Online Metric TSP: Beyond the $\sqrt{n}$ Barrier
We study an online variant of the Traveling Salesperson Problem (TSP) in which $n$ points arrive sequentially and must be inserted into an evolving tour. In the classical setting where arbitrary insertions are allowed, an $O(\log n)$-competitive algorithm has been known since the 1970s (Rosenkrantz, Stearns and Lewis 1977, Imase and Waxman 1991). Recently, Abrahamsen, Bercea, Beretta, Klausen, and Kozma [ESA 2024] introduced online metric TSP, a stricter model in which each arriving point must be assigned to a distinct cell of an array of size $m \ge n$, with the final tour order induced by the non-empty cells; the parameter $m$ captures the space usage of the algorithm. When $m = 2^{n}$, this model recovers arbitrary insertions and therefore admits an $O(\log n)$-competitive algorithm. In contrast, when $m = n$, i.e., when each point's position is fixed on arrival, Bertram [ESA 2025] recently showed that the competitive ratio is $\Theta(\sqrt{n})$. We investigate the tradeoff between space usage and competitiveness between these extremes. We note that this tradeoff was previously explored by the authors [SODA 2026] for the online sorting problem, which is the special case of online metric TSP on a line metric. Our main result is a deterministic online metric TSP algorithm using $m = (1+\epsilon) n$ space that achieves a competitive ratio of $O(\log^{3} n / \epsilon)$, for any $\epsilon \le 1$. In particular, increasing the space from $n$ to $2n$ improves the competitive ratio from $\Theta(\sqrt{n})$ to $O(\log^{3} n)$. We complement this with a lower bound showing that for $m = n^{1+\epsilon}$, any deterministic algorithm has a competitive ratio $\Omega(1/\epsilon)$, for all $\epsilon \ge \Omega(\log \log n / \log n)$. Consequently, even with $m = O(n \cdot \mathrm{polylog}(n))$, deterministic algorithms cannot achieve a constant competitive ratio.