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Low Recourse Arborescence Forests Under Uniformly Random Arcs

J Niklas Dahlmeier, D Ellis Hershkowitz

TL;DR

This work investigates maintaining a maximum arborescence forest under incremental arc insertions, focusing on recourse as the total number of arc changes. It proves a fundamental contrast: adversarial arc arrivals necessitate high recourse on the order of $Ω(m\cdot n)$, while arcs arriving uniformly at random allow a polynomial-time algorithm with expected recourse $O(m\log^2 n)$. The approach combines a path-update dynamic strategy—root-to-root feasible paths merge arborescences while controlling deletions—with a deep connection to random directed graphs $D(n,p)$ to exploit structural guarantees on in-component sizes. The results extend the understanding of low-recourse dynamics from matching-like problems to arborescence forests and highlight a promising avenue for leveraging random-graph structure in dynamic graph algorithms. The paper thus provides a clear separation between worst-case and average-case recourse and sets the stage for exploring random-order models and broader arborescence dynamics.

Abstract

In this work, we study how to maintain a forest of arborescences of maximum arc cardinality under arc insertions while minimizing recourse -- the total number of arcs changed in the maintained solution. This problem is the "arborescence version'' of max cardinality matching. On the impossibility side, we observe that even in this insertion-only model, it is possible for $m$ adversarial arc arrivals to necessarily incur $Ω(m \cdot n)$ recourse, matching a trivial upper bound of $O(m \cdot n)$. On the possibility side, we give an algorithm with expected recourse $O(m \cdot \log^2 n)$ if all $m$ arcs arrive uniformly at random.

Low Recourse Arborescence Forests Under Uniformly Random Arcs

TL;DR

This work investigates maintaining a maximum arborescence forest under incremental arc insertions, focusing on recourse as the total number of arc changes. It proves a fundamental contrast: adversarial arc arrivals necessitate high recourse on the order of , while arcs arriving uniformly at random allow a polynomial-time algorithm with expected recourse . The approach combines a path-update dynamic strategy—root-to-root feasible paths merge arborescences while controlling deletions—with a deep connection to random directed graphs to exploit structural guarantees on in-component sizes. The results extend the understanding of low-recourse dynamics from matching-like problems to arborescence forests and highlight a promising avenue for leveraging random-graph structure in dynamic graph algorithms. The paper thus provides a clear separation between worst-case and average-case recourse and sets the stage for exploring random-order models and broader arborescence dynamics.

Abstract

In this work, we study how to maintain a forest of arborescences of maximum arc cardinality under arc insertions while minimizing recourse -- the total number of arcs changed in the maintained solution. This problem is the "arborescence version'' of max cardinality matching. On the impossibility side, we observe that even in this insertion-only model, it is possible for adversarial arc arrivals to necessarily incur recourse, matching a trivial upper bound of . On the possibility side, we give an algorithm with expected recourse if all arcs arrive uniformly at random.

Paper Structure

This paper contains 20 sections, 20 theorems, 8 equations, 4 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Challenges and Techniques
  4. Adversarial Recourse Lower Bound
  5. Randomized Recourse Upper Bound
  6. Basic Idea of the Algorithm.
  7. A Weak Recourse Bound.
  8. Using Random Graph Structure.
  9. Discussion of Techniques
  10. Lower Bound for Adversarial Arc Arrivals
  11. Upper Bound for Uniformly Random Arc Arrivals
  12. Algorithm Description
  13. Correctness of Algorithm
  14. Recourse of Algorithm
  15. Arcs with Value under 2/n
  16. ...and 5 more sections

Key Result

Theorem 1.1

Given integral $n \geq 0$, there exists a fixed $n$ vertex instance of incremental maximum arborescence forest with $O(n)$ arc insertion such that every solution has recourse at least $\Omega(n^2)$.

Figures (4)

  • Figure 1: Red: an arborescence forest where each root is labeled "$r$". Black: all other arcs.
  • Figure 2: Illustration for \ref{['thm:lower']}. The $i$th row is $G^{(i)}$. Red arcs always denote the most recently added arc, turquoise arcs form the current arborescence forest and violet arcs are arcs deleted from the solution and hence count towards recourse.
  • Figure 3: Illustration of the second direction of proof of \ref{['onlyroots']}. Turquoise: arborescences. Red: strongly connected component of $r"$. Arcs dashed to signal that there could be more arcs. Root labels of $s, r, r'$ and $r"$ correspond to roots in proof. Roots $r$ and $r'$ on right in orange since they are no longer roots.
  • Figure 4: Red arcs denote arcs that will be added in the next step(s), the turquoise arcs form the optimal arborescence and black arcs are simply currently not in use. The adversary completes and bi-directs the bottom of the triangle and introduces a weight $0$ arc on the right side of the triangle which flips all arcs from the bottom. This is then repeated by adding two $0$ cost arcs on the left side which again flips all bottom arcs for the optimal arborescence. After the bottom arcs have been added, every 2nd arc will flip all bottom arcs which comprise of $n/3$ of all vertices, so a linear number of arcs get swapped every two arcs which gives $O(n)$ arcs added and $\Omega(n^2)$ changes.

Theorems & Definitions (40)

  • Definition 1: (Maximum) Arborescence Forest
  • Definition 2: Incremental Maximum Arborescence Forest
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Definition 3: Path Update
  • Definition 4: Feasible Paths
  • Lemma 3.1
  • ...and 30 more