The 4/3 Additive Spanner Exponent Is Tight

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 additive error. That is, is it true that for all ε > 0, there is a constant k_ε such that every graph has a spanner on O(n^1+ε) edges that preserves its pairwise distances up to +k_ε? 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 Õ(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 < ε < 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 − ε}) bits so 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 20-year-old +4 emulator on O(n^4/3) edges also cannot be improved in the exponent unless the error allowance is polynomial.

Central to our construction is a new type of graph product, which we call the Obstacle Product. Intuitively, it takes two graphs G, H and produces a new graph G ⊗ H whose shortest paths structure looks locally like H but globally like G.