We study the k-center clustering problem under demographic fairness constraints, where the point set is partitioned into groups, and the aim is to compute clusters that exhibit a given group proportion. Previous work in this direction assumes that the entire point set already respects the desired proportions or uses relaxed notions of fairness. In this work, we propose a model that facilitates the creation of clusters that exactly match given target ratios, even when the input point set does not. We combine the well-known fair clustering model initiated by Chierichetti, Kumar, Lattanzi, and Vassilvitskii [Flavio Chierichetti et al., 2017] with the notion of outliers to obtain a practical combinatorial framework that provides constant-factor approximate solutions for all proportion settings from 1:1 for two groups to t₁:t₂:…:t_m for m ≥ 2 groups, where t₁,…,t_m are integers. We implement and evaluate our algorithms, compare different variants, and provide evidence of the practicability of this approach.

We consider the k-min-sum-radii (k-MSR) clustering problem with fairness constraints. The k-min-sum-radii problem is a mixture of the classical k-center and k-median problems. We are given a set of points P in a metric space and a number k and aim to partition the points into k clusters, each of the clusters having one designated center. The objective to minimize is the sum of the radii of the k clusters (where in k-center we would only consider the maximum radius and in k-median we would consider the sum of the individual points’ costs). Various notions of fair clustering have been introduced lately, and we follow the definitions due to Chierichetti et al. [13] which demand that cluster compositions shall follow the proportions of the input point set with respect to some given sensitive attribute. For the easier case where the sensitive attribute only has two possible values and each is equally frequent in the input, the aim is to compute a clustering where all clusters have a 1:1 ratio with respect to this attribute. We call this the 1:1 case. There has been a surge of FPT-approximation algorithms for the k-MSR problem lately, solving the problem both in the unconstrained case and in several constrained problem variants. We add to this research area by designing an FPT (6 + ϵ)-approximation that works for k-MSR under the mentioned general fairness notion. For the special 1:1 case, we improve our algorithm to achieve a (3 + ϵ)-approximation.

The metric $k$-center clustering problem with $z$ outliers, also known as $(k,z)$-center clustering, involves clustering a given point set $P$ in a metric space $(M,d)$ using at most $k$ balls, minimizing the maximum ball radius while excluding up to $z$ points from the clustering. This problem holds fundamental significance in various domains such as machine learning, data mining, and database systems. This paper addresses the fully dynamic version of the problem, where the point set undergoes continuous updates (insertions and deletions) over time. The objective is to maintain an approximate $(k,z)$-center clustering with efficient update times. We propose a novel fully dynamic algorithm that maintains a $(4+\epsilon)$-approximate solution to the $(k,z)$-center clustering problem that covers all but at most $(1+\epsilon)z$ points at any time in the sequence with probability $1-k/e^{\Omega(\log k)}$. The algorithm achieves an expected amortized update time of $\mathcal{O}(\epsilon^{-2} k^6\log(k) \log(\Delta))$, and is applicable to general metric spaces. Our dynamic algorithm presents a significant improvement over the recent dynamic $(14+\epsilon)$-approximation algorithm by Chan, Lattanzi, Sozio, and Wang for this problem.

Canonical models of Markov decision processes (MDPs) usually consider geometric discounting based on a constant discount factor. While this standard modeling approach has led to many elegant results, some recent studies indicate the necessity of modeling time-varying discounting in certain applications. This paper studies a model of infinite-horizon MDPs with time-varying discount factors. We take a game-theoretic perspective – whereby each time step is treated as an independent decision maker with their own (fixed) discount factor – and we study the subgame perfect equilibrium (SPE) of the resulting game as well as the related algorithmic problems. We present a constructive proof of the existence of an SPE and demonstrate the EXPTIME-hardness of computing an SPE. We also turn to the approximate notion of epsilon-SPE and show that an epsilon-SPE exists under milder assumptions. An algorithm is presented to compute an epsilon-SPE, of which an upper bound of the time complexity, as a function of the convergence property of the time-varying discount factor, is provided.

Given a point set $P\subseteq M$ from a metric space $(M,d)$ and numbers $k, z \in N$, the *metric $k$-center problem with $z$ outliers* is to find a set $C^\ast\subseteq P$ of $k$ points such that the maximum distance of all but at most $z$ outlier points of $P$ to their nearest center in ${C}^\ast$ is minimized. We consider this problem in the fully dynamic model, i.e., under insertions and deletions of points, for the case that the metric space has a bounded doubling dimension $dim$. We utilize a hierarchical data structure to maintain the points and their neighborhoods, which enables us to efficiently find the clusters. In particular, our data structure can be queried at any time to generate a $(3+\varepsilon)$-approximate solution for input values of $k$ and $z$ in worst-case query time $\varepsilon^{-O(dim)}k \log{n} \log\log{\Delta}$, where $\Delta$ is the ratio between the maximum and minimum distance between two points in $P$. Moreover, it allows insertion/deletion of a point in worst-case update time $\varepsilon^{-O(dim)}\log{n}\log{\Delta}$. Our result achieves a significantly faster query time with respect to $k$ and $z$ than the current state-of-the-art by Pellizzoni, Pietracaprina, and Pucci, which uses $\varepsilon^{-O(dim)}(k+z)^2\log{\Delta}$ query time to obtain a $(3+\varepsilon)$-approximation.