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• W3037800180 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 e > 0, there exists an O(n/e^O(1)) time algorithm that returns a (1 + e)-approximate estimate of the MST cost. This result immediately implies an O(n/e^O(1)) time algorithm to estimate the TSP cost to within a (2 + e) factor for any e > 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 + e)-approximate estimation algorithms for metric TSP with O(n) time for any fixed e > 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-e₀) for some e₀ > 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 e > 0, (1 + e)-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 e₀ > 0, such that any algorithm that estimates the cost of graphic TSP ((1,2)-TSP) to within a (1 + e₀)-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 e > 0, any algorithm that estimates the cost of graphic TSP or (1,2)-TSP to within a (1 + e)-factor, can also be used to estimate the size of a maximum matching in a bipartite graph to within an e 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." @default.
• W3037800180 created "2020-07-02" @default.
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• W3037800180 date "2020-01-01" @default.
• W3037800180 modified "2023-09-23" @default.
• W3037800180 title "Sublinear Algorithms and Lower Bounds for Metric TSP Cost Estimation." @default.
• W3037800180 doi "https://doi.org/10.4230/lipics.icalp.2020.30" @default.
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