Designing algorithms for machine learning problems targeting beyond worst-case analysis and, in particular, analyzing the effect of side-information on the complexity of such problems is a very important line of research with many practical applications. In this paper we study the classic k-means clustering problem in the presence of noisy labels. In this problem, in addition to a set of points and parameter \(k\), we receive cluster labels of each point generated by either an adversarial or a random perturbation of the optimal solution. Our main goal is to formally study the effect of this extra information on the complexity of the k-means problem. In particular, in the context of random perturbations, we give an efficient algorithm that finds a clustering of cost within a factor $1+o(1)$ of the optimum even when the label of each point is perturbed with a large probability (think 99%). In contrast, we show that the side-information with adversarial perturbations is as hard as the original problem even if only a small $\epsilon$ fraction of the labels are perturbed. We complement this negative result by giving a simple algorithm in the case when the adversary is only allowed to perturb an $\epsilon$ fraction of the labels per \emph{each cluster}.

We study the problem of explainable clustering in the setting first formalized by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). A $k$-clustering is said to be explainable if it is given by a decision tree where each internal node splits data points with a threshold cut in a single dimension (feature), and each of the $k$ leaves corresponds to a cluster. We give an algorithm that outputs an explainable clustering that loses at most a factor of $O(\log^2 k)$ compared to an optimal (not necessarily explainable) clustering for the $k$-medians objective, and a factor of $O(k \log^2 k)$ for the $k$-means objective. This improves over the previous best upper bounds of $O(k)$ and $O(k^2)$, respectively, and nearly matches the previous $\Omega(\log k)$ lower bound for $k$-medians and our new $\Omega(k)$ lower bound for $k$-means. The algorithm is remarkably simple. In particular, given an initial not necessarily explainable clustering in $\mathbb{R}^d$, it is oblivious to the data points and runs in time $O(dk \log^2 k)$, independent of the number of data points $n$. Our upper and lower bounds also generalize to objectives given by higher $\ell_p$-norms.

Given a metric space $(F \cup C, d)$, we consider star covers of $C$ with balanced loads. A star is a pair $(f, C_f)$ where $f \in F$ and $C_f \subseteq C$, and the load of a star is $\sum_{c \in C_f} d(f, c)$. In minimum load $k$-star cover problem $(\mathrm{MLkSC})$, one tries to cover the set of clients $C$ using $k$ stars that minimize the maximum load of a star, and in minimum size star cover $(\mathrm{MSSC})$ one aims to find the minimum number of stars of load at most $T$ needed to cover $C$, where $T$ is a given parameter. We obtain new bicriteria approximations for the two problems using novel rounding algorithms for their standard LP relaxations. For $\mathrm{MLkSC}$, we find a star cover with $(1+\varepsilon)k$ stars and $O(1/\varepsilon^2)\mathrm{OPT}_{\mathrm{MLk}}$ load where $\mathrm{OPT}_{\mathrm{MLk}}$ is the optimum load. For $\mathrm{MSSC}$, we find a star cover with $O(1/\varepsilon^2) \mathrm{OPT}_{\mathrm{MS}}$ stars of load at most $(2 + \varepsilon) T$ where $\mathrm{OPT}_{\mathrm{MS}}$ is the optimal number of stars for the problem. Previously, non-trivial bicriteria approximations were known only when $F = C$.

The online matching problem was introduced by Karp, Vazirani and Vazirani nearly three decades ago. In that seminal work, they studied this problem in bipartite graphs with vertices arriving only on one side, and presented optimal deterministic and randomized algorithms for this setting. In comparison, more general arrival models, such as edge arrivals and general vertex arrivals, have proven more challenging and positive results are known only for various relaxations of the problem. In particular, even the basic question of whether randomization allows one to beat the trivially-optimal deterministic competitive ratio of 1/2 for either of these models was open. In this paper, we resolve this question for both these natural arrival models, and show the following.1) For edge arrivals, randomization does not help - no randomized algorithm is better than 1/2 competitive. 2)For general vertex arrivals, randomization helps - there exists a randomized (1/2+Ω(1)) -competitive online matching algorithm.

We design a generic method to reduce the task of finding weighted matchings to that of finding short augmenting paths in unweighted graphs. This method enables us to provide efficient implementations for approximating weighted matchings in the massively parallel computation (MPC) model and in the streaming model.

We consider the maximum bipartite matching problem in stochastic settings, namely the query-commit and price-of-information models. In the query-commit model, an edge e independently exists with probability pe. We can query whether an edge exists or not, but if it does exist, then we have to take it into our solution. In the unweighted case, one can query edges in the order given by the classical online algorithm of Karp, Vazirani, and Vazirani [20] to get a (1 – 1/e)-approximation. In contrast, the previously best known algorithm in the weighted case is the (1/2)-approximation achieved by the greedy algorithm that sorts the edges according to their weights and queries in that order. Improving upon the basic greedy, we give a (1 – 1/e)-approximation algorithm in the weighted query-commit model. We use a linear program (LP) to upper bound the optimum achieved by any strategy. The proposed LP admits several structural properties that play a crucial role in the design and analysis of our algorithm. We also extend these techniques to get a (1 – 1/e)-approximation algorithm for maximum bipartite matching in the price-of-information model introduced by Singla [25], who also used the basic greedy algorithm to give a (1/2)-approximation.

We study $k$-means clustering in a semi-supervised setting. Given an oracle that returns whether two given points belong to the same cluster in a fixed optimal clustering, we investigate the following question: how many oracle queries are sufficient to efficiently recover a clustering that, with probability at least $(1 - δ)$, simultaneously has a cost of at most $(1 + ε)$ times the optimal cost and an accuracy of at least $(1 - ε)$? We show how to achieve such a clustering on $n$ points with $O{((k^2 \log n) \cdot m{(Q, ε^4, δ/ (k\log n))})}$ oracle queries, when the $k$ clusters can be learned with an $ε'$ error and a failure probability $δ'$ using $m(Q, ε',δ')$ labeled samples in the supervised setting, where $Q$ is the set of candidate cluster centers. We show that $m(Q, ε', δ')$ is small both for $k$-means instances in Euclidean space and for those in finite metric spaces. We further show that, for the Euclidean $k$-means instances, we can avoid the dependency on $n$ in the query complexity at the expense of an increased dependency on $k$: specifically, we give a slightly more involved algorithm that uses $O(k^4/(ε^2 δ) + (k^{9}/ε^4) \log(1/δ) + k \cdot m(\mathbb{R}^r, ε^4/k, δ))$ oracle queries. We also show that the number of queries needed for $(1 - ε)$-accuracy in Euclidean $k$-means must linearly depend on the dimension of the underlying Euclidean space, and for finite metric space $k$-means, we show that it must at least be logarithmic in the number of candidate centers. This shows that our query complexities capture the right dependencies on the respective parameters.