An approximate algorithm for finding a maximum-weight $d$-homogeneous connected spanning subgraph in a complete graph with random edge weights

An approximation algorithm is suggested for the problem of finding a $d$-regular spanning connected subgraph of maximum weight in a complete undirected weighted $n$-vertex graph. Probabilistic analysis of the algorithm is carried out for the problem with random input data (some weights of edges) in the case of a uniform distribution of the weights of edges and in the case of a minorized type distribution. It is shown that the algorithm finds an asymptotically optimal solution with time complexity $O(n^2)$ when $d=o(n)$. For the minimization version of the problem, an additional restriction on the dispersion of weights of the graph edges is added to the condition of the asymptotical optimality of the modified algorithm.