arXiv subjects

Marco Bressan

Publications and source records attributed to Marco Bressan.

At least 19 recordsLinked to original sources

arXiv2026

Active Learning on Adversarially Corrupted Graphs

Motivated by real-world scenarios where malicious entities tamper with existing networks, we define a model where an adversary seeks to hide a set of \emph{corrupted vertices} inside a graph $G^*$. To this end, the adversary can add edges between the corrupted vertices, as well as edges between the corrupted vertices and $G^*$, and its power is then measured by the size of the \emph{neighborhood} of the corrupted vertices in $G^*$. Our goal is to design an active learning algorithm that efficiently finds the subset of corrupted vertices using a small number of label queries. We devise an efficient algorithm that approximately recovers the corrupted vertices with a query complexity that depends polynomially on both the power of the adversary and the \emph{vertex expansion} of $G^*$, a fundamental measure of graph connectivity. At the heart of this result is a polynomial-time algorithm, obtained by carefully adapting sum-of-squares algorithms for approximating minimum expansion, that finds a set with small vertex expansion subject to cardinality constraints. To the best of our knowledge, this is the first time that the vertex expansion is shown to play a key role in determining the query complexity of active learning algorithms robust to structural adversarial attacks.

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arXiv2026

The Parametrised Complexity of Counting Small Sub-Hypergraphs

Subgraph counting is a fundamental and well-studied problem whose computational complexity is well understood. Quite surprisingly, the hypergraph version of subgraph counting has been almost ignored. In this work, we address this gap by investigating the most basic sub-hypergraph counting problem: given a (small) hypergraph $H$ and a (large) hypergraph $G$, compute the number of sub-hypergraphs of $G$ isomorphic to $H$. Formally, for a family $\mathcal{H}$ of hypergraphs, let #Sub($\mathcal{H}$) be the restriction of the problem to $H \in \mathcal{H}$; the induced variant #IndSub($\mathcal{H}$) is defined analogously. Our main contribution is a complete classification of the complexity of these problems. Assuming the Exponential Time Hypothesis, we prove that #Sub($\mathcal{H}$) is fixed-parameter tractable if and only if $\mathcal{H}$ has bounded fractional co-independent edge-cover number, a novel graph parameter we introduce. Moreover, #IndSub($\mathcal{H}$) is fixed-parameter tractable if and only if $\mathcal{H}$ has bounded fractional edge-cover number. Both results subsume pre-existing results for graphs as special cases. We also show that the fixed-parameter tractable cases of #Sub($\mathcal{H}$) and #IndSub($\mathcal{H}$) are unlikely to be in polynomial time, unless respectively #P = P and Graph Isomorphism $\in$ P. This shows a separation with the special case of graphs, where the fixed-parameter tractable cases are known to actually be in polynomial time.

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arXiv2026

An Efficient Streaming Algorithm for Approximating Graphlet Distributions

In recent years, the problem of computing the frequencies of the induced $k$-vertex subgraphs of a graph, or \emph{$k$-graphlets}, has become central. One approach for this problem is to sample $k$-graphlets randomly. Classic algorithms for $k$-graphlet sampling require loading the entire graph into main memory, making them impractical for massive graphs. To bypass this limitation, Bourreau et al. (NeurIPS 2024) introduced a \emph{streaming} algorithm that through nontrivial techniques makes only $O(\log n)$ passes using $O(n \log n)$ memory. In this work we break their $O(\log n)$-pass bound by giving an algorithm that, for any fixed $c>0$, makes $O(1/c)$ passes using $\tilde O(n^{1+c})$ memory. As a consequence of their lower bound, our algorithm is optimal up to a factor of $\tilde{O}(n^c)$ in the memory usage. We use this sampling algorithm to obtain an efficient method of approximating $k$-graphlet distributions. Experiments on real-world and synthetic graphs show that our algorithm is always at least as good as the one of Bourreau et al., and outperforms it by orders of magnitude on mildly dense graphs.

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arXiv2026

Counting HyperGraphlets via Color Coding: a Quadratic Barrier and How to Break It

We study the problem of counting $k$-hypergraphlets, an interesting but surprisingly ignored primitive, with the aim of understanding whether efficient algorithms exist. To this end, we consider color coding, a well-known technique for approximately counting $k$-graphlets in graphs. Our first result is that, on hypergraphs, color coding encounters a quadratic barrier: under the Orthogonal Vector Conjecture, no implementation can run in sub-quadratic time in the input size. We then introduce a simple property, $(α,β)$-niceness, that hypergraphs from real-world datasets appear to satisfy for small values of $α$ and $β$. Intuitively, an $(α,β)$-nice hypergraph can be split into two sub-hypergraphs having respectively rank at most $α$ and degree at most $β$. By applying different techniques to each sub-hypergraph and carefully combining the outputs, we show how to run color coding in time $2^{O(k)} \cdot (2^β|V| + α^k |E| + α^2 β\|H\|)$, where $H=(V,E)$ is the input hypergraph. Afterwards, we can sample colorful $k$-hypergraphlets uniformly in expected $k^{O(k)} \cdot (β^2 + \ln |V|)$ time per sample. Experiments on real-world hypergraphs show that our algorithm significantly outperforms the naive quadratic algorithm, sometimes by more than an order of magnitude.

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arXiv2026

Learning Conditional Averages

We introduce the problem of learning conditional averages in the PAC framework. The learner receives a sample labeled by an unknown target concept from a known concept class, as in standard PAC learning. However, instead of learning the target concept itself, the goal is to predict, for each instance, the average label over its neighborhood -- an arbitrary subset of points that contains the instance. In the degenerate case where all neighborhoods are singletons, the problem reduces exactly to classic PAC learning. More generally, it extends PAC learning to a setting that captures learning tasks arising in several domains, including explainability, fairness, and recommendation systems. Our main contribution is a complete characterization of when conditional averages are learnable, together with sample complexity bounds that are tight up to logarithmic factors. The characterization hinges on the joint finiteness of two novel combinatorial parameters, which depend on both the concept class and the neighborhood system, and are closely related to the independence number of the associated neighborhood graph.

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arXiv2025

Efficient Algorithms for Learning and Compressing Monophonic Halfspaces in Graphs

Abstract notions of convexity over the vertices of a graph, and corresponding notions of halfspaces, have recently gained attention from the machine learning community. In this work we study monophonic halfspaces, a notion of graph halfspaces defined through closure under induced paths. Our main result is a $2$-satisfiability based decomposition theorem, which allows one to represent monophonic halfspaces as a disjoint union of certain vertex subsets. Using this decomposition, we achieve efficient and (nearly) optimal algorithms for various learning problems, such as teaching, active, and online learning. Most notably, we obtain a polynomial-time algorithm for empirical risk minimization. Independently of the decomposition theorem, we obtain an efficient, stable, and proper sample compression scheme. This makes monophonic halfspaces efficiently learnable with proper learners and linear error rate $1/\varepsilon$ in the realizable PAC setting. Our results answer open questions from the literature, and show a stark contrast with geodesic halfspaces, for which most of the said learning problems are NP-hard.

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arXiv2025

On Finding Randomly Planted Cliques in Arbitrary Graphs

We study a planted clique model introduced by Feige where a complete graph of size $c\cdot n$ is planted uniformly at random in an arbitrary $n$-vertex graph. We give a simple deterministic algorithm that, in almost linear time, recovers a clique of size $(c/3)^{O(1/c)} \cdot n$ as long as the original graph has maximum degree at most $(1-p)n$ for some fixed $p>0$. The proof hinges on showing that the degrees of the final graph are correlated with the planted clique, in a way similar to (but more intricate than) the classical $G(n,\frac{1}{2})+K_{\sqrt{n}}$ planted clique model. Our algorithm suggests a separation from the worst-case model, where, assuming the Unique Games Conjecture, no polynomial algorithm can find cliques of size $Ω(n)$ for every fixed $c>0$, even if the input graph has maximum degree $(1-p)n$. Our techniques extend beyond the planted clique model. For example, when the planted graph is a balanced biclique, we recover a balanced biclique of size larger than the best guarantees known for the worst case.

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arXiv2024

Of Dice and Games: A Theory of Generalized Boosting

Cost-sensitive loss functions are crucial in many real-world prediction problems, where different types of errors are penalized differently; for example, in medical diagnosis, a false negative prediction can lead to worse consequences than a false positive prediction. However, traditional PAC learning theory has mostly focused on the symmetric 0-1 loss, leaving cost-sensitive losses largely unaddressed. In this work, we extend the celebrated theory of boosting to incorporate both cost-sensitive and multi-objective losses. Cost-sensitive losses assign costs to the entries of a confusion matrix, and are used to control the sum of prediction errors accounting for the cost of each error type. Multi-objective losses, on the other hand, simultaneously track multiple cost-sensitive losses, and are useful when the goal is to satisfy several criteria at once (e.g., minimizing false positives while keeping false negatives below a critical threshold). We develop a comprehensive theory of cost-sensitive and multi-objective boosting, providing a taxonomy of weak learning guarantees that distinguishes which guarantees are trivial (i.e., can always be achieved), which ones are boostable (i.e., imply strong learning), and which ones are intermediate, implying non-trivial yet not arbitrarily accurate learning. For binary classification, we establish a dichotomy: a weak learning guarantee is either trivial or boostable. In the multiclass setting, we describe a more intricate landscape of intermediate weak learning guarantees. Our characterization relies on a geometric interpretation of boosting, revealing a surprising equivalence between cost-sensitive and multi-objective losses.

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arXiv2024

Efficient Algorithms for Learning Monophonic Halfspaces in Graphs

We study the problem of learning a binary classifier on the vertices of a graph. In particular, we consider classifiers given by monophonic halfspaces, partitions of the vertices that are convex in a certain abstract sense. Monophonic halfspaces, and related notions such as geodesic halfspaces,have recently attracted interest, and several connections have been drawn between their properties(e.g., their VC dimension) and the structure of the underlying graph $G$. We prove several novel results for learning monophonic halfspaces in the supervised, online, and active settings. Our main result is that a monophonic halfspace can be learned with near-optimal passive sample complexity in time polynomial in $n = |V(G)|$. This requires us to devise a polynomial-time algorithm for consistent hypothesis checking, based on several structural insights on monophonic halfspaces and on a reduction to $2$-satisfiability. We prove similar results for the online and active settings. We also show that the concept class can be enumerated with delay $\operatorname{poly}(n)$, and that empirical risk minimization can be performed in time $2^{ω(G)}\operatorname{poly}(n)$ where $ω(G)$ is the clique number of $G$. These results answer open questions from the literature (González et al., 2020), and show a contrast with geodesic halfspaces, for which some of the said problems are NP-hard (Seiffarth et al., 2023).

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arXiv2024

A Theory of Interpretable Approximations

Can a deep neural network be approximated by a small decision tree based on simple features? This question and its variants are behind the growing demand for machine learning models that are *interpretable* by humans. In this work we study such questions by introducing *interpretable approximations*, a notion that captures the idea of approximating a target concept $c$ by a small aggregation of concepts from some base class $\mathcal{H}$. In particular, we consider the approximation of a binary concept $c$ by decision trees based on a simple class $\mathcal{H}$ (e.g., of bounded VC dimension), and use the tree depth as a measure of complexity. Our primary contribution is the following remarkable trichotomy. For any given pair of $\mathcal{H}$ and $c$, exactly one of these cases holds: (i) $c$ cannot be approximated by $\mathcal{H}$ with arbitrary accuracy; (ii) $c$ can be approximated by $\mathcal{H}$ with arbitrary accuracy, but there exists no universal rate that bounds the complexity of the approximations as a function of the accuracy; or (iii) there exists a constant $κ$ that depends only on $\mathcal{H}$ and $c$ such that, for *any* data distribution and *any* desired accuracy level, $c$ can be approximated by $\mathcal{H}$ with a complexity not exceeding $κ$. This taxonomy stands in stark contrast to the landscape of supervised classification, which offers a complex array of distribution-free and universally learnable scenarios. We show that, in the case of interpretable approximations, even a slightly nontrivial a-priori guarantee on the complexity of approximations implies approximations with constant (distribution-free and accuracy-free) complexity. We extend our trichotomy to classes $\mathcal{H}$ of unbounded VC dimension and give characterizations of interpretability based on the algebra generated by $\mathcal{H}$.

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arXiv2024

Counting Subgraphs in Somewhere Dense Graphs

We study the problems of counting copies and induced copies of a small pattern graph $H$ in a large host graph $G$. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns $H$. In this work, we address the more challenging task of analysing the complexity for restricted patterns and restricted hosts. Specifically we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time $f(H)\cdot |G|^{O(1)}$ for some computable function $f$. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes $\mathcal{G}$ as our central objects of study and establish the following crisp dichotomies as consequences of the Exponential Time Hypothesis: (1) Counting $k$-matchings in a graph $G\in\mathcal{G}$ is fixed-parameter tractable if and only if $\mathcal{G}$ is nowhere dense. (2) Counting $k$-independent sets in a graph $G\in\mathcal{G}$ is fixed-parameter tractable if and only if $\mathcal{G}$ is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if $\mathcal{G}$ is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting $k$-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in $F$-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting $k$-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19).

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arXiv2023

Fully-Dynamic Approximate Decision Trees With Worst-Case Update Time Guarantees

We give the first algorithm that maintains an approximate decision tree over an arbitrary sequence of insertions and deletions of labeled examples, with strong guarantees on the worst-case running time per update request. For instance, we show how to maintain a decision tree where every vertex has Gini gain within an additive $α$ of the optimum by performing $O\Big(\frac{d\,(\log n)^4}{α^3}\Big)$ elementary operations per update, where $d$ is the number of features and $n$ the maximum size of the active set (the net result of the update requests). We give similar bounds for the information gain and the variance gain. In fact, all these bounds are corollaries of a more general result, stated in terms of decision rules -- functions that, given a set $S$ of labeled examples, decide whether to split $S$ or predict a label. Decision rules give a unified view of greedy decision tree algorithms regardless of the example and label domains, and lead to a general notion of $ε$-approximate decision trees that, for natural decision rules such as those used by ID3 or C4.5, implies the gain approximation guarantees above. The heart of our work provides a deterministic algorithm that, given any decision rule and any $ε> 0$, maintains an $ε$-approximate tree using $O\!\left(\frac{d\, f(n)}{n} \operatorname{poly}\frac{h}ε\right)$ operations per update, where $f(n)$ is the complexity of evaluating the rule over a set of $n$ examples and $h$ is the maximum height of the maintained tree.

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arXiv2022

Fully-Dynamic Decision Trees

We develop the first fully dynamic algorithm that maintains a decision tree over an arbitrary sequence of insertions and deletions of labeled examples. Given $ε> 0$ our algorithm guarantees that, at every point in time, every node of the decision tree uses a split with Gini gain within an additive $ε$ of the optimum. For real-valued features the algorithm has an amortized running time per insertion/deletion of $O\big(\frac{d \log^3 n}{ε^2}\big)$, which improves to $O\big(\frac{d \log^2 n}ε\big)$ for binary or categorical features, while it uses space $O(n d)$, where $n$ is the maximum number of examples at any point in time and $d$ is the number of features. Our algorithm is nearly optimal, as we show that any algorithm with similar guarantees uses amortized running time $Ω(d)$ and space $\tildeΩ (n d)$. We complement our theoretical results with an extensive experimental evaluation on real-world data, showing the effectiveness of our algorithm.

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arXiv2022

The Complexity of Pattern Counting in Directed Graphs, Parameterised by the Outdegree

We study the fixed-parameter tractability of the following fundamental problem: given two directed graphs $\vec H$ and $\vec G$, count the number of copies of $\vec H$ in $\vec G$. The standard setting, where the tractability is well understood, uses only $|\vec H|$ as a parameter. In this paper we take a step forward, and adopt as a parameter $|\vec H|+d(\vec G)$, where $d(\vec G)$ is the maximum outdegree of $|\vec G|$. Under this parameterization, we completely characterize the fixed-parameter tractability of the problem in both its non-induced and induced versions through two novel structural parameters, the fractional cover number $ρ^*$ and the source number $α_s$. On the one hand we give algorithms with running time $f(|\vec H|,d(\vec G)) \cdot |\vec G|^{ρ^*\!(\vec H)+O(1)}$ and $f(|\vec H|,d(\vec G)) \cdot |\vec G|^{α_s(\vec H)+O(1)}$ for counting respectively the copies and induced copies of $\vec H$ in $\vec G$; on the other hand we show that, unless the Exponential Time Hypothesis fails, for any class $\vec C$ of directed graphs the (induced) counting problem is fixed-parameter tractable if and only if $ρ^*(\vec C)$ ($α_s(\vec C)$) is bounded. These results explain how the orientation of the pattern can make counting easy or hard, and prove that a classic algorithm by Chiba and Nishizeki and its extensions (Chiba, Nishizeki SICOMP 85; Bressan Algorithmica 21) are optimal unless ETH fails.

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arXiv2022

Active Learning of Classifiers with Label and Seed Queries

We study exact active learning of binary and multiclass classifiers with margin. Given an $n$-point set $X \subset \mathbb{R}^m$, we want to learn any unknown classifier on $X$ whose classes have finite strong convex hull margin, a new notion extending the SVM margin. In the standard active learning setting, where only label queries are allowed, learning a classifier with strong convex hull margin $γ$ requires in the worst case $Ω\big(1+\frac{1}γ\big)^{(m-1)/2}$ queries. On the other hand, using the more powerful seed queries (a variant of equivalence queries), the target classifier could be learned in $O(m \log n)$ queries via Littlestone's Halving algorithm; however, Halving is computationally inefficient. In this work we show that, by carefully combining the two types of queries, a binary classifier can be learned in time $\operatorname{poly}(n+m)$ using only $O(m^2 \log n)$ label queries and $O\big(m \log \frac{m}γ\big)$ seed queries; the result extends to $k$-class classifiers at the price of a $k!k^2$ multiplicative overhead. Similar results hold when the input points have bounded bit complexity, or when only one class has strong convex hull margin against the rest. We complement the upper bounds by showing that in the worst case any algorithm needs $Ω\big(k m \log \frac{1}γ\big)$ seed and label queries to learn a $k$-class classifier with strong convex hull margin $γ$.

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arXiv2021

Efficient and near-optimal algorithms for sampling small connected subgraphs

We study the following problem: given an integer $k \ge 3$ and a simple graph $G$, sample a connected induced $k$-node subgraph of $G$ uniformly at random. This is a fundamental graph mining primitive with applications in social network analysis, bioinformatics, and more. Surprisingly, no efficient algorithm is known for uniform sampling; the only somewhat efficient algorithms available yield samples that are only approximately uniform, with running times that are unclear or suboptimal. In this work we provide: (i) a near-optimal mixing time bound for a well-known random walk technique, (ii) the first efficient algorithm for truly uniform graphlet sampling, and (iii) the first sublinear-time algorithm for $ε$-uniform graphlet sampling.

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arXiv2021

Faster motif counting via succinct color coding and adaptive sampling

We address the problem of computing the distribution of induced connected subgraphs, aka \emph{graphlets} or \emph{motifs}, in large graphs. The current state-of-the-art algorithms estimate the motif counts via uniform sampling, by leveraging the color coding technique by Alon, Yuster and Zwick. In this work we extend the applicability of this approach, by introducing a set of algorithmic optimizations and techniques that reduce the running time and space usage of color coding and improve the accuracy of the counts. To this end, we first show how to optimize color coding to efficiently build a compact table of a representative subsample of all graphlets in the input graph. For $8$-node motifs, we can build such a table in one hour for a graph with $65$M nodes and $1.8$B edges, which is $2000$ times larger than the state of the art. We then introduce a novel adaptive sampling scheme that breaks the "additive error barrier" of uniform sampling, guaranteeing multiplicative approximations instead of just additive ones. This allows us to count not only the most frequent motifs, but also extremely rare ones. For instance, on one graph we accurately count nearly $10.000$ distinct $8$-node motifs whose relative frequency is so small that uniform sampling would literally take centuries to find them. Our results show that color coding is still the most promising approach to scalable motif counting.

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arXiv2021

Exact Recovery of Clusters in Finite Metric Spaces Using Oracle Queries

We investigate the problem of exact cluster recovery using oracle queries. Previous results show that clusters in Euclidean spaces that are convex and separated with a margin can be reconstructed exactly using only $O(\log n)$ same-cluster queries, where $n$ is the number of input points. In this work, we study this problem in the more challenging non-convex setting. We introduce a structural characterization of clusters, called $(β,γ)$-convexity, that can be applied to any finite set of points equipped with a metric (or even a semimetric, as the triangle inequality is not needed). Using $(β,γ)$-convexity, we can translate natural density properties of clusters (which include, for instance, clusters that are strongly non-convex in $\mathbb{R}^d$) into a graph-theoretic notion of convexity. By exploiting this convexity notion, we design a deterministic algorithm that recovers $(β,γ)$-convex clusters using $O(k^2 \log n + k^2 (6/βγ)^{dens(X)})$ same-cluster queries, where $k$ is the number of clusters and $dens(X)$ is the density dimension of the semimetric. We show that an exponential dependence on the density dimension is necessary, and we also show that, if we are allowed to make $O(k^2 + k\log n)$ additional queries to a "cluster separation" oracle, then we can recover clusters that have different and arbitrary scales, even when the scale of each cluster is unknown.

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