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 epsilon > 0, there exists an O^~(n/epsilon^O(1))-time algorithm that returns a (1+epsilon)-approximate estimate of the MST cost. This result immediately
implies an O^~(n/epsilon^O(1)) time algorithm to estimate the TSP cost to within a (2 + epsilon) factor for any
epsilon > 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 existence of (1 + epsilon)-approximate estimation algorithms for metric TSP with O^~ (n) time for any fixed epsilon > 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 O^~(n) time algorithm that estimates the cost of graphic TSP to within a factor of (2 − epsilon_0) for some epsilon_0 > 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 O^~(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 O^~(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 epsilon > 0, (1 + epsilon)-approximate
estimates can be obtained for graphic TSP and (1, 2)-TSP using O^~ (n) queries. We answer this
question in the negative – there exists an epsilon_0 > 0, such that any algorithm that estimates the cost of
graphic TSP ((1, 2)-TSP) to within a (1 + epsilon_0)-factor, necessarily requires
(n^2) 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 epsilon > 0, any
algorithm that estimates the cost of graphic TSP or (1, 2)-TSP to within a (1 + epsilon)-factor, can also
be used to estimate the size of a maximum matching in a bipartite graph to within an epsilon 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.
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Separating subadditive Euclidean functionals
The classical Beardwood-Halton-Hammersly theorem (1959) asserts the existence of an asymptotic formula of the form $\beta \sqrt n$ for the minimum length of a Traveling Salesperson Tour throuh $n$ random points in the unit square, and in the decades since it was proved, the existence of such formulas has been shown for other such \emph{Euclidean functionals} on random points in the unit square as well. Despite more than 50 years of attention, however, it remained unknown whether the minimum length TSP through $n$ random points in $[0,1]^2$ was asymptotically distinct from its natural lower bounds, such as the minimum length spanning tree, the minimum length 2-factor, or, as raised by Goemans and Bertsimas, from its linear programming relaxation.
We prove that the TSP on random points in Euclidean space is indeed asymptotically distinct from these and other natural lower bounds, and show that this separation implies that branch-and-bound algorithms based on these natural lower bounds must take nearly exponential ($e^{\tilde \Omega(n)}$) time to solve the TSP to optimality, even in average case. This is the first average-case superpolynomial lower bound for these branch-and-bound algorithms (a lower bound as strong as $e^{\tilde \Omega (n)}$ was not even been known in worst-case analysis).
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- Award ID(s):
- 1661063
- NSF-PAR ID:
- 10054181
- Date Published:
- Journal Name:
- Random structures & algorithms
- Volume:
- 51
- ISSN:
- 1042-9832
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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