Metric Approximations of Consistent Path Systems

A path system $\mathscr{P}$ in a graph $G=(V,E)$ is a collection of paths, with exactly one path between any two vertices in $V$. A path system is said to be consistent if it is closed under subpaths. We say that a path system $\mathscr{P}$ is $α$-metric if there exists a metric $ρ$ on $V$ such that $\sum_{i=1}^{k}ρ(x_{i-1},x_{i}) \le αρ(x_0,x_k)$ for every path $(x_0,x_1,\dots,x_k)\in \mathscr{P}$. Also, we denote by $Δ(\mathscr{P})$ the infimum of $α$ for which $\mathscr{P}$ is $α$-metric. We construct here infinitely many $n$-point consistent path systems $\mathscr{P}_n$ with $Δ(\mathscr{P}_n) \ge n^{\frac{1}{2}-o(1)}$. We also show how to efficiently compute $Δ(\mathscr{P})$ for a given path system.

💡 Research Summary

The paper introduces a quantitative framework for studying consistent path systems in graphs by defining the notion of an “α‑metric” system and the associated parameter Δ(P), which measures how far a given system P is from being strictly metric. A consistent path system is a collection of unique paths between every pair of vertices that is closed under taking subpaths. If there exists a metric ρ on the vertex set such that for every path (x₀,…,x_k) in the system the sum of the distances along the path satisfies Σ_{i=1}^{k} ρ(x_{i‑1},x_i) ≤ α·ρ(x₀,x_k), the system is called α‑metric. The infimum of all such α is denoted Δ(P). Trivially 1 ≤ Δ(P) ≤ n, where n is the number of vertices.

The authors aim to understand how large Δ(P) can be for an n‑vertex system. To obtain strong lower bounds they construct highly symmetric, group‑invariant path systems. For a finite group G and a generating set S, the Cayley graph Γ(G,S) is considered. By assigning to each group element a word w_g over S that satisfies natural consistency conditions (essentially that subwords correspond to subpaths), a G‑invariant consistent path system is defined (Proposition 2.4). Proposition 2.5 shows that if such a system is α‑metric, one can average any witnessing metric over the group to obtain a G‑invariant metric with the same α‑factor.

The central technical tool is Lemma 3.2. It selects a symmetric subset X⊂G and integers m, d with three properties: (1) every element of X has order > 2m, (2) the only way two powers g^i and h^j (|i|,|j|≤m) can coincide is when i=j=0, and (3) each g∈X satisfies g^m = h₁⋯h_s with s≤d and each h_j∈X. Defining S = X ∪ (G {g^i : g∈X, |i|≤m}) and using the word construction from Proposition 2.4 yields a G‑invariant consistent path system P on Γ(G,S) with Δ(P) ≥ m·d·|X|.

To translate this abstract lemma into concrete lower bounds, the authors work with the cyclic group Z_n where n is prime. Lemma 3.3 shows that a random set X of size O(log n) satisfies the required properties with high probability when m≈√n·log n and d=O(log n). Substituting these parameters into Lemma 3.2 gives Δ(P) ≥ n^{1/2−o(1)} for the constructed system P_n. Thus, for arbitrarily large n there exist consistent path systems whose metric‑approximation factor grows almost as the square root of the number of vertices.

On the algorithmic side, Section 4 proves that Δ(P) is always an algebraic number and can be computed exactly in polynomial time. Observation 4.4 translates the condition “P is t‑metric” into a system of linear inequalities (4) involving variables x_{a,b} that represent candidate distances. Feasibility of this system can be checked by linear programming; a binary search on t yields arbitrarily close rational approximations to Δ(P). Theorem 4.3 (from algorithmic number theory) guarantees that a sufficiently precise rational approximation uniquely determines the algebraic number Δ(P) and that its minimal polynomial can be recovered in polynomial time. Consequently, Δ(P) is always algebraic, and Theorem 4.2 provides explicit examples where Δ(P) is irrational, answering a question posed by Alex Scott.

Overall, the paper makes three major contributions: (1) it introduces the α‑metric approximation framework for consistent path systems, providing a natural quantitative measure Δ(P); (2) it constructs families of group‑invariant path systems with Δ(P) ≥ n^{1/2−o(1)}, showing that the metric‑approximation gap can be substantial; and (3) it gives a polynomial‑time algorithm to compute Δ(P) exactly, establishing that Δ(P) is always algebraic while also exhibiting irrational instances. These results bridge combinatorial graph theory, metric embedding, and computational algebra, and they open avenues for further research on approximation limits, algorithmic applications in network routing, and connections to geometric group theory.