Minimum-Weight Combinatorial Structures Under Random Cost-Constraints

Recall that Janson showed if the edges of complete graph $K_n$ are assigned exponentially distributed independent random weights, then expected length a shortest path between fixed pair vertices is asymptotically equal to $(\log n)/n$. We consider analogous problems where have not only but also cost, and we interested in minimum-length structure whose total cost less than some budget. For several classes structures, determine correct minimum as function cost-budget, up constant factors. Moreover, achieve this even more general setting distribution weights costs arbitrary, so long density $f(x)$ $x\to 0$ behaves like $cx^\gamma$ for $\gamma\geq 0$; previously, case was understood absence constraints. handle each edge has associated it, must simultaneously satisfy budgets on cost. In case, show obtainable essentially controlled by product thresholds.