The 4/3 Additive Spanner Exponent is Tight

TL;DR

This paper proves that the known bounds for additive spanners, specifically the +6 spanner with $O(n^{4/3})$ edges, are tight and cannot be improved, resolving a long-standing open question.

Contribution

It establishes a new information-theoretic lower bound showing the impossibility of improving the exponent for additive spanners below 4/3, even with subpolynomial additive error.

Findings

The +6 spanner bound on $O(n^{4/3})$ edges is tight.

No compression scheme can recover distance information within $+n^{o(1)}$ error using fewer than $O(n^{4/3 - ext{small}})$ bits.

Lower bounds also apply to related structures like the +4 emulator.

Abstract

A spanner is a sparse subgraph that approximately preserves the pairwise distances of the original graph. It is well known that there is a smooth tradeoff between the sparsity of a spanner and the quality of its approximation, so long as distance error is measured multiplicatively. A central open question in the field is to prove or disprove whether such a tradeoff exists also in the regime of \emph{additive} error. That is, is it true that for all $\varepsilon>0$, there is a constant $k_{\varepsilon}$ such that every graph has a spanner on $O(n^{1+\varepsilon})$ edges that preserves its pairwise distances up to $+k_{\varepsilon}$? Previous lower bounds are consistent with a positive resolution to this question, while previous upper bounds exhibit the beginning of a tradeoff curve: all graphs have $+2$ spanners on $O(n^{3/2})$ edges, $+4$ spanners on $\tilde{O}(n^{7/5})$ edges, and $+6$ spanners on $O(n^{4/3})$ edges. However, progress has mysteriously halted at the $n^{4/3}$ bound, and despite significant effort from the community, the question has remained open for all $0 < \varepsilon < 1/3$. Our main result is a surprising negative resolution of the open question, even in a highly generalized setting. We show a new information theoretic incompressibility bound: there is no function that compresses graphs into $O(n^{4/3 - \varepsilon})$ bits so that distance information can be recovered within $+n^{o(1)}$ error. As a special case of our theorem, we get a tight lower bound on the sparsity of additive spanners: the $+6$ spanner on $O(n^{4/3})$ edges cannot be improved in the exponent, even if any subpolynomial amount of additive error is allowed. Our theorem implies new lower bounds for related objects as well; for example, the twenty-year-old $+4$ emulator on $O(n^{4/3})$ edges also cannot be improved in the exponent unless the error allowance is polynomial.