  We investigate the minimum cost of a wide class of combinatorial optimization
problems over random bipartite geometric graphs in $\mathbb{R}^d$ where the
edge cost between two points is given by a $p$-th power of their Euclidean
distance. This includes e.g.\ the travelling salesperson problem and the
bounded degree minimum spanning tree. We establish in particular almost sure
convergence, as $n$ grows, of a suitable renormalization of the random minimum
cost, if the points are uniformly distributed and $d \ge 3$, $1\le p<d$.
Previous results were limited to the range $p<d/2$. Our proofs are based on
subadditivity methods and build upon new bounds for random instances of the
Euclidean bipartite matching problem, obtained through its optimal transport
relaxation and functional analytic techniques.
