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Weronika Wrzos-Kaminska

Publications and source records attributed to Weronika Wrzos-Kaminska.

8 recordsLinked to original sources
arXiv2026

Recovering Planted Colorings in Sublinear Time

We give a sublinear algorithm for the planted $k$-coloring problem. Given an expander $G$ with a planted coloring, the goal is to efficiently determine the color class of a given vertex. We work in the adversarial planted coloring model of David and Feige [STOC 2016], where an adversary chooses a $d$-regular spectral $λ$-expander $G$ on $n$ vertices and plants a balanced $k$-coloring by partitioning the vertices into $k$ equal parts and deleting all edges within each part. This model generalizes the earlier random graph models studied by Blum and Spencer [J. Algorithms 1995] and Alon and Kahale [STOC 1994]. We give the first sublinear-time algorithm for recovering planted colorings in this model. The algorithm has preprocessing time and space $\widetilde O\left(n^{1/2+O(1/\log(d/λ))}\right)$, and produces a data structure that answers color queries in time $\widetilde O\left(n^{1/2+O(1/\log(d/λ))}\right)$, such that the resulting labeling agrees with the planted coloring on all but an $O(\sqrt{λ/d})$ fraction of vertices, up to a permutation of the $k$ colors. The algorithm gives sublinear-time inner product access to the bottom eigenspace of the normalized adjacency matrix, which allows us to adapt the classical spectral approach of Alon and Kahale in sublinear time.

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arXiv2026

Recovering Communities in Structured Random Graphs

The problem of recovering planted community structure in random graphs has received a lot of attention in the literature on the stochastic block model, where the input is a random graph in which edges crossing between different communities appear with smaller probability than edges induced by communities. The communities themselves form a collection of vertex-disjoint sparse cuts in the expected graph, and can be recovered, often exactly, from a sample as long as a separation condition on the intra- and inter-community edge probabilities is satisfied. In this paper, we ask whether the presence of a large number of overlapping sparsest cuts in the expected graph still allows recovery. For example, the $d$-dimensional hypercube graph admits $d$ distinct (balanced) sparsest cuts, one for every coordinate. Can these cuts be identified given a random sample of the edges of the hypercube where each edge is present independently with some probability $p\in (0, 1)$? We show that this is the case, in a very strong sense: the sparsest balanced cut in a sample of the hypercube at rate $p=C\log d/d$ for a sufficiently large constant $C$ is $1/\text{poly}(d)$-close to a coordinate cut with high probability. This is asymptotically optimal and allows approximate recovery of all $d$ cuts simultaneously. Furthermore, for an appropriate sample of hypercube-like graphs recovery can be made exact. The proof is essentially a strong hypercube cut sparsification bound that combines a theorem of Friedgut, Kalai and Naor on boolean functions whose Fourier transform concentrates on the first level of the Fourier spectrum with Karger's cut counting argument.

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arXiv2026

Spectral Clustering in Birthday Paradox Time

Given a vertex in a $(k, φ, ε)$-clusterable graph, i.e. a graph whose vertex set can be partitioned into a disjoint union of $φ$-expanders of size $\approx n/k$ with outer conductance bounded by $ε$, can one quickly tell which cluster it belongs to? This question goes back to the expansion testing problem of Goldreich and Ron'11. For $k=2$ a sample of $\approx n^{1/2+O(ε/φ^2)}$ logarithmic length walks from a given vertex approximately determines its cluster membership by the birthday paradox: two vertices whose random walk samples are `close' are likely in the same cluster. The study of the general case $k>2$ was initiated by Czumaj, Peng and Sohler [STOC'15], and the works of Chiplunkar et al. [FOCS'18], Gluch et al. [SODA'21] showed that $\approx \text{poly}(k)\cdot n^{1/2+O(ε/φ^2)}$ random walk samples suffice for general $k$. This matches the $k=2$ result up to polynomial factors in $k$, but creates a conceptual inconsistency: if the birthday paradox is the guiding phenomenon, then the query complexity should decrease with the number of clusters $k$! Since clusters have size $\approx n/k$, we expect to need $\approx (n/k)^{1/2+O(ε/φ^2)}$ random walk samples, which decreases with $k$. We design a novel representation of vertices in a $(k, φ, ε)$-clusterable graph by a mixture of logarithmic length walks. This representation uses the optimal $\approx (n/k)^{1/2+O(ε/φ^2)}$ walks per vertex, and allows for a fast nearest neighbor search: given $k$ vertices representing the clusters, we can find the cluster of a given query vertex $x$ using nearly linear time in the representation size of $x$. This gives a clustering oracle with query time $\approx (n/k)^{1/2+O(ε/φ^2)}$ and space complexity $k\cdot (n/k)^{1/2+O(ε/φ^2)}$, matching the birthday paradox bound.

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arXiv2025

Spectral Clustering with Side Information

In the graph clustering problem with a planted solution, the input is a graph on $n$ vertices partitioned into $k$ clusters, and the task is to infer the clusters from graph structure. A standard assumption is that clusters induce well-connected subgraphs (i.e. $Ω(1)$-expanders), and form $ε$-sparse cuts. Such a graph defines the clustering uniquely up to $\approx ε$ misclassification rate, and efficient algorithms for achieving this rate are known. While this vanilla version of graph clustering is well studied, in practice, vertices of the graph are typically equipped with labels that provide additional information on cluster ids of the vertices. For example, each vertex could have a cluster label that is corrupted independently with probability $δ$. Using only one of the two sources of information leads to misclassification rate $\min\{ε, δ\}$, but can they be combined to achieve a rate of $\approx εδ$? In this paper, we give an affirmative answer to this question and present a sublinear-time algorithm in the number of vertices $n$. Our key algorithmic insight is a new observation on ``spectrally ambiguous'' vertices in a well-clusterable graph. While our sublinear-time classifier achieves the nearly optimal $\approx \widetilde O(εδ)$ misclassification rate, the approximate clusters that it outputs do not necessarily induce expanders in the graph $G$. In our second result, we give a polynomial-time algorithm that reweights edges of the original $(k, ε, Ω(1))$-clusterable graph to transform it into a $(k, \widetilde O(εδ), Ω(1))$-clusterable one (for constant $k$), improving sparsity of cuts nearly optimally and preserving expansion properties of the communities - an algorithm for refining community structure of the input graph.

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arXiv2025

Approximating Dasgupta Cost in Sublinear Time from a Few Random Seeds

Testing graph cluster structure has been a central object of study in property testing since the foundational work of Goldreich and Ron [STOC'96] on expansion testing, i.e. the problem of distinguishing between a single cluster (an expander) and a graph that is far from a single cluster. More generally, a $(k, ε)$-clusterable graph $G$ is a graph whose vertex set admits a partition into $k$ induced expanders, each with outer conductance bounded by $ε$. A recent line of work initiated by Czumaj, Peng and Sohler [STOC'15] has shown how to test whether a graph is close to $(k, ε)$-clusterable, and to locally determine which cluster a given vertex belongs to with misclassification rate $\approx ε$, but no sublinear time algorithms for learning the structure of inter-cluster connections are known. As a simple example, can one locally distinguish between the `cluster graph' forming a line and a clique? In this paper, we consider the problem of testing the hierarchical cluster structure of $(k, ε)$-clusterable graphs in sublinear time. Our measure of hierarchical clusterability is the well-established Dasgupta cost, and our main result is an algorithm that approximates Dasgupta cost of a $(k, ε)$-clusterable graph in sublinear time, using a small number of randomly chosen seed vertices for which cluster labels are known. Our main result is an $O(\sqrt{\log k})$ approximation to Dasgupta cost of $G$ in $\approx n^{1/2+O(ε)}$ time using $\approx n^{1/3}$ seeds, effectively giving a sublinear time simulation of the algorithm of Charikar and Chatziafratis [SODA'17] on clusterable graphs. To the best of our knowledge, ours is the first result on approximating the hierarchical clustering properties of such graphs in sublinear time.

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arXiv2024

On the Robustness of Spectral Algorithms for Semirandom Stochastic Block Models

In a graph bisection problem, we are given a graph $G$ with two equally-sized unlabeled communities, and the goal is to recover the vertices in these communities. A popular heuristic, known as spectral clustering, is to output an estimated community assignment based on the eigenvector corresponding to the second smallest eigenvalue of the Laplacian of $G$. Spectral algorithms can be shown to provably recover the cluster structure for graphs generated from certain probabilistic models, such as the Stochastic Block Model (SBM). However, spectral clustering is known to be non-robust to model mis-specification. Techniques based on semidefinite programming have been shown to be more robust, but they incur significant computational overheads. In this work, we study the robustness of spectral algorithms against semirandom adversaries. Informally, a semirandom adversary is allowed to ``helpfully'' change the specification of the model in a way that is consistent with the ground-truth solution. Our semirandom adversaries in particular are allowed to add edges inside clusters or increase the probability that an edge appears inside a cluster. Semirandom adversaries are a useful tool to determine the extent to which an algorithm has overfit to statistical assumptions on the input. On the positive side, we identify classes of semirandom adversaries under which spectral bisection using the _unnormalized_ Laplacian is strongly consistent, i.e., it exactly recovers the planted partitioning. On the negative side, we show that in these classes spectral bisection with the _normalized_ Laplacian outputs a partitioning that makes a classification mistake on a constant fraction of the vertices. Finally, we demonstrate numerical experiments that complement our theoretical findings.

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arXiv2024

Weighted Matching in the Random-Order Streaming and Robust Communication Models

We study the maximum weight matching problem in the random-order semi-streaming model and in the robust communication model. Unlike many other sublinear models, in these two frameworks, there is a large gap between the guarantees of the best known algorithms for the unweighted and weighted versions of the problem. In the random-order semi-streaming setting, the edges of an $n$-vertex graph arrive in a stream in a random order. The goal is to compute an approximate maximum weight matching with a single pass over the stream using $O(n\text{ polylog } n)$ space. Our main result is a $(2/3-ε)$-approximation algorithm for maximum weight matching in random-order streams, using space $O(n \log n \log R)$, where $R$ is the ratio between the heaviest and the lightest edge in the graph. Our result nearly matches the best known unweighted $(2/3+ε_0)$-approximation (where $ε_0 \sim 10^{-14}$ is a small constant) achieved by Assadi and Behnezhad [ICALP 2021], and significantly improves upon previous weighted results. Our techniques also extend to the related robust communication model, in which the edges of a graph are partitioned randomly between Alice and Bob. Alice sends a single message of size $O(n\text{ polylog }n)$ to Bob, who must compute an approximate maximum weight matching. We achieve a $(5/6-ε)$-approximation using $O(n \log n \log R)$ words of communication, matching the results of Azarmehr and Behnezhad [ICALP 2023] for unweighted graphs.

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