Shortest Paths, Minimum Weight Perfect Matchings and Travelling Salesperson Tours with Random weights Plus a Random Cost Constraint

We study the problem of estimating the minimum weight of a combinatorial structure $S$ in a graph where each edge has a random weight $w(e)$ and a random cost $c(e)$ and such that the total cost of $S$ is bounded by some value $C$. More generally, we allow several costs $c_1(e),c_2(e),\ldots,c_r(e),\,r=O(1)$. For simplicity, in much of the analysis the random variables are independent uniform $[0,1]$. The aim is to prove high probability upper and lower bounds on $w(S)$ given that $c(S)\leq C_i,i=1,2,\ldots,r$. In the following instances, we establish upper and lower bounds on the minimum weight that are within a constant factor of each other w.h.p. We first consider the case where the combinatorial structure is a path $P$ from 1 to $n$ in the complete graph $K_n$. We then study the case where the structure is a perfect matching in the complete bipartite graph $K_{n,n}$ or a perfect matching in the complete graph $K_n$. We consider the case of the Travelling Salesperson Problem in $K_n$ or the complete digraph $\vec{K}_n$. We show how to generalise many of our results from the uniform distribution through the use of powers of exponential random variables.