Bicriteria Approximate Navigable Graphs in General Metrics

Abstract

We study approximation algorithms for constructing sparse navigable graphs in general metrics. Given an $n$-point metric $(P,d)$ and a parameter $α\ge 1$, an $α$-navigable graph is a directed graph $G=(P,E)$ such that for every ordered pair of distinct points $s,t\in P$, the vertex $s$ has an outgoing edge $(s,p)$ with $d(p,t)<d(s,t)/α$. The objective is to minimize the number of edges. Approximation algorithms for this problem were recently studied by Khanna, Padaki, and Waingarten (SODA 2026), and independently by Conway et al. (SODA 2026). These works show that the single-criterion problem faces a Set Cover barrier: an $O(\log n)$-approximation is achievable, whereas obtaining an $o(\log n)$-approximation is NP-hard. To obtain faster algorithms, Khanna, Padaki, and Waingarten introduced bicriteria approximation, in which the output graph is required to be $α$-navigable, while its size is compared against that of an optimal $β$-navigable graph for some relaxed parameter $β\ge α$. They gave a fast matrix-multiplication-based algorithm that computes an $(α,2(1+\varepsilon)α)$-bicriteria $O(\log n)$-approximation in time $\widetilde O(n^ω\log Δ/\varepsilon)$, where $Δ$ denotes the aspect ratio and $ω$ is the matrix multiplication exponent. Our main result is a simpler near-quadratic-time algorithm with stronger bicriteria guarantees. For every constant $α\ge 1$, we compute in time $\widetilde O(n^2\logΔ)$ an $α$-navigable graph that is simultaneously an $(α,2α+O(1))$-bicriteria $O(1)$-approximation and an $(α,α+O(1))$-bicriteria $O(\log n)$-approximation. We complement the algorithm with a hardness result showing that for every constant $η\in(0,1)$, no polynomial-time algorithm can compute an $(α,2α-η)$-bicriteria $o(\log n)$-approximation unless $P=NP$.