Hardness of Approximation in P via Short Cycle Removal: Cycle Detection, Distance Oracles, and Beyond

TL;DR

This paper introduces a technique to remove almost all short cycles in graphs without affecting triangles, and uses this to establish new hardness results for approximation and cycle detection problems in graph algorithms.

Contribution

It presents a novel method for cycle removal that preserves triangles, and derives tight lower bounds for distance oracles and cycle detection based on conjectures.

Findings

Proves no constant approximation for distance oracles under 3-SUM and APSP conjectures.

Establishes near-linear time lower bounds for k-cycle detection for all k ≥ 4.

Shows that improving cycle detection algorithms would imply breakthroughs in triangle detection.

Abstract

We present a new technique for efficiently removing almost all short cycles in a graph without unintentionally removing its triangles. Consequently, triangle finding problems do not become easy even in almost $k$-cycle free graphs, for any constant $k\geq 4$. Triangle finding is at the base of many conditional lower bounds in P, mainly for distance computation problems, and the existence of many $4$- or $5$-cycles in a worst-case instance had been the obstacle towards resolving major open questions. Hardness of approximation: Are there distance oracles with $m^{1+o(1)}$ preprocessing time and $m^{o(1)}$ query time that achieve a constant approximation? Existing algorithms with such desirable time bounds only achieve super-constant approximation factors, while only $3-\epsilon$ factors were conditionally ruled out (P\u{a}tra\c{s}cu, Roditty, and Thorup; FOCS 2012). We prove that no $O(1)$ approximations are possible, assuming the $3$-SUM or APSP conjectures. In particular, we prove that $k$-approximations require $\Omega(m^{1+1/ck})$ time, which is tight up to the constant $c$. The lower bound holds even for the offline version where we are given the queries in advance, and extends to other problems such as dynamic shortest paths. The $4$-Cycle problem: An infamous open question in fine-grained complexity is to establish any surprising consequences from a subquadratic or even linear-time algorithm for detecting a $4$-cycle in a graph. We prove that $\Omega(m^{1.1194})$ time is needed for $k$-cycle detection for all $k\geq 4$, unless we can detect a triangle in $\sqrt{n}$-degree graphs in $O(n^{2-\delta})$ time; a breakthrough that is not known to follow even from optimal matrix multiplication algorithms.