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# Successive Minimum Spanning Trees

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## Acknowledgements

We thank Oliver Riordan for helpful comments which simplified our proof, and Balázs Mezei for assistance with Julia programming.

## Cite As

Svante Janson and Gregory B. Sorkin. Successive Minimum Spanning Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 60:1-60:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.60

## Abstract

In a complete graph K_n with edge weights drawn independently from a uniform distribution U(0,1) (or alternatively an exponential distribution Exp(1)), let T_1 be the MST (the spanning tree of minimum weight) and let T_k be the MST after deletion of the edges of all previous trees T_i, i<k. We show that each tree’s weight w(T_k) converges in probability to a constant gamma_k with 2k-2 sqrt k < gamma_k < 2k+2 sqrt k, and we conjecture that gamma_k = 2k-1+o(1). The problem is distinct from that of [Alan Frieze and Tony Johansson, 2018], finding k MSTs of combined minimum weight, and the combined cost for two trees in their problem is, asymptotically, strictly smaller than our gamma_1+gamma_2. Our results also hold (and mostly are derived) in a multigraph model where edge weights for each vertex pair follow a Poisson process; here we additionally have E(w(T_k)) -> gamma_k. Thinking of an edge of weight w as arriving at time t=n w, Kruskal’s algorithm defines forests F_k(t), each initially empty and eventually equal to T_k, with each arriving edge added to the first F_k(t) where it does not create a cycle. Using tools of inhomogeneous random graphs we obtain structural results including that C_1(F_k(t))/n, the fraction of vertices in the largest component of F_k(t), converges in probability to a function rho_k(t), uniformly for all t, and that a giant component appears in F_k(t) at a time t=sigma_k. We conjecture that the functions rho_k tend to time translations of a single function, rho_k(2k+x) -> rho_infty(x) as k -> infty, uniformly in x in R. Simulations and numerical computations give estimated values of gamma_k for small k, and support the conjectures stated above.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Random graphs
• Mathematics of computing → Paths and connectivity problems
• Mathematics of computing → Combinatorial optimization
• Mathematics of computing → Matroids and greedoids
##### Keywords
• miminum spanning tree
• second-cheapest structure
• inhomogeneous random graph
• optimization in random structures
• discrete probability
• multi-type branching process
• functional fixed point
• robust optimization
• Kruskal’s algorithm

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