Successive Minimum Spanning Trees
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.
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
Mathematics of computing~Random graphs
Mathematics of computing~Paths and connectivity problems
Mathematics of computing~Combinatorial optimization
Mathematics of computing~Matroids and greedoids
60:1-60:16
RANDOM
The work was partly supported by the Knut and Alice Wallenberg Foundation.
A full version of the paper is available at https://arxiv.org/abs/1906.01533.
We thank Oliver Riordan for helpful comments which simplified our proof, and Balázs Mezei for assistance with Julia programming.
Svante
Janson
Svante Janson
Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden
http://www.math.uu.se/svante-janson/
https://orcid.org/0000-0002-9680-2790
Gregory B.
Sorkin
Gregory B. Sorkin
Department of Mathematics, The London School of Economics and Political Science, Houghton Street, London WC2A 2AE, England
http://personal.lse.ac.uk/sorkin/
https://orcid.org/0000-0003-4935-7820
10.4230/LIPIcs.APPROX-RANDOM.2019.60
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Svante Janson and Gregory B. Sorkin
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