Abstract
We study the minimum spanning tree problem on the complete graph $K_n$ where an edge $e$ has a weight $W_e$ and a cost $C_e$, each of which is an independent copy of the random variable $U^\gamma$ where $\gamma\leq 1$ and $U$ is the uniform $[0,1]$ random variable. There is also a constraint that the spanning tree $T$ must satisfy $C(T)\leq c_0$. We establish, for a range of values for $c_0,\gamma$, the asymptotic value of the optimum weight via the consideration of a dual problem.
Highlights
Let U denote the uniform [0, 1] random variable and let 0 < γ 1
We consider the minimum spanning tree problem in the context of the complete digraph Kn where each edge has an independent copy of U γ for weight We and an independent copy of U γ for cost Ce
N 2 and note that YL is a function of N i.i.d. random variables X1, . . . , XN
Summary
Let U denote the uniform [0, 1] random variable and let 0 < γ 1. [5] proves that if Ln denotes the expected minimum weight of a spanning tree . For the case γ < 1 we will prove the following. The following holds w.h.p. Note that C1 = c1 and this implies that the expression in (12) is consistent with the expression in (6). We note that a preliminary version containing the results for the case γ = 1 appeared in [10]. We use a result of [12] to show in Section 4 that in the cases discussed, the duality gap is negligible w.h.p
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