Abstract
We explore a similarity between the $n$ by $n$ random assignment problem and the random shortest path problem on the complete graph on $n+1$ vertices. This similarity is a consequence of the proof of the Parisi formula for the assignment problem given by C. Nair, B. Prabhakar and M. Sharma in 2003. We give direct proofs of the analogs for the shortest path problem of some results established by D. Aldous in connection with his $\zeta (2)$ limit theorem for the assignment problem.
Highlights
The purpose of this article is to bring together some results on random assignment and shortest path problems, and to clarify how they relate to each other
By establishing this connection we show that some results of Aldous [1] must hold for the shortest path problem
The infection spreads through this edge, and the edges that are chosen in this way form a tree that grows from v0 and eventually reaches all vertices. Instead of fixing another vertex, say vn, and asking for the probability that the i:th shortest edge from v0 belongs to the shortest path from v0 to vn, we ask for the expected number of vertices among v1, . . . , vn whose shortest path from v0 starts with the i:th shortest edge
Summary
The purpose of this article is to bring together some results on random assignment and shortest path problems, and to clarify how they relate to each other. Van Mieghem [4; 5]). By putting the papers [11] and [6] side by side, one can see that there is a close connection between these two random optimization problems. By establishing this connection we show that some results of Aldous [1] must hold for the shortest path problem. These properties are easier to establish directly for the shortest path problem, and by doing so we obtain, via the results of [11], new independent proofs of Aldous’ results
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