Sublinear Algorithms and Lower Bounds for Metric TSP Cost Estimation

Abstract

We consider the problem of designing sublinear time algorithms for estimating the cost of minimum metric traveling salesman (TSP) tour. Specifically, given access to a n × 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 ε > 0, there exists an Õ(n/ε^O(1)) time algorithm that returns a (1 + ε)-approximate estimate of the MST cost. This result immediately implies an Õ(n/ε^O(1)) time algorithm to estimate the TSP cost to within a (2 + ε) factor for any ε > 0. However, no o(n²) 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 existence of (1 + ε)-approximate estimation algorithms for metric TSP with Õ(n) time for any fixed ε > 0. In this paper, we make progress on both algorithms and lower bounds for estimating metric TSP cost.
On the algorithmic side, we first consider the graphic TSP problem where the metric D corresponds to shortest path distances in a connected unweighted undirected graph. We show that there exists an Õ(n) time algorithm that estimates the cost of graphic TSP to within a factor of (2-ε₀) for some ε₀ > 0. This is the first sublinear cost estimation algorithm for graphic TSP that achieves an approximation factor less than 2. We also consider another well-studied special case of metric TSP, namely, (1,2)-TSP where all distances are either 1 or 2, and give an Õ(n^1.5) time algorithm to estimate optimal cost to within a factor of 1.625. Our estimation algorithms for graphic TSP as well as for (1,2)-TSP naturally lend themselves to Õ(n) space streaming algorithms that give an 11/6-approximation for graphic TSP and a 1.625-approximation for (1,2)-TSP. These results motivate the natural question if analogously to metric MST, for any ε > 0, (1 + ε)-approximate estimates can be obtained for graphic TSP and (1,2)-TSP using Õ(n) queries. We answer this question in the negative - there exists an ε₀ > 0, such that any algorithm that estimates the cost of graphic TSP ((1,2)-TSP) to within a (1 + ε₀)-factor, necessarily requires Ω(n²) queries. This lower bound result highlights a sharp separation between the metric MST and metric TSP problems. 
Similarly to many classical approximation algorithms for TSP, our sublinear time estimation algorithms utilize subroutines for estimating the size of a maximum matching in the underlying graph. We show that this is not merely an artifact of our approach, and that for any ε > 0, any algorithm that estimates the cost of graphic TSP or (1,2)-TSP to within a (1 + ε)-factor, can also be used to estimate the size of a maximum matching in a bipartite graph to within an ε n additive error. This connection allows us to translate known lower bounds for matching size estimation in various models to similar lower bounds for metric TSP cost estimation.