arXiv subjects

Juha Harviainen

Publications and source records attributed to Juha Harviainen.

11 recordsLinked to original sources

arXiv2026

The Power of Graph Doubling: Computing Ultrabubbles in a Bidirected Graph by Reducing to Weak Superbubbles

Bidirected graphs are a common generalisation of directed graphs where arcs can also be incoming to both their incident nodes, or outgoing from both their incident nodes. Such arcs allow a walk to change direction. Some algorithms can easily be adapted from directed graphs to bidirected graphs, such as shortest path algorithms. These adaptions are already used in practice, and implicitly use the graph doubling technique to apply an algorithm for directed graphs to bidirected graphs. In other cases, the applicability of graph doubling is not that obvious. For example, superbubbles and their generalisation to bidirected graphs ultrabubbles. Ultrabubbles are a common structure in bidirected biological graphs which carries biological meaning, but also functions as a nested clustering method, since an ultrabubble is separated by only two nodes from the rest of the graph. There is an existing method that enumerates a structure similar to ultrabubbles by enumerating (weak) superbubbles in the doubled graph. However, the literature does not make any direct connection between superbubbles and ultrabubbles except that a superbubble is an ultrabubble in a directed graph. Only a partial result connecting superbubbles and ultrabubbles exists by Harviainen et al. (2026). Graph doubling on the other hand maintains connectivity, and allows to draw a direct connection between ultrabubbles and weak superbubbles. This results in the first linear-time reduction-based algorithm for computing ultrabubbles on any bidirected graph. Together with the fact that graph doubling is already used implicitly in simple cases, our result motivates that graph doubling is a powerful yet simple technique to apply algorithms for directed graphs to bidirected graphs.

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arXiv2026

Exact and Approximate Algorithms for Polytree Learning

Polytrees are a subclass of Bayesian networks that seek to capture the conditional dependencies between a set of $n$ variables as a directed forest and are motivated by their more efficient inference and improved interpretability. Since the problem of learning the best polytree is NP-hard, we study which restrictions make it more tractable by considering for example in-degree bounds, properties of score functions measuring the quality of a polytree, and approximation algorithms. We devise an algorithm that finds the optimal polytree in time $O((2+ε)^n)$ for arbitrarily small $ε> 0$ and any constant in-degree bound $k$, improving over the fastest previously known algorithm of time complexity $O(3^n)$. We further give polynomial-time algorithms for finding a polytree whose score is within a factor of $k$ from the optimal one for arbitrary scores and a factor of $2$ for additive ones. Many of the results are complemented by (nearly) tight lower bounds for either the time complexity or the approximation factors.

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arXiv2026

Identifying bubble-like subgraphs in linear-time via a unified SPQR-tree framework

A fundamental algorithmic problem in computational biology is to find all subgraphs of a given type (superbubbles, snarls, and ultrabubbles) in a directed or bidirected input graph. These correspond to regions of genetic variation and are useful in analyzing collections of genomes. We present the first linear-time algorithms for identifying all snarls and all ultrabubbles, resolving problems open since 2018. The algorithm for snarls relies on a new linear-size representation of all snarls with respect to the number of vertices in the graph. We employ the well-known SPQR-tree decomposition, which encodes all 2-separators of a biconnected graph. After several dynamic-programming-style traversals of this tree, we maintain key properties (such as acyclicity) that allow us to decide whether a given 2-separator defines a subgraph to be reported. A crucial ingredient for linear-time complexity is that acyclicity of linearly many subgraphs can be tested simultaneously via the problem of computing all arcs in a directed graph whose removal renders it acyclic (so-called feedback arcs). As such, we prove a fundamental result for bidirected graphs, that may be of independent interest: all feedback arcs can be computed in linear time for tipless bidirected graphs, while in general this is at least as hard as matrix multiplication, assuming the k-Clique Conjecture. Our results form a unified framework that also yields a completely different linear-time algorithm for finding all superbubbles. Although some of the results are technically involved, the underlying ideas are conceptually simple, and may extend to other bubble-like subgraphs. More broadly, our work contributes to the theoretical foundations of computational biology and advances a growing line of research that uses SPQR-tree decompositions as a general tool for designing efficient algorithms, beyond their traditional role in graph drawing.

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arXiv2026

Learning Bayesian and Markov Networks with an Unreliable Oracle

We study constraint-based structure learning of Markov networks and Bayesian networks in the presence of an unreliable conditional independence oracle that makes at most a bounded number of errors. For Markov networks, we observe that a low maximum number of vertex-wise disjoint paths implies that the structure is uniquely identifiable even if the number of errors is (moderately) exponential in the number of vertices. For Bayesian networks, however, we prove that one cannot tolerate any errors to always identify the structure even when many commonly used graph parameters like treewidth are bounded. Finally, we give algorithms for structure learning when the structure is uniquely identifiable.

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arXiv2025

Identifying all snarls and superbubbles in linear-time, via a unified SPQR-tree framework

Snarls and superbubbles are fundamental pangenome decompositions capturing variant sites. These bubble-like structures underpin key tasks in computational pangenomics, including structural-variant genotyping, distance indexing, haplotype sampling, and variant annotation. Snarls can be quadratically-many in the size of the graph, and since their introduction in 2018 with the vg toolkit, there has been no work on identifying all snarls in linear time. Moreover, while it is known how to find superbubbles in linear time, this result is a highly specialized solution only achieved after a long series of papers. We present the first algorithm identifying all snarls in linear time. This is based on a new representation of all snarls, of size linear in the input graph size, and which can be computed in linear time. Our algorithm is based on a unified framework that also provides a new linear-time algorithm for finding superbubbles. An observation behind our results is that all such structures are separated from the rest of the graph by two vertices (except for cases which are trivially computable), i.e. their endpoints are a 2-separator of the underlying undirected graph. Based on this, we employ the well-known SPQR tree decomposition, which encodes all 2-separators, to guide a traversal that finds the bubble-like structures efficiently. We implemented our algorithms in C++ (available at https://github.com/algbio/BubbleFinder) and evaluated them on various pangenomic datasets. Our algorithms outcompete or they are on the same level of existing methods. For snarls, we are up to two times faster than vg, while identifying all snarls. When computing superbubbles, we are up to 50 times faster than BubbleGun. Our SPQR tree framework provides a unifying perspective on bubble-like structures in pangenomics, together with a template for finding other bubble-like structures efficiently.

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arXiv2025

Scaling Up Bayesian DAG Sampling

Bayesian inference of Bayesian network structures is often performed by sampling directed acyclic graphs along an appropriately constructed Markov chain. We present two techniques to improve sampling. First, we give an efficient implementation of basic moves, which add, delete, or reverse a single arc. Second, we expedite summing over parent sets, an expensive task required for more sophisticated moves: we devise a preprocessing method to prune possible parent sets so as to approximately preserve the sums. Our empirical study shows that our techniques can yield substantial efficiency gains compared to previous methods.

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arXiv2025

Improving Decision Trees through the Lens of Parameterized Local Search

Algorithms for learning decision trees often include heuristic local-search operations such as (1) adjusting the threshold of a cut or (2) also exchanging the feature of that cut. We study minimizing the number of classification errors by performing a fixed number of a single type of these operations. Although we discover that the corresponding problems are NP-complete in general, we provide a comprehensive parameterized-complexity analysis with the aim of determining those properties of the problems that explain the hardness and those that make the problems tractable. For instance, we show that the problems remain hard for a small number $d$ of features or small domain size $D$ but the combination of both yields fixed-parameter tractability. That is, the problems are solvable in $(D + 1)^{2d} \cdot |I|^{O(1)}$ time, where $|I|$ is the size of the input. We also provide a proof-of-concept implementation of this algorithm and report on empirical results.

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arXiv2025

Graph Reconstruction with a Connected Components Oracle

In the Graph Reconstruction (GR) problem, the goal is to recover a hidden graph by utilizing some oracle that provides limited access to the structure of the graph. The interest is in characterizing how strong different oracles are when the complexity of an algorithm is measured in the number of performed queries. We study a novel oracle that returns the set of connected components (CC) on the subgraph induced by the queried subset of vertices. Our main contributions are as follows: 1. For a hidden graph with $n$ vertices, $m$ edges, maximum degree $Δ$, and treewidth $k$, GR can be solved in $O(\min\{m / \log m, Δ^2, k^2\} \cdot \log n)$ CC queries by an adaptive randomized algorithm. 2. For a hidden graph with $n$ vertices and degeneracy $d$, GR can be solved in $O(d^2 \log^2 n)$ CC queries by an adaptive randomized algorithm. 3. For a hidden graph with $n$ vertices, $m$ edges, maximum degree $Δ$, and treewidth $k$, no algorithm can solve GR in $o(\min\{m, Δ^2, k^2\})$ CC queries.

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arXiv2025

Quantum Speedups for Bayesian Network Structure Learning

The Bayesian network structure learning (BNSL) problem asks for a directed acyclic graph that maximizes a given score function. For networks with $n$ nodes, the fastest known algorithms run in time $O(2^n n^2)$ in the worst case, with no improvement in the asymptotic bound for two decades. Inspired by recent advances in quantum computing, we ask whether BNSL admits a polynomial quantum speedup, that is, whether the problem can be solved by a quantum algorithm in time $O(c^n)$ for some constant $c$ less than $2$. We answer the question in the affirmative by giving two algorithms achieving $c \le 1.817$ and $c \le 1.982$ assuming the number of potential parent sets is, respectively, subexponential and $O(1.453^n)$. Both algorithms assume the availability of a quantum random access memory. We also prove that one presumably cannot lower the base $2$ for any classical algorithm, as that would refute the strong exponential time hypothesis.

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arXiv2025

Optimal Decision Tree Pruning Revisited: Algorithms and Complexity

We present a comprehensive classical and parameterized complexity analysis of decision tree pruning operations, extending recent research on the complexity of learning small decision trees. Thereby, we offer new insights into the computational challenges of decision tree simplification, a crucial aspect of developing interpretable and efficient machine learning models. We focus on fundamental pruning operations of subtree replacement and raising, which are used in heuristics. Surprisingly, while optimal pruning can be performed in polynomial time for subtree replacement, the problem is NP-complete for subtree raising. Therefore, we identify parameters and combinations thereof that lead to fixed-parameter tractability or hardness, establishing a precise borderline between these complexity classes. For example, while subtree raising is hard for small domain size $D$ or number $d$ of features, it can be solved in $D^{2d} \cdot |I|^{O(1)}$ time, where $|I|$ is the input size. We complement our theoretical findings with preliminary experimental results, demonstrating the practical implications of our analysis.

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arXiv2021

Approximating the Permanent with Deep Rejection Sampling

We present a randomized approximation scheme for the permanent of a matrix with nonnegative entries. Our scheme extends a recursive rejection sampling method of Huber and Law (SODA 2008) by replacing the upper bound for the permanent with a linear combination of the subproblem bounds at a moderately large depth of the recursion tree. This method, we call deep rejection sampling, is empirically shown to outperform the basic, depth-zero variant, as well as a related method by Kuck et al. (NeurIPS 2019). We analyze the expected running time of the scheme on random $(0, 1)$-matrices where each entry is independently $1$ with probability $p$. Our bound is superior to a previous one for $p$ less than $1/5$, matching another bound that was known to hold when every row and column has density exactly $p$.

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