Hardness Results for Approximate Distance Labeling in Sparse Graphs

Distance labeling schemes assign short labels to the vertices of a graph so that distances between pairs of vertices can be recovered, exactly or approximately, from the two corresponding labels alone. For sparse graphs, Kosowski, Uznański, and Viennot [PODC'2019] showed that, assuming the hardness of the Sum-Index communication problem, exact distance labeling for sparse unweighted graphs requires near-linear label size. Their construction also gives lower bounds for stretch $1+o(1)$, but does not rule out substantially smaller labels for stretch $1+\varepsilon$, where $\varepsilon>0$ is fixed. We prove lower bounds for this $(1+\varepsilon)$-stretch regime through a density/stretch tradeoff. For every constant integer $k\ge 1$, we show that any approximate distance labeling scheme with stretch strictly smaller than $1+1/k$ for unweighted graphs with $m=O(n^{1+1/k})$ edges requires maximum label size at least $\mathrm{SumIndex}(n)/2^{O_k(\sqrt{\log n})}$ for infinitely many values of $n$. Thus, under the conjecture that $\mathrm{SumIndex}(n)=n^{1-o(1)}$, the maximum label size must be $n^{1-o(1)}$. The main new ingredient is an unweighted graph construction based on $3$-AP-free sets, which replaces the weighted uniqueness mechanism of the previous Sum-Index reduction by an additive uniqueness mechanism and keeps the relevant critical paths short enough to obtain a constant multiplicative distance gap. The same construction also yields near-tight lower bounds for related $2$-hop structures, including hub labeling schemes, shortcut sets, and approximate distance hopsets.