OxTalks will soon move to the new Halo platform and will become 'Oxford Events.' There will be a need for an OxTalks freeze. This was previously planned for Friday 14th November – a new date will be shared as soon as it is available (full details will be available on the Staff Gateway).
In the meantime, the OxTalks site will remain active and events will continue to be published.
If staff have any questions about the Oxford Events launch, please contact halo@digital.ox.ac.uk
In a complete bipartite graph on $m$ by $n$ vertices, the $m\cdot n$ edges are assigned random costs, and we ask for the minimum total cost of a set of $k$ edges where no two share a vertex. This random model of combinatorial optimisation has been studied extensively as a test case for algorithms and as a toy model of a physical system.
It has been known since the 1990’s that taking the edge-costs to be independent mean 1 exponential permits explicit formulas for things like the expected minimum cost, the most famous case being the $\pi^2/6$ limit for the case $k=m=n$ (perfect matching).
In this talk I will show how the distribution, not just its expectation, can be computed explicitly and efficiently from a certain recursion. The recursion implies a Gaussian limit (after proper rescaling) of the cost of matching when a positive proportion of the vertices remain unmatched, although such a “central limit theorem” for perfect matching remains elusive.
As time permits, I will describe how the results can be generalised to other optimisation problems like minimum 2-factor and minimum cover.
