Minimum-weight combinatorial structures under random cost-constraints

TL;DR

This paper investigates the minimal combinatorial structures with constrained total costs in random weighted graphs, extending classical shortest path results to include multiple cost constraints and arbitrary weight distributions.

Contribution

It generalizes shortest path analysis to structures with random costs and weights, providing asymptotic bounds under broad distributional assumptions and multiple cost constraints.

Findings

Determined minimal structure lengths as a function of cost budgets for various classes.

Extended results to arbitrary weight and cost distributions with specific density behavior near zero.

Analyzed structures with multiple independent costs per edge, linking minimal length to the product of cost thresholds.

Abstract

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