Strategy Proof Mechanisms for Facility Location at Limited Locations

09/17/2020 ∙ by Toby Walsh, et al. ∙ UNSW 4

Facility location problems often permit facilities to be located at any position. But what if this is not the case in practice? What if facilities can only be located at particular locations like a highway exit or close to a bus stop? We consider here the impact of such constraints on the location of facilities on the performance of strategy proof mechanisms for locating facilities.We study four different performance objectives: the total distance agents must travel to their closest facility, the maximum distance any agent must travel to their closest facility, and the utilitarian and egalitarian welfare.We show that constraining facilities to a limited set of locations makes all four objectives harder to approximate in general.

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Introduction

The facility location problem has been studied using tools from a wide variety of fields such as Artificial Intelligence, Operations Research, and Game Theory (e.g.

Drezner and Hamacher (2002); Lu et al. (2010); Fotakis and Tzamos (2010); Escoffier et al. (2011); Procaccia and Tennenholtz (2013); Serafino and Ventre (2015); Feigenbaum and Sethuraman (2015); Golowich, Narasimhan, and Parkes (2018); Jagtenberg and Mason (2020)). Our goal here is to design mechanisms that locate one or more facilities to serve a set of agents. In particular, we look to design strategy proof mechanisms where the agents have no incentive to mis-report their true location. Facility location problems model many real world problems including locating bus stops, telephone exchanges, fire stations and hospitals to serve a community. In many of these real world settings, facilities may be limited in where they can be located. For example, a warehouse might need to be constrained to be near to a highway, or a hospital close to a significant transport hub.

One of our contribution is to demonstrate that such constraints on the location of facilities can make it harder to design strategy proof mechanisms which provide high quality solutions. We measure the quality of the solution in four different ways: total distance of the agents to nearest facility, maximum distance of any agent to nearest facility, and the utilitarian and egalitarian welfare of the agents (where the utility of an individual agent is negatively correlated to their distance to the nearest facility). Another of our contributions is to show that guarantees about the performance of strategy proof mechanisms with respect to utilitarian or egalitarian welfare can be very different to guarantees on the total or maximum distance.

As in much previous work on mechanism design for facility location (e.g. Procaccia and Tennenholtz (2013)), we consider the one-dimensional setting. This models a number of real world problems such as locating shopping centres along a highway, or ferry stops along a river. There are also various non-geographical settings that can be viewed as one-dimensional facility location problems (e.g. choosing the temperature of a classroom). In addition, we can use mechanisms for the one-dimensional problem in more complex settings (e.g. we can decompose the 2-d rectilinear problem into a pair of 1-d problems). Finally, results about mechanisms for the one-dimensional problem can inform the results about mechanisms for more complex metrics. For instance, lower bounds on the performance of strategy proof mechanisms for the 1-d problem provide lower bounds for the 2-d problem.

Formal Background

We have agents located on the real line, and wish to locate facilities also on the real line to serve all the agents. Each agent is at location . We suppose agents are ordered so that . A solution is a location for each facility . An agent is served by the nearest facility. We let be the facility serving agent . We consider four different performance measures: total distance, maximum distance, utilitarian welfare and egalitarian welfare. The total distance is the sum of distances agents travel to be served, . The maximum distance is the maximum distance any agent travels to be served, . As in Mei et al. (2016, 2019), we suppose the utility of an individual agent is negatively correlated with the normalized distance they must travel to the nearest facility. In particular, we suppose the utility of agent is where is the maximum possible distance agent may need to travel. For instance, with a single facility in the interval , . Note that individual utilities are, by definition, constrained to be in . In Mei et al. (2016, 2019), these normalized utilities are called “happiness factors” but utility is perhaps a more standard nomenclature. The utilitarian welfare is then the sum of the utilities of the individual agents, . The egalitarian welfare is the minimum utility of any agent, . Our goal is either to minimize one of the two distance objectives, or to maximize one of the two welfare objectives.

We consider a number of mechanisms for facility location problem, many based on the median function which returns where and . With parameters to representing “phantom” agents, the GenMedian mechanism locates a facility at Moulin (1980). The Leftmost  mechanism has parameters for and locates the facility at the location of the leftmost agent. The Rightmost  mechanism has parameters for and locates the facility at the location of the rightmost agent. The Median  mechanism has parameters for and otherwise, and locates the facility at the median agent if

is odd, and leftmost median agent if

is even. The EndPoint mechanism for the two facility location problem locates one facility with the Leftmost mechanism and a second with the Rightmost mechanism.

We consider three desirable properties of mechanisms for facility location problems: anonymity, Pareto optimality and strategy proofness. Anonymity is a fundamental fairness property that requires all agents to be treated alike. Pareto optimality is one of the most fundamental normative properties in economics. It demands that we cannot improve the solution so one agent is better off without other agents being worse off. Finally, strategy proofness is a fundamental game theoretic property that ensures agents have no incentive to act strategically and try to manipulate the mechanisms by mis-reporting their location.

More formally, a mechanism is anonymous iff permuting the agents does not change the solution. A mechanism is Pareto optimal iff it returns solutions that are always Pareto optimal. A solution is Pareto optimal iff there is no other solution in which one agent travels a strictly shorter distance to the facility serving them, and all other agents travel no greater distance. A mechanism is strategy proof iff no agent can mis-report and reduce their distance to travel to the facility serving them. For instance, the Median mechanism is anonymous, Pareto optimal and strategy proof. Finally, we will consider strategy proof mechanisms that may approximate the optimal distance or welfare. A mechanism achieves an approximation ratio of the total distance/ maximum distance/ utilitarian welfare/ egalitarian welfare iff the distance/ welfare in any solution it returns is at most times the optimal. In this case, we say that the mechanism -approximates the optimal.

Procaccia and Tennenholtz initiated the study of designing approximate and strategy proof mechanisms for locating facilities on a line Procaccia and Tennenholtz (2013). With just one facility, they argue that the Median mechanism is strategy proof and optimal for the total distance, while the Leftmost mechanism is strategy proof and 2-approximates the optimal maximum distance, and that no deterministic strategy proof mechanism can do better than this. For the 2-facility game on a line, Procaccia and Tennenholtz prove that EndPoints mechanism is strategy proof and -approximates the optimal total distance Procaccia and Tennenholtz (2013). Lu et al. prove this is asymptotically optimal by providing an lower bound of for the approximation ratio of the optimal total distance Lu et al. (2010). Fotakis and Tzamos prove that no deterministic, anonymous and strategy proof mechanism for locating two facilities on the line has a bounded approximation ratio for the optimal total distance except for the EndPoints mechanism (Theorem 3.1 in Fotakis and Tzamos (2013)). The EndPoints mechanism is thus optimal in the sense that no other deterministic, anonymous and strategy proof mechanism can better approximate the optimal total distance. With three or more facilities, no deterministic and strategy proof mechanism at all has a bounded approximation ratio for the total distance (Theorem 7.1 in Fotakis and Tzamos (2013)). Procaccia and Tennenholtz also prove that the EndPoints mechanism is a -approximation of the optimal maximum distance, and that no deterministic and strategy proof mechanism can do better than this Procaccia and Tennenholtz (2013).

Limited locations for facilities

As argued earlier, in practice facilities may occur only in limited locations. For instance, Sui and Boutilitier assume facilities are constrained in their location and consider how this impacts on strategy proofness in one and higher dimensional facility location problems Sui and Boutilier (2015). There are multiple reasons why such an assumption may be needed. First, facilities might need to be in one of a set of possible locations. For example, it might be that a bus stop might need to be at an intersection or road crossing, while a power station should be within a short distance of a river or of the coast for cooling. Second, facilities might be prohibited from certain other locations. For example, we might not want to build a prison next to a school, while there is little reason to construct a new play ground close to any existing play ground. Third, it might not be possible to locate multiple facilities at the same location. The optimal location for a new hospital might be at the location of an existing hospital. In practice, however, it might not be physically impossible to locate another hospital at the same place. Fourth, some facilities might need to be located close to each other. For example, a bus interchange and train station might need to located near to each other. Fifth, other facilities might need to be distant from each other. For example, two nuclear power stations might need to be far apart in case of any accident.

We therefore study a model for facility location on the line where each facility has a finite set of feasible sub-intervals, and the facility must be located within one of these sub-intervals. For example, a problem for locating one facility might have agents located on and the facility must occur in the sub-intervals or . As a second example, a two facility problem might have agents located on with one facility limited to the sub-interval , and the other to . Our goal is to see how restricting the locations for facilities in this way impacts on the performance of strategy proof mechanisms.

Single facility at limited locations

For a single facility on the line, Moulin proved a seminal result that any mechanism that is anonymous, Pareto optimal and strategy proof is a generalized median mechanism, GenMedian Moulin (1980). This locates the facility at the median location of the agents and “phantom” agents. We cannot apply Moulin’s result directly to our setting as a GenMedian mechanism may select an infeasible location for the facility. Instead, we consider the  mechanism which locates a facility at the nearest feasible location to that returned by a GenMedian mechanism. If there are two nearest and equi-distant feasible locations, then the  mechanism uses a fixed tie-breaking rule for each infeasible interval (e.g,. always use the leftmost of the two nearest locations). We suppose a fixed tie-breaking rule to ensure that the modified mechanism retains anonymity and strategy proofness. However, none of our results on performance guarantees depend on the exact choice of tie-breaking rule.

The  mechanism is an instance of  which locates the facility at the median agent if it is a feasible location, and otherwise at the nearest feasible location to the median agent, tie-breaking with a fixed rule for each infeasible interval. Massó and Moreno de Barreda prove that, when locating a single facility at limited locations, a mechanism is anonymous, Pareto efficient and strategy proof iff it is a  mechanism (Corollary 2 in Massó and de Barreda (2011)). It follows that the  mechanism is anonymous, Pareto efficient and strategy proof.

Total distance

We first consider the objective of minimizing the total distance agents travel to be served.

Theorem 1

The  mechanism 3-approximates the optimal total distance for a facility location problem with limited locations.

Proof:  The total distance is minimized by locating the facility at the median agent. There are two cases when the facility is not located at the median agent. In the first, the median agent is at an infeasible location, the facility is located at the closest feasible location which is to the left of the median agent, and the optimal feasible location for the facility is at the closest feasible location to the median agent but on the right. A worst such case for the approximation ratio is when the only feasible locations for the facility are at 0 or , agents are at , the other agents are at , and the tie-breaking rule locates the facility at 0. This gives a total distance . However, the optimal total distance of just requires the facility to be located at . We therefore have an approximation ratio of . Letting tend to infinity,  at worst 3-approximates the optimal total distance. Note that we can eliminate the dependency on the tie-breaking rule by having the agents at for some small tending to zero. The second case where the facility is located to the right of the median agent and the optimal feasible location is to the left is symmetric.

In fact, the  mechanism is optimal. No deterministic strategy proof mechanism can better approximate the total distance in general.

Theorem 2

For a facility location problem with limited locations, any deterministic and strategy proof mechanism at best 3-approximates the optimal total distance.

Proof:  Suppose the only feasible locations for the facility are at 0 or , and there are two agents, one at and the other at for . There are two cases. In the first case, the mechanism locates the facility at 0. Suppose the agent at mis-reports their location as . Since the mechanism is strategy proof, the location of the facility cannot change. Consider this new problem with one agent at and the other at . The optimal total distance is now . The solution returned by the mechanism has a total distance of . The approximation ratio is then . As increases, this tends to 3. Hence the mechanism at best 3-approximates the total distance. The second case where the facility is placed at is symmetric.

We contrast this with the setting where the facility is unconstrained and can be located anywhere. In this unconstrained setting, the Median mechanism is strategy proof and returns the optimal total distance. Limiting the feasible locations of a facility therefore worsens the performance of the best possible deterministic and strategy proof mechanism. In particular, the best possible mechanism goes from returning an optimal solution to 3-approximating the optimal total distance.

Maximum distance

We consider next the objective of minimizing the maximum distance any agent travels to be served. The  mechanism also 3-approximates the optimal maximum distance.

Theorem 3

The  mechanism 3-approximates the optimal maximum distance for a facility location problem with limited locations.

Proof:  There are three cases. In the first case, the median agent is at a feasible location. The most an agent needs to travel to the facility at the median agent is then at most the distance of the rightmost agent from the leftmost. This is at most twice the optimal. Hence, this is a 2-approximation of the maximum distance. In the second case, the median agent is not at a feasible location, and the facility is located to the left at the nearest feasible location to the median agent. There are two sub-cases. In the first sub-case, the optimal location for the facility is at this point or even further to the left. But this means the facility is located between the leftmost and rightmost agents. Hence, this is again at worst a 2-approximation of the maximum distance. In the second sub-case, the optimal location for the facility is to the right, within some feasible interval to the right of the median agent. The worst such sub-case for the approximation ratio is when the only feasible locations for the facility are at 0 or , two agents are at with , and one agent is at . In this sub-case, the  mechanism locates the facility at 0. This gives a maximum distance of . However, the optimal maximum distance of just requires the facility to be located at . The approximation ratio is thus which approaches 3 from below as increases. The solution returned is therefore at worst a 3-approximation of the optimal maximum distance. The third and final case locates the facility to the right of the median agent and is symmetric to the second case.

We contrast this with the setting where the facility can be located anywhere, and the Median mechanism 2-approximates the optimal maximum distance. In fact, when there are no constraints on where facilities can be located, the Median mechanism is optimal as no deterministic and strategy proof mechanism can do better than 2-approximate the optimal maximum distance (Theorem 3.2 of Procaccia and Tennenholtz (2013)). Restricting the feasible locations of a facility therefore worsens the performance of a median mechanism from a 2-approximation of the optimal maximum distance to a 3-approximation.

Can any strategy proof mechanism do better than 3-approximate the maximum distance when we limit the feasible locations of the facility? We show that no deterministic and strategy proof mechanism has a smaller approximation ratio for the optimal maximum distance.

Theorem 4

For a facility location problem with limited locations, no deterministic and strategy proof mechanism can be better than a 3-approximation of the optimal maximum distance.

Proof:  Suppose the only feasible locations for the facility are at or , and there are two agents, one at and the other at for . There are two cases. In the first case, the mechanism locates the facility at . Suppose the agent at mis-reports their location as . Since the mechanism is strategy proof, the location of the facility cannot change. Consider this new problem with one agent at and the other at . The optimal maximum distance is now with the facility at . However, the solution returned by the mechanism has a total distance of . The approximation ratio is . As increases, this approaches 3 from below. Hence the mechanism at best 3-approximates the total distance. The second case where the facility is located at rather than is symmetric.

Hence the  mechanism is optimal. No deterministic and strategy proof mechanism can do better than 3-approximate the optimal maximum distance. We compare this with the setting where the facility can be located anywhere. In this setting, deterministic and strategy proof mechanisms can 2-approximate the optimal maximum distance, and no such mechanism can do better.

Small infeasible intervals

When infeasible intervals for the facility are small, we can provide stronger approximation guarantees on the optimal total or maximum distance. We begin with the total distance. We will show that the  mechanism has just an additive error in approximating the total distance. We say that a mechanism -approximates the optimal total distance iff the total distance agents travel to the nearest facility in the solution returned is less than or equal to where is the optimal total distance. We prove that when the longest infeasible interval is small, the  mechanism provides a solution which approximates the total distance well with only a small additive error.

Theorem 5

For a facility location problem with limited locations and agents, if the longest infeasible interval is , then the  mechanism -approximates the optimal total distance.

Proof:  If the median agent is at a feasible location then the  mechanism locates the facility here, and this gives a solution with optimal total distance. We therefore need to consider just the setting in which the median agent is located within an infeasible interval. A worst such case for the additive approximation error is when the only feasible locations for the facility are at 0 or , agents are at , the other agents are at , and the tie-breaking rule locates the facility at 0. This gives a total distance . However, the optimal total distance of just requires the facility to be located at . This gives an additive error of . Note that we can eliminate the dependency on the tie-breaking rule by having the agents at for some small tending to zero.

To give a tighter approximation guarantee on the maximum distance, we introduce a novel approximation measure that bounds the additive or multiplicative error. We say that a mechanism provides an -approximation of the optimal maximum distance iff the maximum distance any agent travels to the nearest facility in the solution returned is less than or equal to where is the optimal maximum distance. We can use this measure to provide a tighter guarantee on approximating the optimal maximum distance.

Theorem 6

For a facility location problem with limited locations, if the longest infeasible interval is , then the  mechanism provides a -approximation of the optimal maximum distance.

Proof:  In the proof of Theorem 3, the  mechanism 2-approximates the optimal maximum distance in every case considered except for one sub-case of the second case (and its reflection symmetry). In this sub-case, the median agent is at an infeasible location, the facility is located to the left at the nearest feasible location to the median agent, and the optimal location for the facility is to the right, within some feasible interval to the right of the median agent. A worst setting is when the only feasible locations for the facility are at 0 or , two agents are at , one agent is at , and the tie-breaking rule locates the facility at 0. This gives a maximum distance of . However, the optimal maximum distance of just requires the facility to be located at . The maximum distance in solution is then from the optimal. Note that we can again eliminate the dependency on the tie-breaking rule by having the two agents at for some small tending to zero.

Thus, if is small, even if there are many infeasible intervals, the  mechanism provides close to an 2-approximation of the optimal maximum distance. This is similar to the performance guarantee the mechanism provides when facilities can be located anywhere.

Multiple facilities at limited locations

With two facilities that can be located anywhere, there are multiple mechanisms which are anonymous, Pareto efficient and strategy proof. For example, the EndPoints and TwoLeftPeaks mechanisms are anonymous, Pareto efficient and strategy proof. The TwoLeftPeaks mechanism locates one facility at the leftmost agent, and the second at the nearest agent at a distinct location. Procaccia and Tenenholtz prove that the EndPoints mechanism provides a -approximation of the optimal total distance and 2-approximation of the optimal maximum distance Procaccia and Tennenholtz (2013). Indeed, it is the only deterministic, anonymous and strategy proof mechanism with a bounded approximation ratio of the optimal total (and hence maximum) distance (Theorem 3.1 in Fotakis and Tzamos (2013)).

We cannot apply these results directly to our setting with facilities at limited locations as the EndPoints mechanism may locate a facility at an infeasible location. Instead, we consider the  mechanism which modifies the EndPoints mechanism to locate facilities at the nearest feasible locations. More precisely, the  mechanism locates one facility at the nearest feasible location to the leftmost agent, and the other at the nearest feasible location to the rightmost agent. If there are two nearest and equi-distant feasible locations to the leftmost or rightmost agents, then the  mechanism tie-breaks with some fixed rule for each infeasible interval (e.g. always to the right of the two nearest feasible locations). The tie-breaking rule does not impact on the approximation ratios we report here. It needs, however, to have a specific form to ensure strategy proofness and anonymity.

Theorem 7

For a two facility location problem with limited locations, the  mechanism is deterministic, anonymous, and strategy proof, but does not bound the approximation ratio of the optimal total or maximum distance.

Proof:  Determinism and anonymity are immediate. Agents can, at best, only move the nearest facility further away by misreporting their location. Hence, the mechanism is strategy proof. For the approximation ratio, consider a problem where the leftmost facility can be at 0 or , and the rightmost at or for some . Suppose there are just two agents, one at and another at . Then the mechanism locates the facilities at 0 and giving a total distance of and a maximum distance of , while the optimal total distance is 2 and maximum distance is 1 with facilities at and . Hence the approximation ratio of both the total and maximum distances is . As tends to infinity, the approximation ratios are unbounded. Note we can limit agents and facilities to the interval by scaling every coordinate by .

In fact, while there exist deterministic, anonymous and strategy proof mechanism for locating two facilities at limited locations (e.g. the  mechanism), we can prove that no such mechanism has a bounded approximation ratio of the optimal total or maximum distance.

Theorem 8

No mechanism for locating two or more facilities at limited locations is deterministic, anonymous, strategy proof and bounds the approximation ratio of the optimal total or maximum distance.

Proof:  For two facilities, Theorem 3.1 in Fotakis and Tzamos (2013) demonstrates that the EndPoints mechanism is the only deterministic, anonymous and strategy proof mechanism with a bounded approximation ratio of the total (and hence also of the maximum) distance when facilities can be allocated anywhere. To retain anonymity and strategy proofness, when facilities can only be located at limited locations, the only possible mechanism we need consider is therefore the  mechanism. But this does not have a bounded approximation ratio of the total or maximum distance.

When three or more facilities can be located anywhere, Theorem 7.1 in Fotakis and Tzamos (2013) demonstrates that no deterministic, anonymous and strategy proof mechanism has a bounded approximation ratio of the optimal total distance (and hence also of the optimal maximum distance). Hence, the approximation ratios are also unbounded with three or more facilities at limited locations.

We contrast this result with the setting where facilities can be anywhere. In this case, with two facilities, the EndPoints mechanism is deterministic, anonymous, strategy proof and has a bounded approximation ratio of the optimal total or maximum distance Procaccia and Tennenholtz (2013). Note that, unlike the setting of a single facility, when the longest infeasible interval is small, we cannot bound the approximation ratio of the optimal total or maximum distance. In particular, we cannot guarantee just an additive error.

Orderly feasible locations

It is disappointing that, in general, there is no deterministic, anonymous and strategy proof mechanism for locating two or more facilities with a bounded approximation ratio of the optimal total or maximum distance. For two facilities, we can identify a simple condition on the feasible locations for the facilities which ensures that the two approximation ratios are bounded. For instance, this condition guarantees a bounded approximation ratio when the two facilities have identical feasible locations.

We designate one facility as the leftmost facility and the other as the rightmost. We say that the feasible locations for the two facilities are orderly iff there are no locations which are feasible for only the leftmost facility that are to the right of a feasible location for the rightmost facility, and dually there are no locations which are feasible for only the rightmost facility that are to the left of a feasible location for the leftmost facility. For example, the feasible locations in the proof of Theorem 9 are not orderly as the location is only feasible for the leftmost facility but this is to the right of the location which is feasible for the rightmost facility. As a second example, if the leftmost facility is feasible just for the interval and the rightmost for just the interval then the feasible locations are orderly. As a third example, if two facilities have identical feasible locations then the feasible locations are orderly. In this case, there are no locations which are feasible for only one facility. We now prove that when the feasible locations are orderly, the  mechanism has a bounded approximation ratio of the optimal total or maximum distance.

Theorem 9

For a two facility location problem with orderly feasible locations, the  mechanism -approximates the optimal total distance, and 3-approximates the optimal maximum distance.

Proof:  For total distance, if the leftmost and rightmost agents are at feasible locations, then the  mechanism locates facilities at these end points. As in the setting where facilities can be located anywhere, this -approximates the optimal total distance. Therefore we consider the case where the leftmost or rightmost agent is at an infeasible location. A worst such sub-case has the leftmost facility limited to 0 or , the rightmost limited to , one agent at , an additional agents at , a final agent at , and tie-breaking locating the leftmost facility at 0. Then the total distance is compared to an optimal total distance of when the leftmost facility is at . Hence the approximation ratio is . Note that we can again eliminate the dependency on the tie-breaking rule by having the leftmost agent at for some small tending to zero.

For maximum distance, if the leftmost and rightmost agents are at feasible locations, then the  mechanism locates facilities at these end points. As in the setting where facilities can be located anywhere, this -approximates the optimal maximum distance. Therefore we consider the case where the leftmost or rightmost agent is at an infeasible location. A worst such sub-case has the leftmost facility limited to 0 or , the rightmost limited to , one agent at , an additional agent at , a final agent at , and tie-breaking locating the leftmost facility at 0. Then the maximum distance is compared to an optimal total distance of when the leftmost facility is at . Hence the approximation ratio is . Note that we can again eliminate the dependency on the tie-breaking rule by having the leftmost agent at for some small tending to zero.

We contrast this with the setting in which every location is feasible. In this setting, the approximation ratios are significantly smaller. The EndPoints mechanism -approximates the optimal total distance which is twice as good as in the setting when facilities are at limited but orderly locations. Similarly, the EndPoints mechanism 2-approximates the optimal maximum distance which is one third better than in the setting when facilities are at limited but orderly locations.

Welfare approximations

We switch now from considering how well strategy proof mechanisms approximate the total or maximum distance to how well such mechanisms approximate the utilitarian or egalitarian welfare.

Utilitarian welfare

Mei et al. prove that the Median mechanism is strategy proof and is a -approximation of the optimal utilitarian welfare (Theorem 1 of Mei et al. (2019)). In addition, they show that no deterministic and strategy proof mechanism has an approximation ratio of the optimal utilitarian welfare of less than which is approximately 1.07 (Theorem 3 of Mei et al. (2019)). Indeed, they give a mechanism for two agents that provides an -approximation of the optimal utilitarian welfare. This mechanism locates the facility at the midpoint if the two agents are on opposite sides of the midpoint, otherwise locates the facility at the agent nearest to the midpoint.

We cannot apply these results directly to our setting as the mechanisms considered in Mei et al. (2019) may select an infeasible location for the facility. Instead, we consider the  mechanism which locates the facility at the nearest feasible location to the median agent.

Theorem 10

For a facility location problem with limited locations, the  mechanism is strategy proof and has an unbounded approximation ratio of the optimal utilitarian welfare.

Proof:  Suppose agents are in the interval , the only feasible locations for the facility are at 0 or 1, two agents are at , a third agent is at 1, and the tie-breaking rule locates the facility at 0. This gives an utilitarian welfare of zero units of utility. However, the optimal utilitarian welfare of 1 unit of utility requires the facility to be located at . We therefore have an unbounded approximation ratio. Note that we can eliminate the dependency on the tie-breaking rule by having the two agents not at but at for some small tending to zero.

Note that bounding the additive error in meaningful ways looks problematic. The example in the proof shows that we might need to add every possible unit of utility to get to the optimal utilitarian welfare. We contrast this with the setting where facilities can be located anywhere. In this unconstrained setting, the Median mechanism -approximates the optimal utilitarian welfare.

Unfortunately, we cannot better approximate the utilitarian welfare when facilities are constrained to limited locations. No deterministic and strategy proof mechanism has in general an unbounded approximation ratio of the optimal utilitarian welfare.

Theorem 11

For a facility location problem with limited locations, any deterministic and strategy proof mechanism has an unbounded approximation ratio of the optimal utilitarian welfare.

Proof:  Suppose agents are in the interval , the only feasible locations for the facility are at or , and there are two agents, one at and the other at for small that tends to zero. There are two cases. In the first case, the mechanism locates the facility at 0. Suppose the agent at mis-reports their location as . Since the mechanism is strategy proof, the location of the facility cannot change. Consider this new problem with one agent at and the other at . The solution returned by the mechanism has an utilitarian welfare that tends to zero units of utility as tends to zero. But the optimal utilitarian welfare is 1 unit of utility when the facility is at . Hence the mechanism has an unbounded approximation ratio for the optimal utilitarian welfare. The second case where the facility is placed at is symmetric.

Egalitarian welfare

The MidPoint mechanism locates the facility at the midpoint of the interval irrespective of the reports of the agents (i.e. at when agents are constrained to the interval ). This mechanism is trivially deterministic and anonymous. Mei et al. argue that the mechanism is strategy proof and -approximates the optimal egalitarian welfare (Section 3.3 of Mei et al. (2019)). In addition, they demonstrate that no deterministic and strategy proof mechanism has an approximation ratio of the optimal egalitarian welfare of less than (also Section 3.3 of Mei et al. (2019)).

We again cannot apply these results directly to our setting as the MidPoint mechanism may, for example, select an infeasible location for the facility. Instead, we consider the  mechanism which locates the facility at the nearest feasible location to the midpoint.

Theorem 12

For a facility location problem with limited locations, the  mechanism is strategy proof but does not bound the approximation ratio of the optimal egalitarian welfare.

Proof:  Suppose agents are in the interval , the only feasible locations for the facility are at 0 or , and we have a single agent at . The  mechanism locates the facility at since this is the nearest feasible location to the midpoint at . This gives an egalitarian welfare of zero units of utility. However, the optimal egalitarian welfare of 1 unit of utility requires the facility to be located at . We therefore have an unbounded approximation ratio.

We might hope perhaps to do better with the  mechanism which, unlike the  mechanism, is at least responsive to the location of the agents. In addition, the  mechanism performs relatively well on the related task of minimizing the maximum distance, providing a 3-approximation of the optimal. However, this is not the case. The  mechanism may approximate the optimal egalitarian welfare poorly.

Theorem 13

For a facility location problem with limited locations, the  mechanism does not bound the approximation ratio of the optimal egalitarian welfare.

Proof:  Suppose facilities can be at 0, or 1. If there are two agents at 0 and one at 1 then the  mechanism locates the facility at 0. The egalitarian welfare of this solution has zero units of utility. However, the optimal egalitarian welfare has unit of utility and is achieved by locating the facility at . The approximation ratio is therefore unbounded.

It is not hard to see that, with any other  mechanism, the approximation ratio of the optimal egalitarian welfare is also not bounded. It remains an interesting open question if there is any deterministic and strategy proof mechanism with a bounded approximation ratio.

Conclusions

We have studied the impact of constraints on the location of facilities on the performance of strategy proof mechanisms for locating facilities. We considered four different performance objectives: the total distance agents must travel to their closest facility, the maximum distance any agent must travel to their closest facility, and the utilitarian and egalitarian welfare. In general, constraining facilities to a limited set of locations makes all four objectives harder to approximate in general. For example, with one facility that can only be located at limited locations on the line, we proved that a modified median mechanism is strategy proof and 3-approximates both the optimal total distance and the optimal maximum distance. This is optimal since no deterministic and strategy proof mechanism can do better. This contrasts with the setting in which there are no restrictions on where facilities can be located and the median mechanism returns a solution which has the optimal total distance, and which 2-approximates the optimal maximum distance. With two or more facilities that can only be located at limited locations on the line, no mechanism is deterministic, anonymous, strategy proof and has a bounded approximation ratio of the optimal total or maximum distance. Again, this contrasts with the setting in which there are no restrictions on where facilities can be located and a mechanism exists that is deterministic, anonymous, strategy proof and has a bounded approximation ratio of the optimal total or maximum distance. Interestingly, we observed that results for approximating the utilitarian and egalitarian welfare could be very different to approximating the total and maximum distance. For instance, while the modified median mechanism 3-approximates the total distance, no deterministic and strategy proof mechanism has a bounded approximation ratio for the utilitarian welfare.

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