A Note on Hardness of Diameter Approximation

We revisit the hardness of approximating the diameter of a network. In the CONGEST model of distributed computing, $\tilde \Omega (n)$ rounds are necessary to compute the diameter [Frischknecht et al. SODA'12], where $\tilde \Omega (\cdot)$ hides polylogarithmic factors. Abboud et al. [DISC 2016] extended this result to sparse graphs and, at a more fine-grained level, showed that, for any integer $1 \leq \ell \leq \operatorname{polylog} (n)$, distinguishing between networks of diameter $4 \ell + 2$ and $6 \ell + 1$ requires $\tilde \Omega (n)$ rounds. We slightly tighten this result by showing that even distinguishing between diameter $2 \ell + 1$ and $3 \ell + 1$ requires $\tilde \Omega (n)$ rounds. The reduction of Abboud et al. is inspired by recent conditional lower bounds in the RAM model, where the orthogonal vectors problem plays a pivotal role. In our new lower bound, we make the connection to orthogonal vectors explicit, leading to a conceptually more streamlined exposition.Comment: Accepted to Information Processing Letter

Approximation Hardness of Graphic TSP on Cubic Graphs

We prove explicit approximation hardness results for the Graphic TSP on cubic and subcubic graphs as well as the new inapproximability bounds for the corresponding instances of the (1,2)-TSP. The proof technique uses new modular constructions of simulating gadgets for the restricted cubic and subcubic instances. The modular constructions used in the paper could be also of independent interest