TY - JOUR

T1 - On the random 2-stage minimum spanning tree

AU - Flaxman, Abraham D.

AU - Frieze, Alan

AU - Krivelevich, Michael

PY - 2006/1

Y1 - 2006/1

N2 - It is known [A. M. Frieze, Discrete Appl Math 10 (1985), 47-56] that if the edge costs of the complete graph K n are independent random variables, uniformly distributed between 0 and 1, then the expected cost of the minimum spanning tree is asymptotically equal to ζ (3) = ∑ i=l ∈, i -3. Here we consider the following stochastic two-stage version of this optimization problem. There are two sets of edge costs C M: E → ℝ and c T: E → ℝ, called Monday's prices and Tuesday's prices, respectively. For each edge e, both costs C M (e) and c T (e) are independent random variables, uniformly distributed in [0, 1]. The Monday costs are revealed first. The algorithm has to decide on Monday for each edge e whether to buy it at Monday's price c M(e), or to wait until its Tuesday price c T (e) appears. The set of edges X M bought on Monday is then completed by the set of edges X T bought on Tuesday to form a spanning tree. If both Monday's and Tuesday's prices were revealed simultaneously, then the optimal solution would have expected cost ζ(3)/2 + σ(1). We show that, in the case of two-stage optimization, the expected value of the optimal cost exceeds ζ(3)/2 by an absolute constant ε > 0. We also consider a threshold heuristic, where the algorithm buys on Monday only edges of cost less than α and completes them on Tuesday in an optimal way, and show that the optimal choice for α is α = 1/n with the expected cost ζ (3)- 1/2+0(1). The threshold heuristic is shown to be sub-optimal. Finally we discuss the directed version of the problem, where the task is to construct a spanning outarborescence rooted at a fixed vertex r, and show, somewhat surprisingly, that in this case a simple variant of the threshold heuristic gives the asymptotically optimal value 1 - 1/e + 0 (1).

AB - It is known [A. M. Frieze, Discrete Appl Math 10 (1985), 47-56] that if the edge costs of the complete graph K n are independent random variables, uniformly distributed between 0 and 1, then the expected cost of the minimum spanning tree is asymptotically equal to ζ (3) = ∑ i=l ∈, i -3. Here we consider the following stochastic two-stage version of this optimization problem. There are two sets of edge costs C M: E → ℝ and c T: E → ℝ, called Monday's prices and Tuesday's prices, respectively. For each edge e, both costs C M (e) and c T (e) are independent random variables, uniformly distributed in [0, 1]. The Monday costs are revealed first. The algorithm has to decide on Monday for each edge e whether to buy it at Monday's price c M(e), or to wait until its Tuesday price c T (e) appears. The set of edges X M bought on Monday is then completed by the set of edges X T bought on Tuesday to form a spanning tree. If both Monday's and Tuesday's prices were revealed simultaneously, then the optimal solution would have expected cost ζ(3)/2 + σ(1). We show that, in the case of two-stage optimization, the expected value of the optimal cost exceeds ζ(3)/2 by an absolute constant ε > 0. We also consider a threshold heuristic, where the algorithm buys on Monday only edges of cost less than α and completes them on Tuesday in an optimal way, and show that the optimal choice for α is α = 1/n with the expected cost ζ (3)- 1/2+0(1). The threshold heuristic is shown to be sub-optimal. Finally we discuss the directed version of the problem, where the task is to construct a spanning outarborescence rooted at a fixed vertex r, and show, somewhat surprisingly, that in this case a simple variant of the threshold heuristic gives the asymptotically optimal value 1 - 1/e + 0 (1).

UR - http://www.scopus.com/inward/record.url?scp=33645287007&partnerID=8YFLogxK

U2 - 10.1002/rsa.20079

DO - 10.1002/rsa.20079

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AN - SCOPUS:33645287007

SN - 1042-9832

VL - 28

SP - 24

EP - 36

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 1

ER -