Abstract
We consider the design of sublinear space and query complexity algorithms for estimating the cost of a minimum spanning tree (MST) and the cost of a minimum traveling salesman (TSP) tour in a metric on n points. We start by exploring this estimation task in the regime of o(n) space, when the input is presented as a stream of all binom(n,2) entries of the metric in an arbitrary order (a metric stream). For any α ≥ 2, we show that both MST and TSP cost can be αapproximated using Õ(n/α) space, and moreover, Ω(n/α²) space is necessary for this task. We further show that even if the streaming algorithm is allowed p passes over a metric stream, it still requires Ω̃(√{n/α p²}) space.
We next consider the wellstudied semistreaming regime. In this regime, it is straightforward to compute MST cost exactly even in the case where the input stream only contains the edges of a weighted graph that induce the underlying metric (a graph stream), and the main challenging problem is to estimate TSP cost to within a factor that is strictly better than 2. We show that in graph streams, for any ε > 0, any onepass (2ε)approximation of TSP cost requires Ω(ε² n²) space. On the other hand, we show that there is an Õ(n) space twopass algorithm that approximates the TSP cost to within a factor of 1.96.
Finally, we consider the query complexity of estimating metric TSP cost to within a factor that is strictly better than 2 when the algorithm is given access to an n × n matrix that specifies pairwise distances between n points. The problem of MST cost estimation in this model is wellunderstood and a (1+ε)approximation is achievable by Õ(n/ε^{O(1)}) queries. However, for estimating TSP cost, it is known that an analogous result requires Ω(n²) queries even for (1,2)TSP, and for general metrics, no algorithm that achieves a better than 2approximation with o(n²) queries is known. We make progress on this task by designing an algorithm that performs Õ(n^{1.5}) distance queries and achieves a strictly better than 2approximation when either the metric is known to contain a spanning tree supported on weight1 edges or the algorithm is given access to a minimum spanning tree of the graph. Prior to our work, such results were only known for the special cases of graphic TSP and (1,2)TSP.
In terms of techniques, our algorithms for metric TSP cost estimation in both streaming and query settings rely on estimating the cover advantage which intuitively measures the cost needed to turn an MST into an Eulerian graph. One of our main algorithmic contributions is to show that this quantity can be meaningfully estimated by a sublinear number of queries in the query model. On one hand, the fact that a metric stream reveals pairwise distances for all pairs of vertices provably helps algorithmically. On the other hand, it also seems to render useless techniques for proving space lower bounds via reductions from wellknown hard communication problems. Our main technical contribution in lower bounds is to identify and characterize the communication complexity of new problems that can serve as canonical starting point for proving metric stream lower bounds.
BibTeX  Entry
@InProceedings{chen_et_al:LIPIcs.ICALP.2023.37,
author = {Chen, Yu and Khanna, Sanjeev and Tan, Zihan},
title = {{Sublinear Algorithms and Lower Bounds for Estimating MST and TSP Cost in General Metrics}},
booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
pages = {37:137:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772785},
ISSN = {18688969},
year = {2023},
volume = {261},
editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18089},
URN = {urn:nbn:de:0030drops180892},
doi = {10.4230/LIPIcs.ICALP.2023.37},
annote = {Keywords: Minimum spanning tree, travelling salesman problem, streaming algorithms}
}
Keywords: 

Minimum spanning tree, travelling salesman problem, streaming algorithms 
Collection: 

50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) 
Issue Date: 

2023 
Date of publication: 

05.07.2023 