problems.

Abstract

Many real-world networks, such as transportation or trade networks, are dynamic in the sense that the edge set may change over time, but these changes are known in advance. This behavior is captured by the temporal graphs model, which has recently become a trending topic in theoretical computer science. A core open problem in the field is to prove the existence of linear-size temporal spanners in temporal cliques, i.e., sparse subgraphs of complete temporal graphs that ensure all-pairs reachability via temporal paths. So far, the best known result is the existence of temporal spanners with $\mathcal{O}(n\log n)$ many edges. We present significant progress towards proving that linear-size temporal spanners exist in all temporal cliques. We adapt techniques used in previous works and heavily expand and generalize them to provide a simpler and more intuitive proof of the $\mathcal{O}(n\log n)$ bound. Moreover, we use our novel approach to show that a large class of temporal cliques, called edge-pivot graphs, admit linear-size temporal spanners. To contrast this, we investigate other classes of temporal cliques that do not belong to the class of edge-pivot graphs. We introduce two such graph classes and we develop novel techniques for establishing the existence of linear temporal spanners in these graph classes as well.

Figures (5)

• Figure 1: Examples for the sets $\mathop{\mathrm{In}}\nolimits$ and $\mathop{\mathrm{Out}}\nolimits$.
• Figure 2: Our algorithm for computing $\mathop{\mathrm{\mathcal{O}}}\nolimits(n\log n)$ bi-spanners based on \ref{['lem:biclique-recurrence']}. The function dismount takes a bi-clique, exhaustively dismounts all dismountable vertices, and returns the remaining instance and the included edges. By \ref{['lem:equal-size-sides']}, this dismounts at least until $|A| = |B|$.
• Figure 3: $\mathop{\mathrm{SM}}\nolimits\mathopen{}\mathclose{\left({4}\right)$ with vertices $v \in A \sqcup B$ and all their neighbors, ordered by edge label.
• Figure 4: If at least one of the green or purple paths is temporal, $\{a', b'\}$ is $\mathopen{}\mathclose{\left\{{a}, {b}\right\}$-reverted.
• Figure 5: Product graphs and how to find small spanners.

Theorems & Definitions (23)

• Remark 2
• Definition 3: dismountable
• Lemma 4
• Lemma 5
• Corollary 6
• Theorem 7: Extremal Matching
• Lemma 8
• Corollary 9
• Theorem 10
• Theorem 11