Typical values of extremal-weight combinatorial structures with independent symmetric weights
Yun Cheng, Yixue Liu, Tomasz Tkocz, Albert Xu
TL;DR
The paper investigates extremal-weighted combinatorial structures on complete graphs with i.i.d. symmetric edge weights that have regular upper tails, establishing asymptotically tight high-probability bounds for problems such as perfect matchings, spanning trees, Hamilton cycles, and fixed-balanced subgraphs. The authors develop a framework based on Chernoff-type large deviations and threshold subgraph constructions, anchored by the rate function $\Lambda_*$ and its inverse $\Lambda_*^{-1}$, to show that the maximal total gain $W_n$ scales as $(1+o(1))$ times $n$ or $\ell$ times $\Lambda_*^{-1}$ of an appropriate log term. In the Gaussian case, these bounds yield explicit asymptotics, e.g., $W_n=(1+o(1))\,n\sqrt{2\log n}$ for matchings and $W_n=(1+o(1))\ell\sqrt{2 d^{-1}\log n}$ for fixed balanced subgraphs, with corresponding results for expectations via concentration of Gaussian processes. This work extends the understanding of typical extremal values under symmetric distributions, offering a unified approach and pointing to future directions on limit theorems and fluctuation analysis.
Abstract
Suppose that the edges of a complete graph are assigned weights independently at random and we ask for the weight of the minimal-weight spanning tree, or perfect matching, or Hamiltonian cycle. For these and several other common optimisation problems, we establish asymptotically tight bounds when the weights are independent copies of a symmetric random variable (satisfying a mild condition on tail probabilities), in particular when the weights are Gaussian.