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A practical algorithm for 2-admissibility

Christine Awofeso, Patrick Greaves, Oded Lachish, Felix Reidl

TL;DR

This work targets the practical computation of the sparse-graph measure called $2$-admissibility. It presents a polynomial-time, oracle-based algorithm that decides if $ ext{adm}_2(G) \le p$ in $O(p^4|G|)$ time and $O(|E(G)|+p^2)$ space, and it provides a concrete Rust implementation evaluated on 214 real-world networks, demonstrating scalability to large graphs with modest memory footprints. The theoretical construction combines $(2,L)$-path packings with an online oracle to greedily build an admissible ordering, while the implementation shows many networks exhibit small $2$-admissibility and thus can benefit from such structure-aware algorithms. The results suggest 2-admissibility is a practically useful sparseness measure for real networks and lay groundwork for extending the approach to higher admissibility levels ($3$- and $4$-admissibility) in future work.

Abstract

The $2$-admissibility of a graph is a promising measure to identify real-world networks which have an algorithmically favourable structure. In contrast to other related measures, like the weak/strong $2$-colouring numbers or the maximum density of graphs that appear as $1$-subdivisions, the $2$-admissibility can be computed in polynomial time. However, so far these results are theoretical only and no practical implementation to compute the $2$-admissibility exists. Here we present an algorithm which decides whether the $2$-admissibility of an input graph $G$ is at most $p$ in time $O(p^4 |V(G)|)$ and space $O(|E(G)| + p^2)$. The simple structure of the algorithm makes it easy to implement. We evaluate our implementation on a corpus of 214 real-world networks and find that the algorithm runs efficiently even on networks with millions of edges, that it has a low memory footprint, and that indeed many networks have a small $2$-admissibility.

A practical algorithm for 2-admissibility

TL;DR

This work targets the practical computation of the sparse-graph measure called -admissibility. It presents a polynomial-time, oracle-based algorithm that decides if in time and space, and it provides a concrete Rust implementation evaluated on 214 real-world networks, demonstrating scalability to large graphs with modest memory footprints. The theoretical construction combines -path packings with an online oracle to greedily build an admissible ordering, while the implementation shows many networks exhibit small -admissibility and thus can benefit from such structure-aware algorithms. The results suggest 2-admissibility is a practically useful sparseness measure for real networks and lay groundwork for extending the approach to higher admissibility levels (- and -admissibility) in future work.

Abstract

The -admissibility of a graph is a promising measure to identify real-world networks which have an algorithmically favourable structure. In contrast to other related measures, like the weak/strong -colouring numbers or the maximum density of graphs that appear as -subdivisions, the -admissibility can be computed in polynomial time. However, so far these results are theoretical only and no practical implementation to compute the -admissibility exists. Here we present an algorithm which decides whether the -admissibility of an input graph is at most in time and space . The simple structure of the algorithm makes it easy to implement. We evaluate our implementation on a corpus of 214 real-world networks and find that the algorithm runs efficiently even on networks with millions of edges, that it has a low memory footprint, and that indeed many networks have a small -admissibility.
Paper Structure (8 sections, 11 theorems, 4 equations, 3 figures)

This paper contains 8 sections, 11 theorems, 4 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Admissibility and related measures
  4. The theoretical algorithm
  5. The Oracle
  6. Implementation details and experiments
  7. Conclusion
  8. Appendix

Key Result

Lemma 0

For every graph $G$ and integers $p, r \geq 1$, if $\|G\| > p|G|$ then $\operatorname{adm}_{r} (G) > p$.

Figures (3)

  • Figure 1: Running time dependence on $2$-admissibility. Marker sizes represent the number of edges. The dotted lines and percentages indicate on how many networks the algorithm finished below the specified time.
  • Figure 2: Peak memory consumption dependence on $2$-admissibility. Marker sizes represent the number of edges. The dotted lines and percentages indicate on how many networks the algorithm used less than the indicate amount of memory. On average, a third of the memory consumption is due to storing the network.
  • Figure 3: Comparison of $2$-admissibility and degeneracy. Marker sizes represent the number of edges. The line represents a best-fit polynomial $\operatorname{adm}_{2} \approx \deg^{1.25}$, the red markers are outliers whose residual has an absolute z-score $\geq 3$.

Theorems & Definitions (11)

  • Lemma 0
  • Lemma 0
  • Lemma 0
  • Lemma 0
  • Lemma 0
  • Lemma 1
  • Lemma 5
  • Lemma 6
  • Lemma 6
  • Lemma 8
  • ...and 1 more