Truthful constant-approximation mechanisms for four or more facilities
Develop a strategyproof-in-expectation mechanism with a constant-factor social-cost approximation guarantee for every number of facilities \(k\geq 4\).
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The most intriguing future direction is to obtain a truthful in expectation mechanism with constant social cost approximation bound for k\geq 4. One possible approach is to modify the adjacent-gap volume used by . In particular, one could replace it by a more general weight of the form \widetilde{\Delta}p(\mathbf{x}_S)=\phi(z_0,z_1,...,z{k-1}) where z_0,z_1,...,z_{k-1} are the locations of the anchor p and the reported locations in \mathbf{x}_S, arranged in nondecreasing order. It would be interesting to determine whether an appropriate choice of \phi can restore strategyproofness in expectation for k\geq 4 while retaining a constant-factor approximation guarantee. We leave this as an open direction for future work.
The most immediate question is whether, for every fixed $k\geq 4$, there exists a randomized strategyproof mechanism with an approximation ratio independent of the number of agents on the line.
It would therefore be interesting either to modify Product-Gap so as to restore strategyproofness, or to identify a fundamentally different mechanism for larger $k$.
In particular, it remains open to understand whether, for fixed $k$ and $d$, there are randomized strategyproof mechanisms for $k$-facility location in $Rd$ whose approximation ratios depend only on $k$ and $d$, rather than on the number of agents.