arXiv subjects

Marek Sokołowski

Publications and source records attributed to Marek Sokołowski.

At least 19 recordsLinked to original sources

arXiv2026

Time-Optimal APSP and Matrix Multiplication in Classes of Linear Neighborhood Complexity

The notion of linear neighborhood complexity is a very general structural assumption on a graph class, covering most classes of sparse graphs such as planar graphs, graphs excluding a fixed (topological) minor, or bounded expansion graphs, as well as many structured classes of dense graphs, such as graphs of bounded clique-width, twin-width, merge-width, or flip-width. In this work, we present $O(n^2)$-time optimal algorithms for $n$-vertex graphs coming from a class of linear neighborhood complexity for the following problems: $\bullet$ All-Pairs Shortest Paths, $\bullet$ the multiplication of the adjacency matrix $M$ of the input graph with any $n \times n$ matrix. More specifically, after a quadratic preprocessing, we can multiply $M$ with any $n$-vector in $O(n)$ time. This solves several questions raised in [Bonnet, Kim, Geniet, Moon; ICALP '26], and improves and generalizes results in several other recent papers [Bonnet, Giocanti, Ossona de Mendez, Thomassé; STACS '23], [Bannach, Marwitz, Tantau; STACS '24], [Anand, van den Brand, McCarty; NeurIPS '26], [Kozma, Opler '26], and [Cardinal, McCarty, Yuditsky '26]. We also extend our results to classes of bounded VC density. In classes of linear neighborhood complexity, we also give a triangle-detection algorithm in randomized linear time $O(n+m)$ in $n$-vertex $m$-edge graphs, a $K_4$-detection algorithm in randomized $O(n \log^5 n + m \log n)$ or deterministic $O(n^2)$ time, and a $K_5$-detection algorithm in randomized $O(n \log^9 n + m \log^5 n)$ time.

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arXiv2026

A tight lower bound for malicious online bipartite matching with limited recourse budget

We study one-sided online bipartite matching with recourse. In this setting, one side of a bipartite graph is known in advance, while vertices on the other side arrive online together with their incident edges. After each arrival, the algorithm must maintain a maximum-cardinality matching while minimizing the total number of reallocations, also known as the recourse budget. Despite extensive work, the exact recourse complexity of the problem remains unsettled: the best lower bound is $Ω(n \log n)$, whereas the best upper bound is $\mathcal{O}(n \log^2 n)$, where $n$ denotes the number of online vertices. Tight upper bounds of $\mathcal{O}(n \log n)$ are known only for restricted graph classes, such as forests. The best known upper bounds are attained by a very simple and natural algorithm SAP, which after each arrival applies a shortest augmenting path, and it is conjectured to be optimal. All known upper bound analyses of this algorithm do not depend on the particular maximum matching maintained by the algorithm. Consequently, they also apply to a more difficult problem, which we call the malicious matching setting: after each arrival, the maintained matching is replaced by a worst-case maximum matching for the next step. This led to the conjecture that the malicious setting still admits an $\mathcal{O}(n \log n)$ recourse bound, in line with the conjectured optimal complexity of the original model. Our main result is an $Ω(n \log^2 n)$ lower bound for the malicious matching setting, thus disproving the conjecture. Together with the previous upper bound, this settles the asymptotic recourse complexity of the malicious variant of the problem. We complement our lower bound with an upper bound of $\mathcal{O}(n \log n)$ for expander graphs.

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arXiv2026

Multiway $f$-Cut is fixed-parameter tractable

A connectivity function on a finite set $E$ is a function $f\colon 2^E\to\mathbb Z$ that is submodular and symmetric, with $f(\varnothing)=0$. Given a connectivity function $f$ via a value oracle, terminals $t_1,\ldots,t_r\in E$, and an integer $k$, the Multiway $f$-Cut problem asks whether $E$ has a partition $(P_1,\ldots,P_r)$ with $t_i\in P_i$ for every $i$ and $\sum_{i=1}^r f(P_i)\le k$. We prove that Multiway $f$-Cut is fixed-parameter tractable parameterized by $k$. Cut functions of graphs are connectivity functions, so as a special case we recover the classical result that Edge Multiway Cut in graphs is fixed-parameter tractable. Our proof of correctness is completely elementary, and is arguably the simplest known proof of this fact.

cs.DM↗

arXiv2026

An Erdős-Pósa theorem for cycles and faces of distinct lengths

We show that for every $k \in \mathbb{N}$, every graph $G$ contains $k$ vertex-disjoint cycles of different lengths, or there exists a set $X \subseteq V(G)$ with $|X| \in \mathcal{O}(k^6\mathsf{polylog}(k))$ such that $G-X$ has at most $k-1$ cycle lengths. We also prove analogous results for facial lengths of embedded graphs. Let $G$ be a graph with a closed 2-cell embedding $ψ$ on a surface $Σ$ of Euler genus $g$, let $c$ be a colouring of the faces $\mathcal{F}(ψ)$ of $ψ$, and let $R(G,ψ)$ be the radial graph of $(G, ψ)$. Then there exist $k$ faces $F_1, \ldots , F_k \in \mathcal{F}(ψ)$ that are given pairwise distinct colours by $c$ and are pairwise at distance at least $d$ in $ψ$, or there exists a set $X \subseteq V(G)$ of order at most $\mathcal{O}(k^2dg)$ such that $|\{ c(F) \mid F \in \mathcal{F}(ψ) \text{ and } V(F) \cap \bigcup_{x \in X} N^d_{R(G,ψ)}(x) = \emptyset \}| \leq k(k+2)$. Finally, using a result from additive combinatorics, we show that there are subdivided ladders with only a small number of cycle lengths. This suggests that it may be difficult to improve our bounds.

math.CO↗

arXiv2026

Fast decremental tree sums in forests

We study two fundamental decremental dynamic graph problems. In both problems, we need to maintain a vertex-weighted forest of size $n$ under edge deletions, weight updates, and a certain information-retrieval query. Both problems can be solved in $O(\log n)$ time per update/query using standard dynamic forest data structures like top trees, even if additionally edge insertions are allowed. We investigate whether the deletion-only problem can be solved faster. First, we consider $\texttt{tree-sum}$ queries, where we ask for the sum of vertex weights in one of the connected components (i.e., trees) in the forest. We give a data structure with $O(n)$ preprocessing time and $O(\log^* n)$ time per operation, based on a micro-macro tree decomposition (Alstrup et al., 1997). If the forest is unweighted (i.e., all weights are 1 and cannot be changed), then the operation time can be improved to $O(1)$. Additionally, we give an asymptotically universally optimal algorithm. More specifically, our algorithm works in the group model, and processes $m$ operations on an initial forest $F$ in running time $O( \mathrm{OPT}(F, m) )$. Here $\mathrm{OPT}(F, m)$ is the number of weight additions and subtractions that a best possible algorithm performs to handle a worst-case instance for a fixed initial forest $F$ and a fixed number $m$ of operations. We achieve this with a combination of the aforementioned decomposition technique, precomputation of optimal data structures for very small instances, and some insights into the behavior of $\mathrm{OPT}$. Note that even the worst-case complexity of this algorithm remains unknown to us. Second, we consider $\texttt{subtree-sum}$ queries. Here, the forest is rooted, and a query $\texttt{subtree-sum}(v)$ returns the sum of weights in the subtree rooted at $v$. We show tight bounds for several variants of this problem. [...]

cs.DS↗

arXiv2026

Dynamic Detours

Fix a parameter $k\in \mathbf{N}$. We give dynamic data structures that for a fully dynamic undirected graph $G$, updated over time by edge insertions and edge deletions, can answer the following queries: - Long $(u,v)$-path: Given $u,v\in V(G)$, is there a path from $u$ to $v$ of length at least $k$? - Long $(u,v)$-detour: Given $u,v\in V(G)$, is there a path from $u$ to $v$ of length at least $\text{dist}_G(u,v)+k$? - Even/odd $(u,v)$-path: Given $u,v\in V(G)$, is there a path from $u$ to $v$ of even/odd length? The amortized time of executing an update or answering a query is $2^{O(k^3)} \log n + O(\log^2 n \log^2 \log n)$ in the first two cases, and $O(\log^2 n \log^2 \log n)$ in the last, where $n$ is the number of vertices of $G$. The first result is in sharp contrast with known conditional lower bounds for reporting paths of length at most $k$. Specifically, there is no data structure supporting queries about $(u,v)$-paths of length at most two in time $n^{o(1)}$ unless the Triangle Conjecture fails. Our main technical contribution is a mechanism of "delayed edge insertion" that works locally on the level of biconnected components.

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arXiv2026

Polynomial-size encoding of all cuts of small value in integer-valued symmetric submodular functions

We study connectivity functions, that is, integer-valued symmetric submodular functions on a finite ground set attaining $0$ on the empty set. For a connectivity function $f$ on an $n$-element set $V$ and an integer $k\ge 0$, we show that the family of all sets $X\subseteq V$ with $f(X)=k$ admits a polynomial-size representation: it can be described by a list of at most $O(n^{4k})$ items, each consisting of a set to be included, another set to be excluded, and a partition of remaining elements, such that the union of some members of the partition and the set to be included are precisely all sets $X$ with $f(X)=k$. We also give an algorithm that constructs this representation in time $O(n^{2k+7}γ+n^{2k+8}+n^{4k+2})$, where $γ$ is the oracle time to evaluate $f$. This generalizes the low rank structure theorem of Bojańczyk, Pilipczuk, Przybyszewski, Sokołowski, and Stamoulis [Low rank MSO, arXiv, 2025] on cut-rank functions on graphs to general connectivity functions. As an application, for fixed $k$, we obtain a polynomial-time algorithm for finding a set $A$ with $f(A)=k$ and a prescribed cardinality constraint on $A$.

math.CO↗

arXiv2025

Strongly Polynomial Parallel Work-Depth Tradeoffs for Directed SSSP

In this paper, we show new strongly polynomial work-depth tradeoffs for computing single-source shortest paths (SSSP) in non-negatively weighted directed graphs in parallel. Most importantly, we prove that directed SSSP can be solved within $\tilde{O}(m+n^{2-ε})$ work and $\tilde{O}(n^{1-ε})$ depth for some positive $ε>0$. In particular, for dense graphs with non-negative real weights, we provide the first nearly work-efficient strongly polynomial algorithm with sublinear depth. Our result immediately yields improved strongly polynomial parallel algorithms for min-cost flow and the assignment problem. It also leads to the first non-trivial strongly polynomial dynamic algorithm for minimum mean cycle. Moreover, we develop efficient parallel algorithms in the Word RAM model for several variants of SSSP in graphs with exponentially large edge weights.

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arXiv2025

Treedepth Inapproximability and Exponential ETH Lower Bound

Treedepth is a central parameter to algorithmic graph theory. The current state-of-the-art in computing and approximating treedepth consists of a $2^{O(k^2)} n$-time exact algorithm and a polynomial-time $O(\text{OPT} \log^{3/2} \text{OPT})$-approximation algorithm, where the former algorithm returns an elimination forest of height $k$ (witnessing that treedepth is at most $k$) for the $n$-vertex input graph $G$, or correctly reports that $G$ has treedepth larger than $k$, and $\text{OPT}$ is the actual value of the treedepth. On the complexity side, exactly computing treedepth is NP-complete, but the known reductions do not rule out a polynomial-time approximation scheme (PTAS), and under the Exponential Time Hypothesis (ETH) only exclude a running time of $2^{o(\sqrt n)}$ for exact algorithms. We show that 1.0003-approximating treedepth is NP-hard, and that exactly computing the treedepth of an $n$-vertex graph requires time $2^{Ω(n)}$, unless the ETH fails. We further derive that there exist absolute constants $δ, c > 0$ such that any $(1+δ)$-approximation algorithm requires time $2^{Ω(n / \log^c n)}$. We do so via a simple direct reduction from Satisfiability to Treedepth, inspired by a reduction recently designed for Treewidth [STOC '25].

cs.CC↗

arXiv2025

Elementary first-order model checking for sparse graphs

It is known that for subgraph-closed graph classes the first-order model checking problem is fixed-parameter tractable if and only if the class is nowhere dense [Grohe, Kreutzer, Siebertz, STOC 2014]. However, the dependency on the formula size is non-elementary, and in fact, this is unavoidable even for the class of all trees [Frick and Grohe, LICS 2002]. On the other hand, it is known that the dependency is elementary for classes of bounded degree [Frick and Grohe, LICS 2002] as well as for classes of bounded pathwidth [Lampis, ICALP 2023]. In this paper we generalise these results and almost completely characterise subgraph-closed graph classes for which the model checking problem is fixed-parameter tractable with an elementary dependency on the formula size. Those are the graph classes for which there exists a number $d$ such that for every $r$, some tree of depth $d$ and size bounded by an elementary function of $r$ is avoided as an $({\leq} r)$-subdivision in all graphs in the class. In particular, this implies that if the class in question excludes a fixed tree as a topological minor, then first-order model checking for graphs in the class is fixed-parameter tractable with an elementary dependency on the formula size.

cs.LO↗

arXiv2025

Fully dynamic biconnectivity in $\tilde{\mathcal{O}}(\log^2 n)$ time

We present a deterministic fully-dynamic data structure for maintaining information about the cut-vertices in a graph; i.e. the vertices whose removal would disconnect the graph. Our data structure supports insertion and deletion of edges, as well as queries to whether a pair of connected vertices are either biconnected, or can be separated by a cutvertex, and in the latter case we support access to separating cutvertices. All update operations are supported in amortized $O(\log^2 n \log^2 \log n)$ time, and queries take worst-case $O(\log n \log^2 \log n)$ time. Note that these time bounds match the current best for deterministic dynamic connectivity up to $\log \log n$ factors. We obtain our improved running time by a series of reductions from the original problem into well-defined data structure problems. While we do apply the well-known techniques for improving running time of two-edge connectivity [STOC'00, SODA'18], these techniques alone do not lead to an update time of $\tilde{O}(\log^3 n)$, let alone the $\tilde{O}(\log^2 n)$ we give as a final result. Our contributions include a formally defined transient expose operation, which can be thought of as a cheaper read-only expose operation on a top tree. For each vertex in the graph, we maintain a data structure over its neighbors, and in this data structure we apply biasing (twice) to save two $\tilde{O}(\log n)$ factors. One of these biasing techniques is a new biased disjoint sets data structure, which may be of independent interest. Moreover, in this neighborhood data structure, we facilitate that the vertex can select two VIP neighbors that get special treatment, corresponding to its potentially two neighbors on an exposed path, improving a $\log n$-time operation down to constant time. It is this combination of VIP neighbors with the transient expose that saves an $\tilde{O}(\log n)$-factor from another bottleneck.

cs.DS↗

arXiv2025

Flipper games for monadically stable graph classes

A class of graphs $\mathscr{C}$ is monadically stable if for any unary expansion $\widehat{\mathscr{C}}$ of $\mathscr{C}$, one cannot interpret, in first-order logic, arbitrarily long linear orders in graphs from $\widehat{\mathscr{C}}$. It is known that nowhere dense graph classes are monadically stable; these encompass most of the studied concepts of sparsity in graphs, including graph classes that exclude a fixed topological minor. On the other hand, monadic stability is a property expressed in purely model-theoretic terms and hence it is also suited for capturing structure in dense graphs. For several years, it has been suspected that one can create a structure theory for monadically stable graph classes that mirrors the theory of nowhere dense graph classes in the dense setting. In this work we provide a step in this direction by giving a characterization of monadic stability through the Flipper game: a game on a graph played by Flipper, who in each round can complement the edge relation between any pair of vertex subsets, and Connector, who in each round localizes the game to a ball of bounded radius. This is an analog of the Splitter game, which characterizes nowhere dense classes of graphs (Grohe, Kreutzer, and Siebertz, J.ACM'17). We give two different proofs of our main result. The first proof uses tools from model theory, and it exposes an additional property of monadically stable graph classes that is close in spirit to definability of types. Also, as a byproduct, we give an alternative proof of the recent result of Braunfeld and Laskowski (arXiv 2209.05120) that monadic stability for graph classes coincides with existential monadic stability. The second proof relies on the recently introduced notion of flip-wideness (Dreier, Mählmann, Siebertz, and Toruńczyk, ICALP 2023) and provides an efficient algorithm to compute Flipper's moves in a winning strategy.

cs.LO↗

arXiv2025

Low rank MSO

We introduce a new logic for describing properties of graphs, which we call low rank MSO. This is the fragment of monadic second-order logic in which set quantification is restricted to vertex sets of bounded cutrank. We prove the following statements about the expressive power of low rank MSO. - Over any class of graphs that is weakly sparse, low rank MSO has the same expressive power as separator logic. This equivalence does not hold over all graphs. - Over any class of graphs that has bounded VC dimension, low rank MSO has the same expressive power as flip-connectivity logic. This equivalence does not hold over all graphs. - Over all graphs, low rank MSO has the same expressive power as flip-reachability logic. Here, separator logic is an extension of first-order logic by basic predicates for checking connectivity, which was proposed by Bojańczyk [ArXiv 2107.13953] and by Schirrmacher, Siebertz, and Vigny [ACM ToCL 2023]. Flip-connectivity logic and flip-reachability logic are analogues of separator logic suited for non-sparse graphs, which we propose in this work. In particular, the last statement above implies that every property of undirected graphs expressible in low rank MSO can be decided in polynomial time.

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arXiv2025

Maintaining $\mathsf{CMSO}_2$ properties on dynamic structures with bounded feedback vertex number

Let $φ$ be a sentence of $\mathsf{CMSO}_2$ (monadic second-order logic with quantification over edge subsets and counting modular predicates) over the signature of graphs. We present a dynamic data structure that for a given graph $G$ that is updated by edge insertions and edge deletions, maintains whether $φ$ is satisfied in $G$. The data structure is required to correctly report the outcome only when the feedback vertex number of $G$ does not exceed a fixed constant $k$, otherwise it reports that the feedback vertex number is too large. With this assumption, we guarantee amortized update time ${\cal O}_{φ,k}(\log n)$. If we additionally assume that the feedback vertex number of $G$ never exceeds $k$, this update time guarantee is worst-case. By combining this result with a classic theorem of Erdős and Pósa, we give a fully dynamic data structure that maintains whether a graph contains a packing of $k$ vertex-disjoint cycles with amortized update time ${\cal O}_{k}(\log n)$. Our data structure also works in a larger generality of relational structures over binary signatures.

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arXiv2024

Half-integral Erdős-Pósa property for non-null $S$-$T$ paths

For a group $Γ$, a $Γ$-labelled graph is an undirected graph $G$ where every orientation of an edge is assigned an element of $Γ$ so that opposite orientations of the same edge are assigned inverse elements. A path in $G$ is non-null if the product of the labels along the path is not the neutral element of $Γ$. We prove that for every finite group $Γ$, non-null $S$-$T$ paths in $Γ$-labelled graphs exhibit the half-integral Erdős-Pósa property. More precisely, there is a function $f$, depending on $Γ$, such that for every $Γ$-labelled graph $G$, subsets of vertices $S$ and $T$, and integer $k$, one of the following objects exists: a family $\cal F$ consisting of $k$ non-null $S$-$T$ paths in $G$ such that every vertex of $G$ participates in at most two paths of $\cal F$; or a set $X$ consisting of at most $f(k)$ vertices that meets every non-null $S$-$T$ path in $G$. This in particular proves that in undirected graphs $S$-$T$ paths of odd length have the half-integral Erdős-Pósa property.

math.CO↗

arXiv2024

Almost-linear time parameterized algorithm for rankwidth via dynamic rankwidth

We give an algorithm that given a graph $G$ with $n$ vertices and $m$ edges and an integer $k$, in time $O_k(n^{1+o(1)}) + O(m)$ either outputs a rank decomposition of $G$ of width at most $k$ or determines that the rankwidth of $G$ is larger than $k$; the $O_k(\cdot)$-notation hides factors depending on $k$. Our algorithm returns also a $(2^{k+1}-1)$-expression for cliquewidth, yielding a $(2^{k+1}-1)$-approximation algorithm for cliquewidth with the same running time. This improves upon the $O_k(n^2)$ time algorithm of Fomin and Korhonen [STOC 2022]. The main ingredient of our algorithm is a fully dynamic algorithm for maintaining rank decompositions of bounded width: We give a data structure that for a dynamic $n$-vertex graph $G$ that is updated by edge insertions and deletions maintains a rank decomposition of $G$ of width at most $4k$ under the promise that the rankwidth of $G$ never grows above $k$. The amortized running time of each update is $O_k(2^{\sqrt{\log n} \log \log n})$. The data structure furthermore can maintain whether $G$ satisfies some fixed ${\sf CMSO}_1$ property within the same running time. We also give a framework for performing ``dense'' edge updates inside a given set of vertices $X$, where the new edges inside $X$ are described by a given ${\sf CMSO}_1$ sentence and vertex labels, in amortized $O_k(|X| \cdot 2^{\sqrt{\log n} \log \log n})$ time. Our dynamic algorithm generalizes the dynamic treewidth algorithm of Korhonen, Majewski, Nadara, Pilipczuk, and Sokołowski [FOCS 2023].

cs.DS↗

arXiv2023

Exact Shortest Paths with Rational Weights on the Word RAM

Exact computation of shortest paths in weighted graphs has been traditionally studied in one of two settings. First, one can assume that the edge weights are real numbers and all the performed operations on reals (typically comparisons and additions) take constant time. Classical Dijkstra's and Bellman-Ford algorithms have been described in this setting. More efficient exact shortest paths algorithms have been obtained for integer-weighted graphs. Integrality assumption not only enables faster algorithms but also allows implementing the aforementioned algorithms in a much more realistic word RAM model where only arithmetic operations on $O(\log{n})$-bit integers are performed in constant time. On the word RAM one can as efficiently exactly encode even \emph{rational-weighted} instances with $O(\log{n})$-bit numerators and denominators. However, the known exact real-weighted shortest paths algorithms, run on such a rational input, can easily encounter intermediate values of $Θ(n)$ bits if represented exactly. This leads to a factor-$Ω(n)$ slowdown on the word RAM. At the same time, the scaling algorithms suited for integer weights do not produce exact solutions for rational inputs without dramatically increasing their accuracy. In this paper, we design randomized exact single-source shortest paths algorithms for rational-weighted graphs on the word RAM. Most importantly, in the non-negative case, we obtain a near-linear time algorithm matching Dijkstra's algorithm running time up to polylogarithmic factors. In presence of negative weights, we give an $\tilde{O}(n^{2.5})$-time algorithm breaking through the best known strongly polynomial bound attained by Bellman-Ford for sufficiently dense graphs.

cs.DS↗

arXiv2023

Fully dynamic approximation schemes on planar and apex-minor-free graphs

The classic technique of Baker [J. ACM '94] is the most fundamental approach for designing approximation schemes on planar, or more generally topologically-constrained graphs, and it has been applied in a myriad of different variants and settings throughout the last 30 years. In this work we propose a dynamic variant of Baker's technique, where instead of finding an approximate solution in a given static graph, the task is to design a data structure for maintaining an approximate solution in a fully dynamic graph, that is, a graph that is changing over time by edge deletions and edge insertions. Specifically, we address the two most basic problems -- Maximum Weight Independent Set and Minimum Weight Dominating Set -- and we prove the following: for a fully dynamic $n$-vertex planar graph $G$, one can: * maintain a $(1-\varepsilon)$-approximation of the maximum weight of an independent set in $G$ with amortized update time $f(\varepsilon)\cdot n^{o(1)}$; and, * under the additional assumption that the maximum degree of the graph is bounded at all times by a constant, also maintain a $(1+\varepsilon)$-approximation of the minimum weight of a dominating set in $G$ with amortized update time $f(\varepsilon)\cdot n^{o(1)}$. In both cases, $f(\varepsilon)$ is doubly-exponential in $\mathrm{poly}(1/\varepsilon)$ and the data structure can be initialized in time $f(\varepsilon)\cdot n^{1+o(1)}$. All our results in fact hold in the larger generality of any graph class that excludes a fixed apex-graph as a minor.

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