Facility Reallocation on the Line

TL;DR

This paper studies multi-stage facility reallocation on a line, deriving optimal algorithms for offline and online cases, and proposing strategy-proof mechanisms with competitive ratios, including extensions to multiple facilities and weighted agents.

Contribution

It introduces optimal algorithms for both offline and online facility reallocation problems, and proposes strategy-proof mechanisms with competitive ratios, extending to multiple facilities and weighted agents.

Findings

Optimal offline and online algorithms derived.

Online algorithm has competitive ratio of (n+2)/(n+1).

Strategy-proof mechanism with competitive ratios of (n+3)/(n+1) and (n+4)/n.

Abstract

We consider a multi-stage facility reallocation problems on the real line, where a facility is being moved between time stages based on the locations reported by $n$ agents. The aim of the reallocation algorithm is to minimise the social cost, i.e., the sum over the total distance between the facility and all agents at all stages, plus the cost incurred for moving the facility. We study this problem both in the offline setting and online setting. In the offline case the algorithm has full knowledge of the agent locations in all future stages, and in the online setting the algorithm does not know these future locations and must decide the location of the facility on a stage-per-stage basis. We derive the optimal algorithm in both cases. For the online setting we show that its competitive ratio is $(n+2)/(n+1)$. As neither of these algorithms turns out to yield a strategy-proof mechanism, we propose another strategy-proof mechanism which has a competitive ratio of $(n+3)/(n+1)$ for odd $n$ and $(n+4)/n$ for even $n$, which we conjecture to be the best possible. We also consider a generalisation with multiple facilities and weighted agents, for which we show that the optimum can be computed in polynomial time for a fixed number of facilities.