eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-09-17
60:1
60:16
10.4230/LIPIcs.APPROX-RANDOM.2019.60
article
Successive Minimum Spanning Trees
Janson, Svante
1
https://orcid.org/0000-0002-9680-2790
Sorkin, Gregory B.
2
https://orcid.org/0000-0003-4935-7820
Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden
Department of Mathematics, The London School of Economics and Political Science, Houghton Street, London WC2A 2AE, England
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol145-approx-random2019/LIPIcs.APPROX-RANDOM.2019.60/LIPIcs.APPROX-RANDOM.2019.60.pdf
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