Abstract
AbstractIn a complete graph with independent uniform (or exponential) edge weights, let be the minimum‐weight spanning tree (MST), and the MST after deleting the edges of all previous trees. We show that each tree's weight converges in probability to a constant , with , and we conjecture that . The problem is distinct from Frieze and Johansson's minimum combined weight of k edge‐disjoint spanning trees; indeed, . With an edge of weight w “arriving” at time , Kruskal's algorithm defines forests , initially empty and eventually equal to , each edge added to the first possible . Using tools of inhomogeneous random graphs we obtain structural results including that the fraction of vertices in the largest component of converges to some . We conjecture that the functions tend to time translations of a single function.
Highlights
1.1 Problem definition and main resultsConsider the complete graph Kn with edge costs that are i.i.d. random variables, with a uniform distribution U (0, 1) or, alternatively, an exponential distribution Exp(1)
We show in Theorem 19 that, at least for k = 2, the two problems asymptotically have different answers, in the sense that the limiting values of the minimum cost – which exist for both problems – are different. (as discussed in Section 3.1, we improve on the upper bound from [6, Section 3] on the cost of the net cheapest k trees, since our upper bound (3.1) on the cost of the first k trees is smaller.)
We again buy the tree that is cheapest according to the posted prices w, but for each edge e purchased, we pay an amount that is a function of w−e, i.e., of all posted prices except that of e; for details see for example [16, Chapter 9]
Summary
Consider the complete graph Kn with edge costs that are i.i.d. random variables, with a uniform distribution U (0, 1) or, alternatively, an exponential distribution Exp(1). For each k 1, there exists a constant γk such that, as n → ∞, w(Tk) −p→ γk (for both uniform and exponential cost distributions). The result extends to other distributions of the edge costs (see full version for details), but we consider in this paper only the uniform and exponential cases. In the main part of the paper we avoid this problem completely by modifying the model: we assume that we have a multigraph, which we denote by Kn∞, with an infinite number of copies of each edge in Kn, and that each edge’s copies’ costs are given by the points in a Poisson process with intensity 1 on [0, ∞). For the Poisson multigraph model, E w(Tk) → γk for each k 1 as n → ∞
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