Engineering Fully Dynamic Exact $Δ$-Orientation Algorithms

TL;DR

This work tackles the problem of maintaining an optimal $\Delta$-orientation in fully dynamic graphs under arbitrary edge insertions and deletions. It introduces three invariant based algorithms NaiveDynOpt, StrongDynOpt, and ImprovedDynOpt that adapt static optimal approaches to the dynamic setting and proves their correctness. Empirically, the authors show ImprovedDynOpt outperforms prior nearly exact methods by about 32% and achieves update times up to six orders of magnitude faster than static exact solvers, demonstrating practical viability for large evolving graphs. The results highlight the potential of invariant driven dynamic optimization for fundamental graph problems and point to directions like parallelization and batching to further improve scalability.

Abstract

A (fully) dynamic graph algorithm is a data structure that supports edge insertions, edge deletions, and answers specific queries pertinent to the problem at hand. In this work, we address the fully dynamic edge orientation problem, also known as the fully dynamic $Δ$-orientation problem. The objective is to maintain an orientation of the edges in an undirected graph such that the out-degree of any vertex remains low. When edges are inserted or deleted, it may be necessary to reorient some edges to prevent vertices from having excessively high out-degrees. In this paper, we introduce the first algorithm that maintains an optimal edge orientation during both insertions and deletions. In experiments comparing with recent nearly exact algorithms, we achieve a 32% lower running time. The update time of our algorithm is up to 6 orders of magnitude faster than static exact algorithms.

Figures (3)

• Figure 1: Visualization of the two (most important) cases in the proof for Lemma \ref{['lem:ins']}. The path $u \to z \to w$ is the improving path found after inserting the edge, the path $x \to z \to y$ is the assumed improving path in $\overline{G}_{i+1}$. Case 1 depicts the case in which $\textsc{odeg}(x, \overline{G}_{i+1}) > \textsc{odeg}(u, \overline{G}_{i+1})$. In this case an improving path $x \to w$ can be found in $\overline{G}_i$ if there is an improving path in $\overline{G}_{i+1}$. Case 2 shows the case in which $\textsc{odeg}(x, \overline{G}_{i+1}) \leq \textsc{odeg}(u, \overline{G}_{i+1})$. In this case an improving path $u \to y$ can be found in $\overline{G}_i$ if there is an improving path in $\overline{G}_{i+1}$. In both cases, this yields a contradiction to Invariant \ref{['strong_invariant']}.
• Figure 2: Performance Profiles.
• Figure 3: Optimal maximum out-degree ($\Delta$) over time for the fully dynamic instances.

Theorems & Definitions (9)

• Lemma 3.1: Venkateswaran venkateswaran2004minimizing
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Theorem 3.2
• Lemma 3.6
• Lemma 3.7