For metric k-facility location, all pairs of the four sum/max objectives admit a single solution that is within a constant (or sqrt(k)) factor of optimal for both.
Recent developments in approximation algorithms for facilitylocationandclusteringproblems
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Compatibility of Max and Sum Objectives for Committee Selection and $k$-Facility Location
For metric k-facility location, all pairs of the four sum/max objectives admit a single solution that is within a constant (or sqrt(k)) factor of optimal for both.