We study the problem of fair k-median where each cluster is required to have a fair representation of individuals from different groups. In the fair representation k-median problem, we are given a set of points X in a metric space. Each point x ∈ X belongs to one of ℓ groups. Further, we are given fair representation parameters αj and β_j for each group j ∈ [ℓ]. We say that a k-clustering C_1, ⋅⋅⋅, C_k fairly represents all groups if the number of points from group j in cluster C_i is between α_j |C_i| and β_j |C_i| for every j ∈ [ℓ] and i ∈ [k]. The goal is to find a set of k centers and an assignment such that the clustering defined by fairly represents all groups and minimizes the ℓ_1-objective ∑_{x ∈ X} d(x, ϕ(x)).
We present an O(log k)-approximation algorithm that runs in time n^{O(ℓ)}. Note that the known algorithms for the problem either (i) violate the fairness constraints by an additive term or (ii) run in time that is exponential in both k and ℓ. We also consider an important special case of the problem where and for all j ∈ [ℓ]. For this special case, we present an O(log k)-approximation algorithm that runs in time.
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Approximation Algorithms for Socially Fair Clustering
We present an $e^{O(p)} (\log \ell) / (\log \log \ell)$-approximation algorithm for socially fair clustering with the $\ell_p$-objective. In this problem, we are given a set of points in a metric space. Each point belongs to one (or several) of $\ell$ groups. The goal is to find a $k$-medians, $k$-means, or, more generally, $\ell_p$-clustering that is simultaneously good for all of the groups. More precisely, we need to find a set of $k$ centers $C$ so as to minimize the maximum over all groups $j$ of $\sum_{u \text{ in group } j} d(u, C)^p$. The socially fair clustering problem was independently proposed by Abbasi, Bhaskara, and Venkatasubramanian (2021) and Ghadiri, Samadi, and Vempala (2021). Our algorithm improves and generalizes their $O(\ell)$-approximation algorithms for the problem. The natural LP relaxation for the problem has an integrality gap of $\Omega(\ell)$. In order to obtain our result, we introduce a strengthened LP relaxation and show that it has an integrality gap of $\Theta((\log \ell) / (\log \log \ell))$ for a fixed p. Additionally, we present a bicriteria approximation algorithm, which generalizes the bicriteria approximation of Abbasi et al. (2021).
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- NSF-PAR ID:
- 10336944
- Editor(s):
- Belkin, Mikhail; Kpotufe, Samor
- Date Published:
- Journal Name:
- Proceedings of the Conference on Learning Theory, PMLR
- Volume:
- 134
- Page Range / eLocation ID:
- 3246-3264
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract We consider the problem of covering multiple submodular constraints. Given a finite ground set
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null (Ed.)Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph G = (V, E) with edge costs c ∈ ℝ_{≥ 0}^E, a root r ∈ V and k terminals K ⊆ V, we need to output a minimum-cost arborescence in G that contains an rrightarrow t path for every t ∈ K. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasi-polynomial time O(log²k/log log k)-approximation algorithms for the problem, which are tight under popular complexity assumptions. In this paper, we consider the more general Degree-Bounded Directed Steiner Tree (DB-DST) problem, where we are additionally given a degree bound d_v on each vertex v ∈ V, and we require that every vertex v in the output tree has at most d_v children. We give a quasi-polynomial time (O(log n log k), O(log² n))-bicriteria approximation: The algorithm produces a solution with cost at most O(log nlog k) times the cost of the optimum solution that violates the degree constraints by at most a factor of O(log²n). This is the first non-trivial result for the problem. While our cost-guarantee is nearly optimal, the degree violation factor of O(log²n) is an O(log n)-factor away from the approximation lower bound of Ω(log n) from the Set Cover hardness. The hardness result holds even on the special case of the Degree-Bounded Group Steiner Tree problem on trees (DB-GST-T). With the hope of closing the gap, we study the question of whether the degree violation factor can be made tight for this special case. We answer the question in the affirmative by giving an (O(log nlog k), O(log n))-bicriteria approximation algorithm for DB-GST-T.more » « less
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Given a set of facilities and clients, and costs to open facilities, the classic facility location problem seeks to open a set of facilities and assign each client to one open facility to minimize the cost of opening the chosen facilities and the total distance of the clients to their assigned open facilities. Such an objective may induce an unequal cost over certain socioeconomic groups of clients (i.e., total distance traveled by clients in such a group). This is important when planning the location of socially relevant facilities such as emergency rooms. In this work, we consider a fair version of the problem where we are given 𝑟 clients groups that partition the set of clients, and the distance of a given group is defined as the average distance of clients in the group to their respective open facilities. The objective is to minimize the Minkowski 𝑝-norm of vector of group distances, to penalize high access costs to open facilities across 𝑟 groups of clients. This generalizes classic facility location (𝑝 = 1) and the minimization of the maximum group distance (𝑝 = ∞). However, in practice, fairness criteria may not be explicit or even known to a decision maker, and it is often unclear how to select a specific "𝑝" to model the cost of unfairness. To get around this, we study the notion of solution portfolios where for a fixed problem instance, we seek a small portfolio of solutions such that for any Minkowski norm 𝑝, one of these solutions is an 𝑂(1)-approximation. Using the geometric relationship between various 𝑝-norms, we show the existence of a portfolio of cardinality 𝑂(log 𝑟), and a lower bound of (\sqrt{log r}). There may not be common structure across different solutions in this portfolio, which can make planning difficult if the notion of fairness changes over time or if the budget to open facilities is disbursed over time. For example, small changes in 𝑝 could lead to a completely different set of open facilities in the portfolio. Inspired by this, we introduce the notion of refinement, which is a family of solutions for each 𝑝-norm satisfying a combinatorial property. This property requires that (1) the set of facilities open for a higher 𝑝-norm must be a subset of the facilities open for a lower 𝑝-norm, and (2) all clients assigned to an open facility for a lower 𝑝-norm must be assigned to the same open facility for any higher 𝑝-norm. A refinement is 𝛼-approximate if the solution for each 𝑝-norm problem is an 𝛼-approximation for it. We show that it is sufficient to consider only 𝑂(log 𝑟) norms instead of all 𝑝-norms, 𝑝 ∈ [1, ∞] to construct refinements. A natural greedy algorithm for the problem gives a poly(𝑟)-approximate refinement, which we improve to poly(r^1/\sqrt{log 𝑟})-approximate using a recursive algorithm. We improve this ratio to 𝑂(log 𝑟) for the special case of tree metric for uniform facility open cost. Our recursive algorithm extends to other settings, including to a hierarchical facility location problem that models facility location problems at several levels, such as public works departments and schools. A full version of this paper can be found at https://arxiv.org/abs/2211.14873.more » « less
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