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1. Near-optimal Algorithms for Explainable k-Medians and k-Means

   Makarychev, Konstantin ; Shan, Liren ( July 2021 , International Conference on Machine Learning)

   null (Ed.)

   We consider the problem of explainable k-medians and k-means introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). In this problem, our goal is to find a threshold decision tree that partitions data into k clusters and minimizes the k-medians or k-means objective. The obtained clustering is easy to interpret because every decision node of a threshold tree splits data based on a single feature into two groups. We propose a new algorithm for this problem which is O(log k) competitive with k-medians with ℓ1 norm and O(k) competitive with k-means. This is an improvement over the previous guarantees of O(k) and O(k^2) by Dasgupta et al (2020). We also provide a new algorithm which is O(log^{3}{2}k) competitive for k-medians with ℓ2 norm. Our first algorithm is near-optimal: Dasgupta et al (2020) showed a lower bound of Ω(log k) for k-medians; in this work, we prove a lower bound of Ω(k) for k-means. We also provide a lower bound of Ω(log k) for k-medians with ℓ2 norm. 

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2. Explainable k-means: don’t be greedy, plant bigger trees!

   https://doi.org/10.1145/3519935.3520056  

   Makarychev, Konstantin ; Shan, Liren ( June 2022 , STOC 2022)

   We provide a new bi-criteria O(log2k) competitive algorithm for explainable k-means clustering. Explainable k-means was recently introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). It is described by an easy to interpret and understand (threshold) decision tree or diagram. The cost of the explainable k-means clustering equals to the sum of costs of its clusters; and the cost of each cluster equals the sum of squared distances from the points in the cluster to the center of that cluster. The best non bi-criteria algorithm for explainable clustering O(k) competitive, and this bound is tight. Our randomized bi-criteria algorithm constructs a threshold decision tree that partitions the data set into (1+δ)k clusters (where δ∈(0,1) is a parameter of the algorithm). The cost of this clustering is at most O(1/δ⋅log2k) times the cost of the optimal unconstrained k-means clustering. We show that this bound is almost optimal. 

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3. Online Hypergraph Matching with Delays

   Pavone, Marco ; Saberi, Amin. ; Schiffer, Maximilian ; Tsao, Matthew ( December 2020 , Conference on Web and Internet Economics)

   null (Ed.)

   We study an online hypergraph matching problem with delays, motivated by ridesharing applications. In this model, users enter a marketplace sequentially, and are willing to wait up to $d$ timesteps to be matched, after which they will leave the system in favor of an outside option. A platform can match groups of up to $k$ users together, indicating that they will share a ride. Each group of users yields a match value depending on how compatible they are with one another. As an example, in ridesharing, $k$ is the capacity of the service vehicles, and $d$ is the amount of time a user is willing to wait for a driver to be matched to them. We present results for both the utility maximization and cost minimization variants of the problem. In the utility maximization setting, the optimal competitive ratio is $\frac{1}{d}$ whenever $k \geq 3$, and is achievable in polynomial-time for any fixed $k$. In the cost minimization variation, when $k = 2$, the optimal competitive ratio for deterministic algorithms is $\frac{3}{2}$ and is achieved by a polynomial-time thresholding algorithm. When $k>2$, we show that a polynomial-time randomized batching algorithm is $(2 - \frac{1}{d}) \log k$-competitive, and it is NP-hard to achieve a competitive ratio better than $\log k - O (\log \log k)$. 

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4. Efficient Algorithms and Hardness Results for the Weighted k-Server Problem

   https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2023.12  

   Gupta, Anupam ; Kumar, Amit ; Panigrahi, Debmalya ( January 2023 , Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Megow, Nicole ; Smith, Adam (Ed.)

   In this paper, we study the weighted k-server problem on the uniform metric in both the offline and online settings. We start with the offline setting. In contrast to the (unweighted) k-server problem which has a polynomial-time solution using min-cost flows, there are strong computational lower bounds for the weighted k-server problem, even on the uniform metric. Specifically, we show that assuming the unique games conjecture, there are no polynomial-time algorithms with a sub-polynomial approximation factor, even if we use c-resource augmentation for c < 2. Furthermore, if we consider the natural LP relaxation of the problem, then obtaining a bounded integrality gap requires us to use at least 𝓁 resource augmentation, where 𝓁 is the number of distinct server weights. We complement these results by obtaining a constant-approximation algorithm via LP rounding, with a resource augmentation of (2+ε)𝓁 for any constant ε > 0. In the online setting, an exp(k) lower bound is known for the competitive ratio of any randomized algorithm for the weighted k-server problem on the uniform metric. In contrast, we show that 2𝓁-resource augmentation can bring the competitive ratio down by an exponential factor to only O(𝓁² log 𝓁). Our online algorithm uses the two-stage approach of first obtaining a fractional solution using the online primal-dual framework, and then rounding it online. 

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5. Online Spanners in Metric Spaces

   Bhore, Sujoy ; Filtser, Arnold ; Khodabandeh, Hadi ; Toth, Csaba D. ( September 2022 , Proc. 30th Annual European Symposium on Algorithms (ESA))

   Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric t-spanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a t-spanner G_i for S_i for all i, while minimizing the number of edges, and their total weight. Under the L₂-norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)-spanner algorithm with competitive ratio O_d(ε^{-d} log n), improving the previous bound of O_d(ε^{-(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1-d}log ε^{-1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1-d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{-3/2}logε^{-1}log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{-d}) lower bound for the competitive ratio for online (1+ε)-spanner algorithms in ℝ^d under the L₁-norm. Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k-1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{-1}logε^{-1})⋅ n^{1+1/k} edges and O(ε^{-1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)-spanner for ultrametrics with O(ε^{-1}logε^{-1})⋅ n edges and O(ε^{-2}) lightness. 

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