Facility Location with Fair Outliers

• Facility Location with Fair Outliers is a robust optimization problem that balances service cost with fairness by enforcing group-specific and individual outlier constraints.
• It leverages mathematical formulations such as LP relaxations, thresholding, and renormalization to translate fairness requirements into tractable facility assignment models.
• Bicriteria approximation and local search methods are employed to ensure practical cost efficiency while meeting relaxed fairness guarantees in both group-wise and individual settings.

Facility location with fair outliers denotes a family of robust service-allocation problems in which some clients may be excluded from service, but the exclusion mechanism itself is constrained by fairness. In the current literature, fairness appears in two distinct forms. The first is group-wise fairness: clients are partitioned into groups, each group has its own outlier budget, and the objective is classical facility-opening plus assignment cost, or assignment cost alone under a $k$-facility bound (Dabas et al., 4 Aug 2025). The second is individual fairness: every retained point must have a center within a locally defined neighborhood radius, while a bounded set of points may be discarded as outliers (Han et al., 2022). LP-based and local-search-based formulations for fair $k$-means and $k$-median with outliers extend this second view to cost-sensitive clustering (Maity et al., 2024).

1. Formal problem family

In the group-wise formulation, classical Facility Location is specified by a metric space $(\mathcal M,d)$, a facility set $\mathcal F\subseteq \mathcal M$, a client set $C\subseteq \mathcal M$, and opening costs $f_i\ge 0$ for $i\in\mathcal F$. One opens a subset $F\subseteq \mathcal F$ and assigns each client to its nearest open facility, minimizing

$$\sum_{i\in F} f_i+\sum_{j\in C} d_{j,F}, \qquad d_{j,F}=\min_{i\in F} d(i,j).$$

Facility Location with Fair Outliers (FLFO) extends this by partitioning clients into $k$0 disjoint groups

$k$1

and associating to each group $k$2 an outlier budget $k$3. The algorithm chooses opened facilities $k$4 and, for each group, an outlier set $k$5 with $k$6, minimizing

$k$7

Equivalently, at least $k$8 clients from each group must be served. The model in this line of work uses only disjoint groups and does not introduce overlapping groups or multiple protected attributes per client beyond that partition model (Dabas et al., 4 Aug 2025).

The corresponding $k$9-median formulation replaces opening costs by a cardinality constraint $k$0. With the same groups and outlier budgets, the objective becomes

$k$1

This yields a common template: serve clients cheaply while controlling outliers per group (Dabas et al., 4 Aug 2025).

A different fairness line studies outliers through local service entitlements rather than group quotas. In individually fair $k$2-center with outliers, one may choose at most $k$3 centers and at most $k$4 outliers, and each served vertex $k$5 is evaluated against its outlier-related neighborhood radius $k$6, defined as the distance to its $k$7-th nearest neighbor. The objective is to minimize the maximum fairness ratio

$k$8

Here fairness is not demographic; it is individual and density-sensitive (Han et al., 2022).

LP-based individually fair $k$9-means and $(\mathcal M,d)$0-median with outliers use the same general idea. Given an outlier set $(\mathcal M,d)$1 with $(\mathcal M,d)$2, every non-outlier $(\mathcal M,d)$3 must be assigned to a center within a client-specific radius $(\mathcal M,d)$4, where $(\mathcal M,d)$5 is the distance to the $(\mathcal M,d)$6-th nearest neighbor. The optimization then minimizes $(\mathcal M,d)$7 over non-outliers, with $(\mathcal M,d)$8 for $(\mathcal M,d)$9-means and $\mathcal F\subseteq \mathcal M$0 for $\mathcal F\subseteq \mathcal M$1-median (Maity et al., 2024).

2. Mathematical formulations and structural obstacles

For FLFO, a natural integer program uses binary variables $\mathcal F\subseteq \mathcal M$2 for opening facility $\mathcal F\subseteq \mathcal M$3, $\mathcal F\subseteq \mathcal M$4 for assigning client $\mathcal F\subseteq \mathcal M$5 to facility $\mathcal F\subseteq \mathcal M$6, and $\mathcal F\subseteq \mathcal M$7 for declaring client $\mathcal F\subseteq \mathcal M$8 an outlier. The formulation is

$\mathcal F\subseteq \mathcal M$9

subject to

$C\subseteq \mathcal M$0

$C\subseteq \mathcal M$1

$C\subseteq \mathcal M$2

with integrality relaxed to $C\subseteq \mathcal M$3 in the LP relaxation (Dabas et al., 4 Aug 2025).

This LP is natural but structurally weak. Even when $C\subseteq \mathcal M$4, it has an unbounded integrality gap. The canonical example has one facility of opening cost $C\subseteq \mathcal M$5, $C\subseteq \mathcal M$6 co-located clients, and outlier budget $C\subseteq \mathcal M$7. Fractionally, the LP can open the facility to extent $C\subseteq \mathcal M$8 and serve every client fractionally at total cost $C\subseteq \mathcal M$9. Any integral solution that serves one client must open the facility fully and pay $f_i\ge 0$0. As $f_i\ge 0$1, the gap is unbounded. This eliminates straightforward exact-budget LP-rounding as a generic approach for FLFO (Dabas et al., 4 Aug 2025).

The individual-fairness line inherits a different structural difficulty from the neighborhood-radius benchmark itself. In the no-outlier setting, “A Center in Your Neighborhood” defines

$f_i\ge 0$2

and optimizes

$f_i\ge 0$3

That problem admits a universal factor-2 algorithm in arbitrary metrics, but deciding whether $f_i\ge 0$4-fairness is achievable is NP-complete (Jung et al., 2019). The outlier-aware individual-fairness models therefore start from an already nontrivial fairness geometry.

Taken together, these results explain why fair-outlier algorithms frequently become bicriteria. Exact compliance with outlier budgets or exact fairness radii interacts poorly with the natural LPs and with the combinatorial geometry of local service constraints. This suggests that controlled relaxation of either cost, fairness, or outlier counts is not merely an implementation artifact, but a recurring structural response.

3. Group-wise facility location and $f_i\ge 0$5-median with fair outliers

The main current group-wise result is a bicriteria approximation for FLFO. For any fixed $f_i\ge 0$6, there is a polynomial-time $f_i\ge 0$7-approximation algorithm that violates each group’s outlier budget by at most a factor $f_i\ge 0$8. The starting point is the LP optimum $f_i\ge 0$9. Clients with $i\in\mathcal F$0 are placed into

$i\in\mathcal F$1

and are committed as integral outliers. The remaining clients form $i\in\mathcal F$2. This thresholding yields the key fairness inequality

$i\in\mathcal F$3

while not increasing LP cost (Dabas et al., 4 Aug 2025).

The thresholding step is then followed by renormalization. Every client $i\in\mathcal F$4 has residual assignment mass at least $i\in\mathcal F$5, so assignments can be rescaled to obtain a feasible fractional solution to the standard Facility Location LP on the reduced client set. The resulting cost increases by at most $i\in\mathcal F$6. Any $i\in\mathcal F$7-approximation for ordinary Facility Location therefore yields an $i\in\mathcal F$8-approximation for FLFO, with the same $i\in\mathcal F$9 factor violation in every group’s outlier budget. Combining this with a constant-factor approximation for classical Facility Location gives the stated $F\subseteq \mathcal F$0 bicriteria guarantee (Dabas et al., 4 Aug 2025).

The same paper develops a parallel bicriteria result for $F\subseteq \mathcal F$1-Median with Fair Outliers. There is a polynomial-time

$F\subseteq \mathcal F$2

factor approximation with $F\subseteq \mathcal F$3 violation in outliers per group. Here the technique is different: the reduction is to $F\subseteq \mathcal F$4-Median with Penalties, with penalties set group-wise as

$F\subseteq \mathcal F$5

This improves earlier fair $F\subseteq \mathcal F$6-median work by removing dependence on $F\subseteq \mathcal F$7 from the outlier-budget violation; the prior result of Almanza et al. allowed a $F\subseteq \mathcal F$8-factor outlier violation per group (Dabas et al., 4 Aug 2025).

The group-wise line is also shaped by practicality. Earlier FLFO results of Inamdar–Varadarajan and Bajpai et al. give proper approximations, but rely on exponentially large LPs solved through ellipsoid-based machinery. The compact-LP bicriteria formulation changes the guarantee—small multiplicative violation in each group’s outlier budget replaces exact compliance—but it is polynomial-time and directly implementable (Dabas et al., 4 Aug 2025).

4. Individual fairness with outliers

The individually fair $F\subseteq \mathcal F$9-center with outliers problem, IF$$\sum_{i\in F} f_i+\sum_{j\in C} d_{j,F}, \qquad d_{j,F}=\min_{i\in F} d(i,j).$$0CO, is a clean min-max formulation. A feasible solution chooses centers $$\sum_{i\in F} f_i+\sum_{j\in C} d_{j,F}, \qquad d_{j,F}=\min_{i\in F} d(i,j).$$1 with $$\sum_{i\in F} f_i+\sum_{j\in C} d_{j,F}, \qquad d_{j,F}=\min_{i\in F} d(i,j).$$2, outliers $$\sum_{i\in F} f_i+\sum_{j\in C} d_{j,F}, \qquad d_{j,F}=\min_{i\in F} d(i,j).$$3 with $$\sum_{i\in F} f_i+\sum_{j\in C} d_{j,F}, \qquad d_{j,F}=\min_{i\in F} d(i,j).$$4, and an assignment $$\sum_{i\in F} f_i+\sum_{j\in C} d_{j,F}, \qquad d_{j,F}=\min_{i\in F} d(i,j).$$5. The fairness ratio of a served vertex $$\sum_{i\in F} f_i+\sum_{j\in C} d_{j,F}, \qquad d_{j,F}=\min_{i\in F} d(i,j).$$6 is

$$\sum_{i\in F} f_i+\sum_{j\in C} d_{j,F}, \qquad d_{j,F}=\min_{i\in F} d(i,j).$$7

and the objective is to minimize the maximum such ratio over served vertices. A greedy algorithm that repeatedly selects the point of minimum $$\sum_{i\in F} f_i+\sum_{j\in C} d_{j,F}, \qquad d_{j,F}=\min_{i\in F} d(i,j).$$8 and removes all points within distance $$\sum_{i\in F} f_i+\sum_{j\in C} d_{j,F}, \qquad d_{j,F}=\min_{i\in F} d(i,j).$$9 is proved to be a 4-approximation. A refined practical variant performs binary search over a parameter $k$00 and retains the same 4-approximation guarantee while often achieving smaller empirical ratios. The center and outlier budgets are respected exactly; there is no bicriteria violation in $k$01 or $k$02 (Han et al., 2022).

A different LP-based line targets individually fair $k$03-means and $k$04-median with outliers. The LP uses assignment variables $k$05, opening variables $k$06, and outlier indicators $k$07, with the service-access constraint

$k$08

After solving the LP with $k$09, the procedure OutRound thresholds outliers, deletes any center mass placed on points designated as outliers, reroutes their assignments to the nearest surviving non-outlier, and then invokes an existing fair-rounding subroutine on the inliers. The key rerouting lemma is

$k$10

which yields

$k$11

The resulting guarantees are a 12-approximation for fair $k$12-means with outliers, a 24-approximation for fair $k$13-median with outliers, and a 16-approximation to the fair radius for inliers. The important caveat is that the final number of detected outliers is not theoretically bounded by $k$14, even though the LP satisfies $k$15 fractionally (Maity et al., 2024).

More recently, local search has been adapted to individually fair clustering with outliers through the anchor-zone framework. The algorithm first constructs fairness-aware anchor zones, discards an initial set of fairness-based outliers, and then performs local search while maintaining the invariant that every anchor zone contains at least one center. The main theorem gives an $k$16-approximation for fixed $k$17, and the analysis bounds the total number of discarded outliers by

$k$18

where $k$19 is the aspect ratio. The appendix states an explicit constant of about $k$20 for the $k$21-means objective, so the guarantee is again bicriteria in practice and in theory (Maity et al., 7 Oct 2025).

These individual-fairness papers show that “fair outliers” need not mean demographic parity in exclusion. It can also mean that once outliers are removed, every retained client must still receive service compatible with a local density benchmark.

5. Complexity and relation to adjacent facility-location models

The group-wise FLFO landscape includes a parameterized hardness barrier. Assuming ETH, Facility Location with Fair Outliers is W[1]-hard when parameterized by the number of groups $k$22, and the same holds for $k$23-Median with Fair Outliers. The reduction proceeds from a multidimensional subset-sum type problem, with the dimension becoming the number of groups. Consequently, one should not expect algorithms of running time $k$24 for arbitrary $k$25 unless ETH fails (Dabas et al., 4 Aug 2025).

The surrounding robust-facility-location literature clarifies what is specific to fairness and what is inherited from outliers alone.

Setting	Representative guarantee	Relevance
UFL/$k$26-Median/$k$27-Means with outliers	PTAS for uniform-cost UFL with outliers on doubling and minor-closed metrics; bicriteria PTAS for $k$28-Median/$k$29-Means there; $k$30 for general-metric $k$31-Means with outliers (Friggstad et al., 2017)	Robust baseline without fairness
CFLPO/C$k$32FLPP	Constant-factor LP-rounding with slight capacity and outlier or cardinality violations (Dabas et al., 2020)	Shows how capacities and outliers interact
$k$33FLO and LBFLO	$k$34-approximation for $k$35FLO with at most $k$36 facilities; tri-criteria tradeoff for LBFLO (Dabas et al., 2021)	Global outlier budgets, but no fair allocation of outliers
Capacitated $k$37-Facility Location with Outliers; $k$38-fair clustering with outliers	FPT reduction framework; $k$39 for capacitated $k$40-facility location with outliers in arbitrary metrics; fair $k$41-facility location with outliers is stated as an extension of the same framework (Dabas et al., 2023)	Closest bridge between capacities, fairness, and outliers

The fair-outlier topic also borders group-fair facility-location models without outliers. Strategyproof single-facility location on the line has been studied under objectives such as maximum total group cost, maximum average group cost, and intergroup/intragroup fairness, but those models do not include an outlier budget or partial service (Zhou et al., 2021). They provide fairness objectives and impossibility patterns that are relevant conceptually, but not direct algorithms for fair outlier control.

A broad lesson from these adjacent models is that robust facility location already becomes bicriteria once capacities, lower bounds, or cardinality limits are added. This suggests that fair-outlier facility location inherits not one source of approximation loss, but several: robustness, fairness, and classical facility-opening structure.

6. Empirical behavior and practical implications

The FLFO experimental results are unusually strong relative to the worst-case theory. On the Adult dataset grouped by sex, the Bank dataset grouped by marital status, and a synthetic two-group dataset, the fair facility-location methods LPR-F and GDF-F achieve unfairness essentially equal to $k$42 in almost all cases; the only reported exception is the Bank dataset, where the maximum unfairness is $k$43. By contrast, the non-fair baselines can be much more skewed: on Adult, unfairness reaches $k$44 for LPR-NF and $k$45 for GDF-NF; on Bank, $k$46 and $k$47; on the synthetic data, as high as $k$48 and $k$49. The cost increase from fairness is reported as negligible: LPR-F remains close to the LP optimum, and GDF-F has cost comparable to the classical greedy baseline. The same section also reports that the standard LP, despite its unbounded worst-case gap, is usually nearly integral in practice (Dabas et al., 4 Aug 2025).

The paper’s $k$50-median experiments show a similar pattern. Using 500-point samples with $k$51, the fair method R+LS-F attains unfairness close to $k$52, while non-fair alternatives can reach $k$53 and $k$54 on Adult, $k$55 and $k$56 on Bank, and $k$57 on the synthetic data. Fairness increases cost somewhat, but not prohibitively (Dabas et al., 4 Aug 2025).

For individually fair $k$58-center with outliers, synthetic experiments report a maximum outlier-related fairness ratio of only $k$59, far below the proven upper bound of $k$60. On the Shenzhen POI dataset, the outlier-aware fair algorithm tends to place centers in dense areas, whereas the naive non-outlier-aware baseline does so less effectively (Han et al., 2022).

The LP-based individually fair $k$61-means/$k$62-median with outliers framework evaluates on sampled UCI datasets with injected outliers and compares against an Isolation Forest plus FairRound baseline. The reported behavior is that optimization-aware outlier identification yields lower cost and smaller maximum fair radius than the baseline, but the number of detected outliers can substantially exceed the nominal budget, empirically confirming the theoretical caveat of uncontrolled final outlier count (Maity et al., 2024).

The local-search-based individually fair clustering method also reports strong empirical performance. On Adult, Bank, and Skin, it achieves lower $k$63-means cost than the listed baselines, while maintaining reported fairness ratios such as $k$64, $k$65, and $k$66. The observed discarded-outlier counts—495, 707, and 14000 in the reported runs—again illustrate that practical success is accompanied by bicriteria outlier behavior rather than exact-budget compliance (Maity et al., 7 Oct 2025).

Across these results, one repeated empirical pattern is clear: fairness constraints often change cost far less than unconstrained robust-clustering intuition would suggest, whereas the main practical tension is more often between exact budget compliance and algorithmic tractability than between fairness and objective value itself.