Juha Harviainen

Sebastian Schmidt, Juha Harviainen, Corentin Moumard et al.

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.

Juha Harviainen, Frank Sommer, Manuel Sorge

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.

Juha Harviainen, Pekka Parviainen, Vidya Sagar Sharma

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.

Francisco Sena, Aleksandr Politov, Corentin Moumard et al.

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.

Juha Harviainen, Frank Sommer, Manuel Sorge et al.

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.

Daniele Nikzad, Alexander Zhilkin, Juha Harviainen et al.

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.

Juha Harviainen, Frank Sommer, Manuel Sorge

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.