arXiv subjects

Viktoriia Korchemna

Publications and source records attributed to Viktoriia Korchemna.

10 recordsLinked to original sources

arXiv2026

A Fixed-Parameter Algorithm for Extending Upward Planar Drawings

An upward planar drawing of a directed acyclic graph is a planar drawing where every edge is pointed upward from its tail to head. Upward planar drawings are among the most natural drawing styles of directed graphs and have been researched in a variety of different settings, recently including that of drawing extension. In the drawing extension setting, one asks: given a graph $G$ and a (typically connected) subgraph $H$ of $G$ with a drawing $Γ(H)$, can we complete $Γ(H)$ to a drawing of $G$? Drawing extension problems have been studied for numerous drawing styles; the vast majority of these are NP-hard and a typical approach aimed at circumventing their intractability is to design parameterized algorithms where the parameter measures "how much" of $G$ is still missing from the pre-drawn graph $H$. Most algorithms obtained within this framework require only a small number of edges to be missing from $H$ in order to remain efficient. In this article, we present a fixed-parameter algorithm for extending upward planar drawings which overcomes this drawback by using the $\textit{vertex+edge deletion distance}$ as the parameter, thus achieving tractability even for instances with many missing edges. A key ingredient towards our result is a novel characterization of "canonical" sets of missing edges which cross a horizontal line segment in the drawing.

cs.CG↗

arXiv2026

The Complexity of Bayesian Network Learning: Revisiting the Superstructure

We investigate the parameterized complexity of Bayesian Network Structure Learning (BNSL), a classical problem that has received significant attention in empirical but also purely theoretical studies. We follow up on previous works that have analyzed the complexity of BNSL w.r.t. the so-called superstructure of the input. While known results imply that BNSL is unlikely to be fixed-parameter tractable even when parameterized by the size of a vertex cover in the superstructure, here we show that a different kind of parameterization - notably by the size of a feedback edge set - yields fixed-parameter tractability. We proceed by showing that this result can be strengthened to a localized version of the feedback edge set, and provide corresponding lower bounds that complement previous results to provide a complexity classification of BNSL w.r.t. virtually all well-studied graph parameters. We then analyze how the complexity of BNSL depends on the representation of the input. In particular, while the bulk of past theoretical work on the topic assumed the use of the so-called non-zero representation, here we prove that if an additive representation can be used instead then BNSL becomes fixed-parameter tractable even under significantly milder restrictions to the superstructure, notably when parameterized by the treewidth alone. Last but not least, we show how our results can be extended to the closely related problem of Polytree Learning.

cs.DS↗

arXiv2024

Efficient Approximation of Fractional Hypertree Width

We give two new approximation algorithms to compute the fractional hypertree width of an input hypergraph. The first algorithm takes as input $n$-vertex $m$-edge hypergraph $H$ of fractional hypertree width at most $ω$, runs in polynomial time and produces a tree decomposition of $H$ of fractional hypertree width $O(ω\log n \log ω)$. As an immediate corollary this yields polynomial time $O(\log^2 n \log ω)$-approximation algorithms for (generalized) hypertree width as well. To the best of our knowledge our algorithm is the first non-trivial polynomial-time approximation algorithm for fractional hypertree width and (generalized) hypertree width, as opposed to algorithms that run in polynomial time only when $ω$ is considered a constant. For hypergraphs with the bounded intersection property we get better bounds, comparable with that recent algorithm of Lanzinger and Razgon [STACS 2024]. The second algorithm runs in time $n^ωm^{O(1)}$ and produces a tree decomposition of $H$ of fractional hypertree width $O(ω\log^2 ω)$. This significantly improves over the $(n+m)^{O(ω^3)}$ time algorithm of Marx [ACM TALG 2010], which produces a tree decomposition of fractional hypertree width $O(ω^3)$, both in terms of running time and the approximation ratio. Our main technical contribution, and the key insight behind both algorithms, is a variant of the classic Menger's Theorem for clique separators in graphs: For every graph $G$, vertex sets $A$ and $B$, family ${\cal F}$ of cliques in $G$, and positive rational $f$, either there exists a sub-family of $O(f \cdot \log^2 n)$ cliques in ${\cal F}$ whose union separates $A$ from $B$, or there exist $f \cdot \log |{\cal F}|$ paths from $A$ to $B$ such that no clique in ${\cal F}$ intersects more than $\log |{\cal F}|$ paths.

cs.DS↗

arXiv2024

EF1 and EFX Orientations

We study the problem of finding fair allocations -- EF1 and EFX -- of indivisible goods with orientations. In an orientation, every agent gets items from their own predetermined set. For EF1, we show that EF1 orientations always exist when agents have monotone valuations, via a pseudopolynomial-time algorithm. This surprisingly positive result is the main contribution of our paper. We complement this result with a comprehensive set of scenarios where our algorithm, or a slight modification of it, finds an EF1 orientation in polynomial time. For EFX, we focus on the recently proposed graph instances, where every agent corresponds to a vertex on a graph and their allowed set of items consists of the edges incident to their vertex. It was shown that finding an EFX orientation is NP-complete in general. We prove that it remains intractable even when the graph has a vertex cover of size 8, or when we have a multigraph with only 10 vertices. We essentially match these strong negative results with a fixed-parameter tractable algorithm that is virtually the best someone could hope for.

cs.GT↗

arXiv2024

The Complexity of Fair Division of Indivisible Items with Externalities

We study the computational complexity of fairly allocating a set of indivisible items under externalities. In this recently-proposed setting, in addition to the utility the agent gets from their bundle, they also receive utility from items allocated to other agents. We focus on the extended definitions of envy-freeness up to one item (EF1) and of envy-freeness up to any item (EFX), and we provide the landscape of their complexity for several different scenarios. We prove that it is NP-complete to decide whether there exists an EFX allocation, even when there are only three agents, or even when there are only six different values for the items. We complement these negative results by showing that when both the number of agents and the number of different values for items are bounded by a parameter the problem becomes fixed-parameter tractable. Furthermore, we prove that two-valued and binary-valued instances are equivalent and that EFX and EF1 allocations coincide for this class of instances. Finally, motivated from real-life scenarios, we focus on a class of structured valuation functions, which we term agent/item-correlated. We prove their equivalence to the ``standard'' setting without externalities. Therefore, all previous results for EF1 and EFX apply immediately for these valuations.

cs.GT↗

arXiv2023

A Structural Complexity Analysis of Synchronous Dynamical Systems

Synchronous dynamic systems are well-established models that have been used to capture a range of phenomena in networks, including opinion diffusion, spread of disease and product adoption. We study the three most notable problems in synchronous dynamic systems: whether the system will transition to a target configuration from a starting configuration, whether the system will reach convergence from a starting configuration, and whether the system is guaranteed to converge from every possible starting configuration. While all three problems were known to be intractable in the classical sense, we initiate the study of their exact boundaries of tractability from the perspective of structural parameters of the network by making use of the more fine-grained parameterized complexity paradigm. As our first result, we consider treewidth - as the most prominent and ubiquitous structural parameter - and show that all three problems remain intractable even on instances of constant treewidth. We complement this negative finding with fixed-parameter algorithms for the former two problems parameterized by treedepth, a well-studied restriction of treewidth. While it is possible to rule out a similar algorithm for convergence guarantee under treedepth, we conclude with a fixed-parameter algorithm for this last problem when parameterized by treedepth and the maximum in-degree.

cs.DS↗

arXiv2023

Counting Vanishing Matrix-Vector Products

Consider the following parameterized counting variation of the classic subset sum problem, which arises notably in the context of higher homotopy groups of topological spaces: Let $\mathbf{v} \in \mathbb{Q}^d$ be a rational vector, $(T_{1}, T_{2} \ldots T_{m})$ a list of $d \times d$ rational matrices, $S \in \mathbb{Q}^{h \times d}$ a rational matrix not necessarily square and $k$ a parameter. The goal is to compute the number of ways one can choose $k$ matrices $T_{i_1}, T_{i_2}, \ldots, T_{i_k}$ from the list such that $ST_{i_k} \cdots T_{i_1}\mathbf{v} = \mathbf{0} \in \mathbb{Q}^h$. In this paper, we show that this problem is $\# W[2]$-hard for parameter $k$. As a consequence, computing the $k$-th homotopy group of a $d$-dimensional 1-connected topological space for $d > 3$ is $\# W[2]$-hard for parameter $k$. We also discuss a decision version of the problem and its several modifications for which we show $W[1]/W[2]$-hardness. This is in contrast to the parameterized $k$-sum problem, which is only $W[1]$-hard (Abboud-Lewi-Williams, ESA'14). In addition, we show that the decision version of the problem without parameter is an undecidable problem, and we give a fixed-parameter tractable algorithm for matrices of bounded size over finite fields, parameterized the matrix dimensions and the order of the field.

cs.CC↗

arXiv2022

The Fine-Grained Complexity of Graph Homomorphism Parameterized by Clique-Width

The generic homomorphism problem, which asks whether an input graph $G$ admits a homomorphism into a fixed target graph $H$, has been widely studied in the literature. In this article, we provide a fine-grained complexity classification of the running time of the homomorphism problem with respect to the clique-width of $G$ (denoted $\operatorname{cw}$) for virtually all choices of $H$ under the Strong Exponential Time Hypothesis. In particular, we identify a property of $H$ called the signature number $s(H)$ and show that for each $H$, the homomorphism problem can be solved in time $\mathcal{O}^*(s(H)^{\operatorname{cw}})$. Crucially, we then show that this algorithm can be used to obtain essentially tight upper bounds. Specifically, we provide a reduction that yields matching lower bounds for each $H$ that is either a projective core or a graph admitting a factorization with additional properties -- allowing us to cover all possible target graphs under long-standing conjectures.

cs.CC↗

arXiv2022

Slim Tree-Cut Width

Tree-cut width is a parameter that has been introduced as an attempt to obtain an analogue of treewidth for edge cuts. Unfortunately, in spite of its desirable structural properties, it turned out that tree-cut width falls short as an edge-cut based alternative to treewidth in algorithmic aspects. This has led to the very recent introduction of a simple edge-based parameter called edge-cut width [WG 2022], which has precisely the algorithmic applications one would expect from an analogue of treewidth for edge cuts, but does not have the desired structural properties. In this paper, we study a variant of tree-cut width obtained by changing the threshold for so-called thin nodes in tree-cut decompositions from 2 to 1. We show that this "slim tree-cut width" satisfies all the requirements of an edge-cut based analogue of treewidth, both structural and algorithmic, while being less restrictive than edge-cut width. Our results also include an alternative characterization of slim tree-cut width via an easy-to-use spanning-tree decomposition akin to the one used for edge-cut width, a characterization of slim tree-cut width in terms of forbidden immersions as well as approximation algorithm for computing the parameter.

cs.CC↗

arXiv2022

Edge-Cut Width: An Algorithmically Driven Analogue of Treewidth Based on Edge Cuts

Decompositional parameters such as treewidth are commonly used to obtain fixed-parameter algorithms for NP-hard graph problems. For problems that are W[1]-hard parameterized by treewidth, a natural alternative would be to use a suitable analogue of treewidth that is based on edge cuts instead of vertex separators. While tree-cut width has been coined as such an analogue of treewidth for edge cuts, its algorithmic applications have often led to disappointing results: out of twelve problems where one would hope for fixed-parameter tractability parameterized by an edge-cut based analogue to treewidth, eight were shown to be W[1]-hard parameterized by tree-cut width. As our main contribution, we develop an edge-cut based analogue to treewidth called edge-cut width. Edge-cut width is, intuitively, based on measuring the density of cycles passing through a spanning tree of the graph. Its benefits include not only a comparatively simple definition, but mainly that it has interesting algorithmic properties: it can be computed by a fixed-parameter algorithm, and it yields fixed-parameter algorithms for all the aforementioned problems where tree-cut width failed to do so.

cs.DS↗