Minimum-weight combinatorial structures under random cost-constraints

Frieze, Alan, Pegden, Wesley, Sorkin, Gregory B. ORCID: 0000-0003-4935-7820 and Tkocz, Tomasz (2021) Minimum-weight combinatorial structures under random cost-constraints. Electronic Journal of Combinatorics, 28 (1). ISSN 1077-8926

	Text (Minimum-weight combinatorial structures under random cost-constraints) - Accepted Version Download (214kB)
	Text (Minimum-weight combinatorial structures under random cost-constraints) - Accepted Version Download (214kB)

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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 cost-budget, 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 minimum-length 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:	17 Apr 2024 02:42
URI:	http://eprints.lse.ac.uk/id/eprint/107136

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