arXiv subjects

Karolina Drabik

Publications and source records attributed to Karolina Drabik.

5 recordsLinked to original sources

arXiv2026

Minimal-Norm Univariate Two-Layer ReLU Classification: Exact Solutions and Global Optimality with Skip Connections

We study minimal-norm interpolation and $\ell_2$-regularized logistic-loss minimization for binary classification by univariate two-layer ReLU networks. We give complete geometric characterizations of the optimal classifiers in function space, resolving how the solutions depend on whether hidden-layer biases are included in the parameter norm. When biases are unpenalized, the minimal-norm interpolators are exactly the continuous piecewise-affine functions that hug every label switch and have kinks of the appropriate convexity. When biases are penalized, the minimizer is unique in function space, has exactly one kink in each intermediate same-label segment, and is therefore a sparsest positive-margin classifier. We further show that adding a free affine skip connection leaves these function-space solutions unchanged but fundamentally improves the parameter-space landscape: every KKT point of the constrained problem becomes globally optimal, whereas suboptimal KKT points can occur without the skip connection. We establish analogous global-optimality and geometric results for sufficiently weak $\ell_2$-regularization of the logistic loss. In the unpenalized-bias case, we identify an additional sparsity-like restriction, implying that most minimal-norm interpolators cannot arise as small-regularization limits of margin-normalized logistic-loss minimizers. Numerical experiments across varying dataset complexity and network width support the predicted landscape and sparsity phenomena.

cs.LG↗

arXiv2026

Variants of Merge-Width and Applications

Merge-width is a recently introduced family of graph parameters that unifies treewidth, clique-width, twin-width, and generalised colouring numbers. We prove the equivalence of several alternative definitions of merge-width, thus demonstrating the robustness of the notion. Our characterisation via definable merge-width uses vertex orderings inspired by generalised colouring numbers from sparsity theory, and enables us to obtain the first non-trivial approximation algorithm for merge-width, running in time $n^{O(1)} \cdot 2^n$. We also obtain a new characterisation of bounded clique-width in terms of vertex orderings, and establish that graphs of bounded merge-width admit sparse quotients with bounded strong colouring numbers, are quasi-isometric to graphs of bounded expansion, and admit neighbourhood covers with constant overlap. We also discuss several other variants of merge-width and connections to adjacency labelling schemes.

math.CO↗

arXiv2026

Flips and Merge-Width in Sparse Graphs

A flip of a graph is obtained by complementing the edge relation within a set of vertices. Flips are typically used to separate vertices in a graph, by increasing the distances between them. We show that in $K_{t,t}$-free graphs, every short sequence of flips can be simulated by a short sequence of vertex deletions that achieves a similar degree of separation: distances in the resulting graph are, up to a factor of three, at least as large as those obtained after the flips. This result provides a simple and uniform explanation of an emerging pattern in structural graph theory and finite model theory: the $K_{t,t}$-free fragment of a tameness notion for dense graphs often coincides with a tameness notion for sparse graphs. As immediate applications, we recover the following known equivalences. In the $K_{t,t}$-free setting, the dense notions (1) bounded shrub-depth, (2) bounded clique-width, (3) bounded flip-width, (4) monadic dependence, respectively, coincide with the sparse notions (1) bounded tree-depth, (2) bounded tree-width, (3) bounded expansion, and (4) no-where dense-ness. Furthermore, we reprove the result by Dreier and Toruńczyk (STOC 2025) stating that $K_{t,t}$-free classes of bounded merge-width have bounded expansion. Our proof provides explicit bounds and is direct, as it shows how to construct strong coloring orders (witnesses of bounded expansion) from merge sequences (witnesses of bounded merge-width). Along the way, we identify a new family of graph parameters, dubbed separation-width, that is sandwiched between the strong and weak coloring numbers, and is closely related to the merge-width parameters. We provide evidence that this family of graph parameters, apparently overlooked in the literature, may play a fundamental role in the study of sparse graphs.

math.CO↗

arXiv2025

Finding Diverse Solutions Parameterized by Cliquewidth

Finding a few solutions for a given problem that are diverse, as opposed to finding a single best solution to solve the problem, has recently become a notable topic in theoretical computer science. Recently, Baste, Fellows, Jaffke, Masařík, Oliveira, Philip, and Rosamond showed that under a standard structural parameterization by treewidth, one can find a set of diverse solutions for many problems with only a very small additional cost [Artificial Intelligence 2022]. In this paper, we investigate a much stronger graph parameter, the cliquewidth, which can additionally describe some dense graph classes. Broadly speaking, it describes graphs that can be recursively constructed by a few operations defined on graphs whose vertices are divided into a bounded number of groups while each such group behaves uniformly with respect to any operation. We show that for any vertex problem, if we are given a dynamic program solving that problem on cliquewidth decomposition, we can modify it to produce a few solutions that are as diverse as possible with as little overhead as in the above-mentioned treewidth paper. As a consequence, we prove that a diverse version of any MSO$_1$ expressible problem can be solved in linear FPT time parameterized by the cliquewidth, the number of sought solutions, and the number of quantifiers in the formula, which was a natural missing piece in the complexity landscape of structural graph parameters and logic for the diverse problems. We prove our results allowing for a more general natural collection of diversity functions compared to only two mostly studied diversity functions previously. That might be of independent interest as a larger pool of different diversity functions can highlight various aspects of different solutions to a problem.

cs.DS↗

arXiv2025

Fine-Grained Complexity of Ambiguity Problems on Automata and Directed Graphs

In the field of computational logic, two classes of finite automata are considered fundamental: deterministic and nondeterministic automata (DFAs and NFAs). In a more fine-grained approach three natural intermediate classes were introduced, defined by restricting the number of accepting runs of the input NFA. The classes are called: unambiguous, finitely ambiguous, and polynomially ambiguous finite automata. It was observed that central problems, like equivalence, become tractable when the input NFA is restricted to some of these classes. This naturally brought interest into problems determining whether an input NFA belongs to the intermediate classes. Our first result is a nearly complete characterization of the fine-grained complexity of these problems. We show that the respective quadratic and cubic running times of Allauzen et al. are optimal under the Orthogonal Vectors hypothesis or the k-Cycle hypothesis, for alphabets with at least two symbols. In contrast, for unary alphabets we show that all aforementioned variants of ambiguity can be decided in almost linear time. Finally, we study determinisability of unambiguous weighted automata. We positively resolve a conjecture of Allauzen and Mohri, proving that their quadratic-time algorithm for verifying determinisability of unambiguous weighted automata is optimal, assuming the Orthogonal Vectors hypothesis or the k-Cycle hypothesis. We additionally show that for unary alphabets, this can be decided in linear time.

cs.FL↗