Frieze, Alan, Pegden, Wesley, Sorkin, Gregory B. ORCID: 0000000349357820 and Tkocz, Tomasz (2021) Minimumweight combinatorial structures under random costconstraints. Electronic Journal of Combinatorics, 28 (1). ISSN 10778926
Text (Minimumweight combinatorial structures under random costconstraints)
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Text (Minimumweight combinatorial structures under random costconstraints)
 Accepted Version
Download (214kB) 
Abstract
Recall that Janson showed that if the edges of the complete graph Kn are assigned exponentially distributed independent random weights, then the expected length of a shortest path between a fixed pair of vertices is asymptotically equal to (log n)/n. We consider analogous problems where edges have not only a random length but also a random cost, and we are interested in the length of the minimumlength structure whose total cost is less than some cost budget. For several classes of structures, we determine the correct minimum length structure as a function of the costbudget, up to constant factors. Moreover, we achieve this even in the more general setting where the distribution of weights and costs are arbitrary, so long as the density f(x) as x → 0 behaves like cxγ for some γ ≥ 0; previously, this case was not understood even in the absence of cost constraints. We also handle the case where each edge has several independent costs associated to it, and we must simultaneously satisfy budgets on each cost. In this case, we show that the minimumlength structure obtainable is essentially controlled by the product of the cost thresholds.
Item Type:  Article 

Official URL:  https://www.combinatorics.org/ 
Additional Information:  © 2021 The Authors 
Divisions:  Mathematics 
Subjects:  Q Science > QA Mathematics 
Date Deposited:  05 Nov 2020 10:50 
Last Modified:  20 Sep 2021 02:19 
URI:  http://eprints.lse.ac.uk/id/eprint/107136 
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