# The 4/3 Additive Spanner Exponent is Tight

## 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.Comment: Updated for journal versio