The 4/3 additive spanner exponent is tight

Amir Abboud, Greg Bodwin
<span title="">2016</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jlc5kugafjg4dl7ozagimqfcbm" style="color: black;">Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2016</a> </i> &nbsp;
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 kε such that every graph
more &raquo; ... s a spanner on O(n 1+ε ) edges that preserves its pairwise distances up to +kε? 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 onÕ(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 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. 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.
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