arXiv subjects

Marco Bressan

Publications and source records attributed to Marco Bressan.

29 records · Page 2Linked to original sources

arXiv2021

On Margin-Based Cluster Recovery with Oracle Queries

We study an active cluster recovery problem where, given a set of $n$ points and an oracle answering queries like "are these two points in the same cluster?", the task is to recover exactly all clusters using as few queries as possible. We begin by introducing a simple but general notion of margin between clusters that captures, as special cases, the margins used in previous work, the classic SVM margin, and standard notions of stability for center-based clusterings. Then, under our margin assumptions we design algorithms that, in a variety of settings, recover all clusters exactly using only $O(\log n)$ queries. For the Euclidean case, $\mathbb{R}^m$, we give an algorithm that recovers arbitrary convex clusters, in polynomial time, and with a number of queries that is lower than the best existing algorithm by $Θ(m^m)$ factors. For general pseudometric spaces, where clusters might not be convex or might not have any notion of shape, we give an algorithm that achieves the $O(\log n)$ query bound, and is provably near-optimal as a function of the packing number of the space. Finally, for clusterings realized by binary concept classes, we give a combinatorial characterization of recoverability with $O(\log n)$ queries, and we show that, for many concept classes in Euclidean spaces, this characterization is equivalent to our margin condition. Our results show a deep connection between cluster margins and active cluster recoverability.

cs.LG↗

arXiv2021

Exact and Approximate Pattern Counting in Degenerate Graphs: New Algorithms, Hardness Results, and Complexity Dichotomies

We study the problems of counting the homomorphisms, counting the copies, and counting the induced copies of a $k$-vertex graph $H$ in a $d$-degenerate $n$-vertex graph $G$. Our main result establishes exhaustive and explicit complexity classifications for counting subgraphs and induced subgraphs. We show that the (not necessarily induced) copies of $H$ in $G$ can be counted in time $f(k,d)\cdot n^{\max(\mathsf{imn}(H),1)}\cdot \log n$, where $f$ is some computable function and $\mathsf{imn}(H)$ is the size of the largest induced matching of $H$. Whenever the class of allowed patterns has unbounded induced matching number, this algorithm is essentially optimal: Unless the Exponential Time Hypothesis (ETH) fails, there is no algorithm running in time $f(k,d)\cdot n^{o(\mathsf{imn}(H)/\log \mathsf{imn}(H))}$ for any function $f$. In case of counting induced subgraphs, we obtain a similar classification along the independence number $α$: we can count the induced copies of $H$ in $G$ in time $f(k,d)\cdot n^{α(H)}\cdot \log n$, and if the class of allowed patterns has unbounded independence number, an algorithm running in time $f(k,d)\cdot n^{o(α(H)/\log α(H))}$ is impossible, unless ETH fails. In the language of parameterized complexity, our results yield dichotomies in fixed-parameter tractable and $\#\mathsf{W}[1]$-hard cases if we parameterize by the size of the pattern and the degeneracy of the host graph. Our results imply that several patterns cannot be counted in time $f(k,d)\cdot n^{o(k/\log k)}$, including $k$-matchings, $k$-independent sets, (induced) $k$-paths, (induced) $k$-cycles, and induced $(k,k)$-bicliques, unless ETH fails. Those lower bounds for exact counting are complemented with new algorithms for approximate counting of subgraphs and induced subgraphs in degenerate graphs.

cs.CC↗

arXiv2020

Exact Recovery of Mangled Clusters with Same-Cluster Queries

We study the cluster recovery problem in the semi-supervised active clustering framework. Given a finite set of input points, and an oracle revealing whether any two points lie in the same cluster, our goal is to recover all clusters exactly using as few queries as possible. To this end, we relax the spherical $k$-means cluster assumption of Ashtiani et al.\ to allow for arbitrary ellipsoidal clusters with margin. This removes the assumption that the clustering is center-based (i.e., defined through an optimization problem), and includes all those cases where spherical clusters are individually transformed by any combination of rotations, axis scalings, and point deletions. We show that, even in this much more general setting, it is still possible to recover the latent clustering exactly using a number of queries that scales only logarithmically with the number of input points. More precisely, we design an algorithm that, given $n$ points to be partitioned into $k$ clusters, uses $O(k^3 \ln k \ln n)$ oracle queries and $\tilde{O}(kn + k^3)$ time to recover the clustering with zero misclassification error. The $O(\cdot)$ notation hides an exponential dependence on the dimensionality of the clusters, which we show to be necessary thus characterizing the query complexity of the problem. Our algorithm is simple, easy to implement, and can also learn the clusters using low-stretch separators, a class of ellipsoids with additional theoretical guarantees. Experiments on large synthetic datasets confirm that we can reconstruct clusterings exactly and efficiently.

cs.LG↗

arXiv2020

Faster algorithms for counting subgraphs in sparse graphs

Given a $k$-node pattern graph $H$ and an $n$-node host graph $G$, the subgraph counting problem asks to compute the number of copies of $H$ in $G$. In this work we address the following question: can we count the copies of $H$ faster if $G$ is sparse? We answer in the affirmative by introducing a novel tree-like decomposition for directed acyclic graphs, inspired by the classic tree decomposition for undirected graphs. This decomposition gives a dynamic program for counting the homomorphisms of $H$ in $G$ by exploiting the degeneracy of $G$, which allows us to beat the state-of-the-art subgraph counting algorithms when $G$ is sparse enough. For example, we can count the induced copies of any $k$-node pattern $H$ in time $2^{O(k^2)} O(n^{0.25k + 2} \log n)$ if $G$ has bounded degeneracy, and in time $2^{O(k^2)} O(n^{0.625k + 1} \log n)$ if $G$ has bounded average degree. These bounds are instantiations of a more general result, parameterized by the degeneracy of $G$ and the structure of $H$, which generalizes classic bounds on counting cliques and complete bipartite graphs. We also give lower bounds based on the Exponential Time Hypothesis, showing that our results are actually a characterization of the complexity of subgraph counting in bounded-degeneracy graphs.

cs.CC↗

arXiv2020

Correlation Clustering with Adaptive Similarity Queries

In correlation clustering, we are given $n$ objects together with a binary similarity score between each pair of them. The goal is to partition the objects into clusters so to minimise the disagreements with the scores. In this work we investigate correlation clustering as an active learning problem: each similarity score can be learned by making a query, and the goal is to minimise both the disagreements and the total number of queries. On the one hand, we describe simple active learning algorithms, which provably achieve an almost optimal trade-off while giving cluster recovery guarantees, and we test them on different datasets. On the other hand, we prove information-theoretical bounds on the number of queries necessary to guarantee a prescribed disagreement bound. These results give a rich characterization of the trade-off between queries and clustering error.

cs.LG↗

arXiv2019

Motivo: fast motif counting via succinct color coding and adaptive sampling

The randomized technique of color coding is behind state-of-the-art algorithms for estimating graph motif counts. Those algorithms, however, are not yet capable of scaling well to very large graphs with billions of edges. In this paper we develop novel tools for the `motif counting via color coding' framework. As a result, our new algorithm, Motivo, is able to scale well to larger graphs while at the same time provide more accurate graphlet counts than ever before. This is achieved thanks to two types of improvements. First, we design new succinct data structures that support fast common color coding operations, and a biased coloring trick that trades accuracy versus running time and memory usage. These adaptations drastically reduce the time and memory requirements of color coding. Second, we develop an adaptive graphlet sampling strategy, based on a fractional set cover problem, that breaks the additive approximation barrier of standard sampling. This strategy gives multiplicative approximations for all graphlets at once, allowing us to count not only the most frequent graphlets but also extremely rare ones. To give an idea of the improvements, in $40$ minutes Motivo counts $7$-nodes motifs on a graph with $65$M nodes and $1.8$B edges; this is $30$ and $500$ times larger than the state of the art, respectively in terms of nodes and edges. On the accuracy side, in one hour Motivo produces accurate counts of $\approx \! 10.000$ distinct $8$-node motifs on graphs where state-of-the-art algorithms fail even to find the second most frequent motif. Our method requires just a high-end desktop machine. These results show how color coding can bring motif mining to the realm of truly massive graphs using only ordinary hardware.

cs.DB↗

arXiv2018

Sublinear algorithms for local graph centrality estimation

We study the complexity of local graph centrality estimation, with the goal of approximating the centrality score of a given target node while exploring only a sublinear number of nodes/arcs of the graph and performing a sublinear number of elementary operations. We develop a technique, that we apply to the PageRank and Heat Kernel centralities, for building a low-variance score estimator through a local exploration of the graph. We obtain an algorithm that, given any node in any graph of $m$ arcs, with probability $(1-δ)$ computes a multiplicative $(1\pmε)$-approximation of its score by examining only $\tilde{O}(\min(m^{2/3} Δ^{1/3} d^{-2/3},\, m^{4/5} d^{-3/5}))$ nodes/arcs, where $Δ$ and $d$ are respectively the maximum and average outdegree of the graph (omitting for readability $\operatorname{poly}(ε^{-1})$ and $\operatorname{polylog}(δ^{-1})$ factors). A similar bound holds for computational complexity. We also prove a lower bound of $Ω(\min(m^{1/2} Δ^{1/2} d^{-1/2}, \, m^{2/3} d^{-1/3}))$ for both query complexity and computational complexity. Moreover, our technique yields a $\tilde{O}(n^{2/3})$ query complexity algorithm for the graph access model of [Brautbar et al., 2010], widely used in social network mining; we show this algorithm is optimal up to a sublogarithmic factor. These are the first algorithms yielding worst-case sublinear bounds for general directed graphs and any choice of the target node.

cs.DS↗

arXiv2018

Simple set cardinality estimation through random sampling

We present a simple algorithm that estimates the cardinality $n$ of a set $V$ when allowed to sample elements of $V$ uniformly and independently at random. Our algorithm with probability $(1-δ)$ returns a $(1\pmε)-$approximation of $n$ drawing $O\big(\sqrt{n} \cdot ε^{-1}\sqrt{\log(δ^{-1})}\big)$ samples (for $ε^{-1}\sqrt{\log(δ^{-1})} = O(\sqrt{n})$).

cs.DM↗

arXiv2017

On approximating the stationary distribution of time-reversible Markov chains

Approximating the stationary probability of a state in a Markov chain through Markov chain Monte Carlo techniques is, in general, inefficient. Standard random walk approaches require $\tilde{O}(τ/π(v))$ operations to approximate the probability $π(v)$ of a state $v$ in a chain with mixing time $τ$, and even the best available techniques still have complexity $\tilde{O}(τ^{1.5}/π(v)^{0.5})$, and since these complexities depend inversely on $π(v)$, they can grow beyond any bound in the size of the chain or in its mixing time. In this paper we show that, for time-reversible Markov chains, there exists a simple randomized approximation algorithm that breaks this "small-$π(v)$ barrier".

cs.DM↗

arXiv2016

The Limits of Popularity-Based Recommendations, and the Role of Social Ties

In this paper we introduce a mathematical model that captures some of the salient features of recommender systems that are based on popularity and that try to exploit social ties among the users. We show that, under very general conditions, the market always converges to a steady state, for which we are able to give an explicit form. Thanks to this we can tell rather precisely how much a market is altered by a recommendation system, and determine the power of users to influence others. Our theoretical results are complemented by experiments with real world social networks showing that social graphs prevent large market distortions in spite of the presence of highly influential users.

cs.SI↗

arXiv2016

The Power of Local Information in PageRank

How large a fraction of a graph must one explore to rank a small set of nodes according to their PageRank scores? We show that the answer is quite nuanced, and depends crucially on the interplay between the correctness guarantees one requires and the way one can access the graph. On the one hand, assuming the graph can be accessed only via "natural" exploration queries that reveal small pieces of its topology, we prove that deterministic and Las Vegas algorithms must in the worst case perform $n - o(n)$ queries and explore essentially the entire graph, independently of the specific types of query employed. On the other hand we show that, depending on the types of query available, Monte Carlo algorithms can perform asymptotically better: if allowed to both explore the local topology around single nodes and access nodes at random in the graph they need $Ω(n^{2/3})$ queries in the worst case, otherwise they still need $Ω(n)$ queries similarly to Las Vegas algorithms. All our bounds generalize and tighten those already known, cover the different types of graph exploration queries appearing in the literature, and immediately apply also to the problem of approximating the PageRank score of single nodes.

cs.DM↗