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We study the problem of fair kmedian where each cluster is required to have a fair representation of individuals from different groups. In the fair representation kmedian 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 kclustering 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_1objective ∑_{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,more »Free, publiclyaccessible full text available June 20, 2023

In the Correlation Clustering problem, we are given a complete weighted graph G with its edges labeled as “similar" and “dissimilar" by a noisy binary classifier. For a clustering C of graph G, a similar edge is in disagreement with C, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with C if its endpoints belong to the same cluster. The disagreements vector, Agree, is a vector indexed by the vertices of G such that the vth coordinate Disagre equals the weight of all disagreeing edges incident on v. The goal is to produce a clustering that minimizes the ℓp norm of the disagreements vector for p≥1. We study the ℓ_p objective in Correlation Clustering under the following assumption: Every similar edge has weight in [αw,w] and every dissimilar edge has weight at least αw (where α≤1 and w>0 is a scaling parameter). We give an O((1/α)^{1/2−1/2p}⋅log(1/α)) approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap.

Buchin, Kevin ; Colin de Verdiere, Eric (Ed.)In this paper, we prove a twosided variant of the Kirszbraun theorem. Consider an arbitrary subset X of Euclidean space and its superset Y. Let f be a 1Lipschitz map from X to ℝ^m. The Kirszbraun theorem states that the map f can be extended to a 1Lipschitz map ̃ f from Y to ℝ^m. While the extension ̃ f does not increase distances between points, there is no guarantee that it does not decrease distances significantly. In fact, ̃ f may even map distinct points to the same point (that is, it can infinitely decrease some distances). However, we prove that there exists a (1 + ε)Lipschitz outer extension f̃:Y → ℝ^{m'} that does not decrease distances more than "necessary". Namely, ‖f̃(x)  f̃(y)‖ ≥ c √{ε} min(‖xy‖, inf_{a,b ∈ X} (‖x  a‖ + ‖f(a)  f(b)‖ + ‖by‖)) for some absolutely constant c > 0. This bound is asymptotically optimal, since no LLipschitz extension g can have ‖g(x)  g(y)‖ > L min(‖xy‖, inf_{a,b ∈ X} (‖x  a‖ + ‖f(a)  f(b)‖ + ‖by‖)) even for a single pair of points x and y. In some applications, one is interested in the distances ‖f̃(x)  f̃(y)‖more »

Belkin, Mikhail ; Kpotufe, Samor (Ed.)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).

In the Correlation Clustering problem, we are given a complete weighted graph G with its edges labeled as “similar" and “dissimilar" by a noisy binary classifier. For a clustering C of graph G, a similar edge is in disagreement with C, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with C if its endpoints belong to the same cluster. The disagreements vector is a vector indexed by the vertices of G such that the vth coordinate of the disagreements vector equals the weight of all disagreeing edges incident on v. The goal is to produce a clustering that minimizes the ℓp norm of the disagreements vector for p≥1. We study the ℓ_p objective in Correlation Clustering under the following assumption: Every similar edge has weight in [αw,w] and every dissimilar edge has weight at least αw (where α ≤ 1 and w > 0 is a scaling parameter). We give an O((1/α)^{1/2−1/(2p)} log 1/α) approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap.

Meila, Marina ; Zhang, Tong (Ed.)In the Correlation Clustering problem, we are given a complete weighted graph $G$ with its edges labeled as “similar" and “dissimilar" by a noisy binary classifier. For a clustering $\mathcal{C}$ of graph $G$, a similar edge is in disagreement with $\mathcal{C}$, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with $\mathcal{C}$ if its endpoints belong to the same cluster. The disagreements vector, $\mathbf{disagree}$, is a vector indexed by the vertices of $G$ such that the $v$th coordinate $\mathbf{disagree}_v$ equals the weight of all disagreeing edges incident on $v$. The goal is to produce a clustering that minimizes the $\ell_p$ norm of the disagreements vector for $p\geq 1$. We study the $\ell_p$ objective in Correlation Clustering under the following assumption: Every similar edge has weight in $[\alpha\mathbf{w},\mathbf{w}]$ and every dissimilar edge has weight at least $\alpha\mathbf{w}$ (where $\alpha \leq 1$ and $\mathbf{w}>0$ is a scaling parameter). We give an $O\left((\frac{1}{\alpha})^{\frac{1}{2}\frac{1}{2p}}\cdot \log\frac{1}{\alpha}\right)$ approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap.

In this paper, we introduce the notion of a certified algorithm. Certified algorithms provide worstcase and beyondworstcase performance guarantees. First, a γcertified algorithm is also a γapproximation algorithm  it finds a γapproximation no matter what the input is. Second, it exactly solves γperturbationresilient instances (γperturbationresilient instances model reallife instances). Additionally, certified algorithms have a number of other desirable properties: they solve both maximization and minimization versions of a problem (e.g. Max Cut and Min Uncut), solve weakly perturbationresilient instances, and solve optimization problems with hard constraints. In the paper, we define certified algorithms, describe their properties, present a framework for designing certified algorithms, provide examples of certified algorithms for Max Cut/Min Uncut, Minimum Multiway Cut, kmedians and kmeans. We also present some negative results.

In the Correlation Clustering problem, we are given a weighted graph $G$ with its edges labelled as "similar" or "dissimilar" by a binary classifier. The goal is to produce a clustering that minimizes the weight of "disagreements": the sum of the weights of "similar" edges across clusters and "dissimilar" edges within clusters. We study the correlation clustering problem under the following assumption: Every "similar" edge $e$ has weight $w_e \in [ \alpha w, w ]$ and every "dissimilar" edge $e$ has weight $w_e \geq \alpha w$ (where $\alpha \leq 1$ and $w > 0$ is a scaling parameter). We give a $(3 + 2 \log_e (1/\alpha))$ approximation algorithm for this problem. This assumption captures well the scenario when classification errors are asymmetric. Additionally, we show an asymptotically matching Linear Programming integrality gap of $\Omega(\log 1/\alpha)$

Consider an instance of Euclidean kmeans or kmedians clustering. We show that the cost of the optimal solution is preserved up to a factor of (1+ε) under a projection onto a random O(log(k /ε) / ε2)dimensional subspace. Further, the cost of every clustering is preserved within (1+ε). More generally, our result applies to any dimension reduction map satisfying a mild subGaussiantail condition. Our bound on the dimension is nearly optimal. Additionally, our result applies to Euclidean kclustering with the distances raised to the pth power for any constant p. For kmeans, our result resolves an open problem posed by Cohen, Elder, Musco, Musco, and Persu (STOC 2015); for kmedians, it answers a question raised by Kannan.