We consider the problem of designing sublinear time algorithms for estimating the cost of a minimum metric traveling salesman (TSP) tour. Specifically, given access to a $n times n$ distance matrix $D$ that specifies pairwise distances between $n$ points, the goal is to estimate the TSP cost by performing only sublinear (in the size of $D$) queries. For the closely related problem of estimating the weight of a metric minimum spanning tree (MST), it is known that for any $varepsilon > 0$, there exists an $tilde{O}(n/varepsilon^{O(1)})$ time algorithm that returns a $(1 + varepsilon)$-approximate estimate of the MST cost. This result immediately implies an $tilde{O}(n/varepsilon^{O(1)})$ time algorithm to estimate the TSP cost to within a $(2 + varepsilon)$ factor for any $varepsilon > 0$. However, no $o(n^2)$ time algorithms are known to approximate metric TSP to a factor that is strictly better than $2$. On the other hand, there were also no known barriers that rule out the existence of $(1 + varepsilon)$-approximate estimation algorithms for metric TSP with $tilde{O}(n)$ time for any fixed $varepsilon > 0$. In this paper, we make progress on both algorithms and lower bounds for estimating metric TSP cost. We also show that the problem of estimating metric TSP cost is closely connected to the problem of estimating the size of a maximum matching in a graph.
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