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We study a variant of the Coordinated Motion Planning problem on undirected graphs, referred to herein as the \textsc{Coordinated Sliding-Motion Planning} (CSMP) problem. In this variant, we are given an undirected graph $G$, $k$ robots $R_1,\dots,R_k$ positioned on distinct vertices of $G$, $p\leq k$ distinct destination vertices for robots $R_1,\dots,R_p$, and $\ell \in \mathbb{N}$. The problem is to decide if there is a serial schedule of at most $\ell$ moves (i.e., of makespan $\ell$) such that at the end of the schedule each robot with a destination reaches it, where a robot's move is a free path (unoccupied by any robots) from its current position to an unoccupied vertex. The problem is known to be NP-hard even on full grids. It has been studied in several contexts, including coin movement and reconfiguration problems, with respect to feasibility, complexity, and approximation. Geometric variants of the problem, in which congruent geometric-shape robots (e.g., unit disk/squares) slide or translate in the Euclidean plane, have also been studied extensively. We investigate the parameterized complexity of CSMP with respect to two parameters: the number $k$ of robots and the makespan $\ell$. As our first result, we present a fixed-parameter algorithm for CSMP parameterized by $k$. For our second result, we present a fixed-parameter algorithm parameterized by $\ell$ for the special case of CSMP in which only a single robot has a destination and the graph is planar, which we prove to be NP-complete. A crucial new ingredient for both of our results is that the solution admits a succinct representation as a small labeled topological minor of the input graph.

Enumeration kernelization for parameterized enumeration problems was defined by Creignou et al. [Theory Comput. Syst. 2017] and was later refined by Golovach et al. [J. Comput. Syst. Sci. 2022, STACS 2021] to polynomial-delay enumeration kernelization. We consider ENUM LONG-PATH, the enumeration variant of the Long-Path problem, from the perspective of enumeration kernelization. Formally, given an undirected graph G and an integer k, the objective of ENUM LONG-PATH is to enumerate all paths of G having exactly k vertices. We consider the structural parameters vertex cover number, dissociation number, and distance to clique and provide polynomial-delay enumeration kernels of polynomial size for each of these parameters.

A tangram is a word in which every letter occurs an even number of times. Thus it can be cut into parts that can be arranged into two identical words. The \emph{cut number} of a tangram is the minimum number of required cuts in this process. Tangrams with cut number one corresponds to squares. For $k\ge1$, let $t(k)$ denote the minimum size of an alphabet over which an infinite word avoids tangrams with cut number at most~$k$. The existence of infinite ternary square-free words shows that $t(1)=t(2)=3$. We show that $t(3)=t(4)=4$, answering a question from D\k{e}bski, Grytczuk, Pawlik, Przyby\l{}o, and \'Sleszy\'nska-Nowak.

Dokow and Holzman determined which predicates over $\{0, 1\}$ satisfy an analog of Arrow's theorem: all unanimous aggregators are dictatorial. Szegedy and Xu, extending earlier work of Dokow and Holzman, extended this to predicates over arbitrary finite alphabets. Mossel extended Arrow's theorem in an orthogonal direction, determining all aggregators without the assumption of unanimity. We bring together both threads of research by extending the results of Dokow-Holzman and Szegedy-Xu to the setting of Mossel. As an application, we determine which symmetric predicates over $\{0,1\}$ are such that all aggregators are dictatorial., Comment: 17 pages; a formally verified version can be found at https://github.com/YuvalFilmus/Polymorphisms/

Let $(X, \mathcal{F})$ be a hypergraph. The Maker-Breaker game on $(X, \mathcal{F})$ is a combinatorial game between two players, Maker and Breaker. Beginning with Maker, the players take turns claiming vertices from $X$ that have not yet been claimed. Maker wins if she manages to claim all vertices of some hyperedge $F \in \mathcal{F}$. Breaker wins if he claims at least one vertex in every hyperedge. M. L. Rahman and Thomas Watson proved in 2021 that, even when only Maker-Breaker games on 6-uniform hypergraphs are considered, the decision problem of determining which player has a winning strategy is PSPACE-complete. They also showed that the problem is NL-hard when considering hypergraphs of rank 5. In this paper, we improve the latter result by showing that deciding who wins Maker-Breaker games on 5-uniform hypergraphs is still a PSPACE-complete problem. We achieve this by polynomial transformation from the problem of solving the generalized geography game on bipartite digraphs with vertex degrees 3 or less, which is known to be PSPACE-complete., Comment: 13 pages, 2 tables containing figures, 5 references

It is known that there is a linear dependence between the treewidth of a graph and its balanced separator number: the smallest integer $k$ such that for every weighing of the vertices, the graph admits a balanced separator of size at most $k$. We investigate whether this connection can be lifted to the setting of coarse graph theory, where both the bags of the considered tree decompositions and the considered separators should be coverable by a bounded number of bounded-radius balls. As the first result, we prove that if an $n$-vertex graph $G$ admits balanced separators coverable by $k$ balls of radius $r$, then $G$ also admits tree decompositions ${\cal T}_1$ and ${\cal T}_2$ such that: - in ${\cal T}_1$, every bag can be covered by $O(k\log n)$ balls of radius $r$; and - in ${\cal T}_2$, every bag can be covered by $O(k^2\log k)$ balls of radius $r(\log k+\log\log n+O(1))$. As the second result, we show that if we additionally assume that $G$ has doubling dimension at most $m$, then the functional equivalence between the existence of small balanced separators and of tree decompositions of small width can be fully lifted to the coarse setting. Precisely, we prove that for a positive integer $r$ and a graph $G$ of doubling dimension at most $m$, the following conditions are equivalent, with constants $k_1,k_2,k_3,k_4,\Delta_3,\Delta_4$ depending on each other and on $m$: - $G$ admits balanced separators consisting of $k_1$ balls of radius $r$; - $G$ has a tree decomposition with bags coverable by $k_2$ balls of radius $r$; - $G$ has a tree-partition of maximum degree $\leq \Delta_3$ with bags coverable by $k_3$ balls of radius $r$; - $G$ is quasi-isometric to a graph of maximum degree $\leq \Delta_4$ and tree-partition width $\leq k_4$.

The notion of graph covers (also referred to as locally bijective homomorphisms) plays an important role in topological graph theory and has found its computer science applications in models of local computation. For a fixed target graph $H$, the {\sc $H$-Cover} problem asks if an input graph $G$ allows a graph covering projection onto $H$. Despite the fact that the quest for characterizing the computational complexity of {\sc $H$-Cover} had been started more than 30 years ago, only a handful of general results have been known so far. In this paper, we present a complete characterization of the computational complexity of covering coloured graphs for the case that every equivalence class in the degree partition of the target graph has at most two vertices. We prove this result in a very general form. Following the lines of current development of topological graph theory, we study graphs in the most relaxed sense of the definition. In particular, we consider graphs that are mixed (they may have both directed and undirected edges), may have multiple edges, loops, and semi-edges. We show that a strong P/NP-complete dichotomy holds true in the sense that for each such fixed target graph $H$, the {\sc $H$-Cover} problem is either polynomial-time solvable for arbitrary inputs, or NP-complete even for simple input graphs., Comment: 38 pages, journal version of our WG'23 paper

Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the task and the data are encoded as axioms of an algebra, and a model is obtained where only these axioms and their logical consequences hold. Although this is not a generalizing model, we show that selecting specific subsets of its breakdown into algebraic atoms obtained via subdirect decomposition gives a model that generalizes. We validate this new learning principle on standard datasets such as MNIST, FashionMNIST, CIFAR-10, and medical images, achieving performance comparable to optimized multilayer perceptrons. Beyond data-driven tasks, the new learning principle extends to formal problems, such as finding Hamiltonian cycles from their specifications and without relying on search. This algebraic foundation offers a fresh perspective on machine intelligence, featuring direct learning from training data without the need for validation dataset, scaling through model additivity, and asymptotic convergence to the underlying rule in the data.

A piecewise affine map is one of the simplest mathematical objects exhibiting complex dynamics. The reachability problem of piecewise affine maps is given as follows: Given two vectors $\mathbf{s}, \mathbf{t} \in \mathbb{Q}^d$ and a piecewise affine map $f$, is there $n\in \mathbb{N}$ such that $f^{n}(\mathbf{s}) = \mathbf{t}$? Koiran, Cosnard, and Garzon show that the reachability problem of piecewise affine maps is undecidable even in dimension 2. Most of the recent progress has been focused on decision procedures for one-dimensional piecewise affine maps, where the reachability problem has been shown to be decidable for some subclasses. However, the general undecidability discouraged research into positive results in arbitrary dimension. In this work, we consider a rich subclass of piecewise affine maps defined by Bellman operators of Markov decision processes (MDPs). We then investigate the restriction of the piecewise affine reachability problem to that with Bellman operators and, in particular, its decidability in any dimension. As one of our primary contributions, we establish the decidability of reachability for two-dimensional Bellman operators, in contrast to the negative result known for general piecewise affine maps., Comment: 17 pages, 4 figures

An upward (resp.~downward) digitally convex word is a binary word that best approximates from below (resp.~from above) an upward (resp.~downward) convex curve in the plane. We study these words from the combinatorial point of view, formalizing their geometrical properties and highlighting connections with Christoffel words and finite Sturmian words. In particular, we study from the combinatorial perspective the operations of inflation and deflation on digitally convex words., Comment: arXiv admin note: text overlap with arXiv:2211.09417

A matching cut is a matching that is also an edge cut. In the problem Minimum Matching Cut, we ask for a matching cut with the minimum number of edges in the matching. We give polynomial-time algorithms for $P_7$-free, $S_{1,1,2}$-free and $(P_6 + P_4)$-free graphs, which also solve several open cases for the well-studied problem Matching Cut. In addition, we show NP-hardness for $3P_3$-free graphs, implying that Minimum Matching Cut and Matching Cut differ in complexity on certain graph classes. We also give complexity dichotomies for both general and bipartite graphs of bounded radius and diameter.

In an $r$-partite graph, an independent transversal of size $s$ (ITS) consists of $s$ vertices from each part forming an independent set. Motivated by a question from Bollob\'as, Erd\H{o}s, and Szemer\'edi (1975), Di Braccio and Illingworth (2024) inquired about the minimum degree needed to ensure an $n \times \cdots \times n$ $r$-partite graph contains $K_r(s)$, a complete $r$-partite graph with $s$ vertices in each part. We reformulate this as finding the smallest $n$ such that any $n \times \cdots \times n$ $r$-partite graph with maximum degree $\Delta$ has an ITS. For any $\varepsilon>0$, we prove the existence of a $\gamma>0$ ensuring that if $G$ is a multipartite graph partitioned as $(V_1, V_2, \ldots, V_r)$, where the average degree of each part $V_i$ is at most $D$, the maximum degree of any vertex to any part $V_i$ is at most $\gamma D$, and the size of each part $V_i$ is at least $(s + \varepsilon)D$, then $G$ possesses an ITS. The constraint $(s + \varepsilon)D$ on the part size is tight. This extends results of Loh and Sudakov (2007), Glock and Sudakov (2022), and Kang and Kelly (2022). We also show that any $n \times \cdots \times n$ $r$-partite graph with minimum degree at least $\left(r-1-\frac{1}{2s^2}\right)n$ contains $K_r(s)$ and provide a relative Tur\'an-type result. Additionally, this paper explores counting ITSs in multipartite graphs., Comment: 28 pages

This paper is the second in a series of studies on developing efficient artificial intelligence-based approaches to pathfinding on extremely large graphs (e.g. $10^{70}$ nodes) with a focus on Cayley graphs and mathematical applications. The open-source CayleyPy project is a central component of our research. The present paper proposes a novel combination of a reinforcement learning approach with a more direct diffusion distance approach from the first paper. Our analysis includes benchmarking various choices for the key building blocks of the approach: architectures of the neural network, generators for the random walks and beam search pathfinding. We compared these methods against the classical computer algebra system GAP, demonstrating that they "overcome the GAP" for the considered examples. As a particular mathematical application we examine the Cayley graph of the symmetric group with cyclic shift and transposition generators. We provide strong support for the OEIS-A186783 conjecture that the diameter is equal to n(n-1)/2 by machine learning and mathematical methods. We identify the conjectured longest element and generate its decomposition of the desired length. We prove a diameter lower bound of n(n-1)/2-n/2 and an upper bound of n(n-1)/2+ 3n by presenting the algorithm with given complexity. We also present several conjectures motivated by numerical experiments, including observations on the central limit phenomenon (with growth approximated by a Gumbel distribution), the uniform distribution for the spectrum of the graph, and a numerical study of sorting networks. To stimulate crowdsourcing activity, we create challenges on the Kaggle platform and invite contributions to improve and benchmark approaches on Cayley graph pathfinding and other tasks., Comment: 28 pages

Recently, there has been interest in representing single graphs by multiple drawings; for example, using graph stories, storyplans, or uncrossed collections. In this paper, we apply this idea to orthogonal graph drawing. Due to the orthogonal drawing style, we focus on plane 4-graphs, that is, planar graphs of maximum degree 4 whose embedding is fixed. Our goal is to represent any plane 4-graph $G$ by an unbent collection, that is, a collection of orthogonal drawings of $G$ that adhere to the embedding of $G$ and ensure that each edge of $G$ is drawn without bends in at least one of the drawings. We investigate two objectives. First, we consider minimizing the number of drawings in an unbent collection. We prove that every plane 4-graph can be represented by a collection with at most three drawings, which is tight. We also give sufficient conditions for a graph to admit an unbent collection of size 2. Second, we consider minimizing the total number of bends over all drawings in an unbent collection. We show that this problem is NP-hard and give a 3-approximation algorithm. For the special case of plane triconnected cubic graphs, we show how to compute minimum-bend collections in linear time., Comment: 42 pages, 18 figures

We extend the study of property testing on hypergraphs initiated by Czumaj and Sohler (Theoretical Computer Science, 2005). Provided oracle access to a hypergraph, our goal is to distinguish between the case it has a certain property and the case it is "far" from having this property. Here, we assume that hypergraphs are represented by bounded-length incidence lists and we measure distances between them as a fraction of the maximum possible number of hyperedges. This contrasts with previous work where representations were given by adjacency matrices and distances by fractions of all possible vertex tuples. Thus, while the previous model is better for studying dense hypergraphs, ours is more effective for testing those of bounded-degree. In particular, our model can be seen as an extension to hypergraphs of the graph testing framework introduced by Goldreich and Ron (Algorithmica, 2002). In this framework, we analyse the query complexity of three fundamental hypergraph properties: colorability, $k$-partiteness, and independence number. We show that $k$-partiteness within families of $k$-uniform $n$-vertex hypergraphs of bounded treewidth is strongly testable. Our algorithm always accepts when the hypergraph is $k$-partite, and rejects with high probability if it is $\varepsilon$-far from $k$-partiteness. In addition, we prove optimal lower bounds of $\Omega(n)$ on the query complexity of testing algorithms for $k$-colorability, $k$-partiteness, and independence number in $k$-uniform $n$-vertex hypergraphs of bounded degree. For each of these properties, as an independently interesting combinatorial question, we consider the problem of explicitly constructing $k$-uniform hypergraphs of bounded degree that differ in $\Theta(n)$ hyperedges from any hypergraph satisfying the property, but where violations of the latter cannot be detected in any neighborhood of $o(n)$ vertices., Comment: 21 pages, 1 figure

The goal of graph inference is to design algorithms for learning properties of a hidden graph using queries to an oracle that returns information about the graph. Graph reconstruction, verification, and property testing are all types of graph inference. In this work, we study graph inference using an oracle that returns the effective resistance (ER) between a pair of vertices. Effective resistance is a distance originating from the study of electrical circuits with many applications. However, ER has received little attention from a graph inference perspective. Indeed, although it is known that an $n$-vertex graph can be uniquely reconstructed from all $\binom{n}{2}$ possible ER queries, little else is known. We address this gap with several new results, including: 1. $O(n)$-query algorithms for testing whether a graph is a tree; deciding whether two graphs are equal assuming one is a subgraph of the other; and testing whether a given vertex (or edge) is a cut vertex (or cut edge). 2. Property testing algorithms, including for testing whether a graph is vertex- or edge-biconnected. We also give a reduction to adapt property testing results from the bounded-degree model to our ER query model. This yields ER-query-based algorithms for testing $k$-connectivity, bipartiteness, planarity, and containment of a fixed subgraph. 3. Graph reconstruction algorithms, including an algorithm for reconstructing a graph from a low-width tree decomposition; a $\Theta(k^2)$-query, polynomial-time algorithm for recovering the adjacency matrix $A$ of a hidden graph, given $A$ with $k$ of its entries deleted; and a $k$-query, exponential-time algorithm for the same task. We also compare the power of ER queries and shortest path queries, which are closely related but better studied. Interestingly, we show that the two query models are incomparable in power.

Recently, the influence of potentially present symmetries has begun to be studied in complex networks. A typical way of studying symmetries is via the automorphism group of the corresponding graph. Since complex networks are often subject to uncertainty and automorphisms are very sensitive to small changes, this characterization needs to be modified to an approximate version for successful application. This paper considers a recently introduced approximate symmetry of complex networks computed as an automorphism with acceptance of small edge preservation error, see Liu 2020. This problem is generally very hard with respect to the large space of candidate permutations, and hence the corresponding computation methods typically lead to the utilization of local algorithms such as the simulated annealing used in the original work. This paper proposes a new heuristic algorithm extending such iterative search algorithm method by using network centralities as heuristics. Centralities are shown to be a good tool to navigate the local search towards more appropriate permutations and lead to better search results.

We introduce merge-width, a family of graph parameters that unifies several structural graph measures, including treewidth, degeneracy, twin-width, clique-width, and generalized coloring numbers. Our parameters are based on new decompositions called construction sequences. These are sequences of ever coarser partitions of the vertex set, where each pair of parts has a specified default connection, and all vertex pairs of the graph that differ from the default are marked as resolved. The radius-$r$ merge-width is the maximum number of parts reached from a vertex by following a path of at most $r$ resolved edges. Graph classes of bounded merge-width -- for which the radius-$r$ merge-width parameter can be bounded by a constant, for each fixed $r=1,2,3,\ldots$ -- include all classes of bounded expansion or of bounded twin-width, thus unifying two central notions from the Sparsity and Twin-width frameworks. Furthermore, they are preserved under first-order transductions, which attests to their robustness. We conjecture that classes of bounded merge-width are equivalent to the previously introduced classes of bounded flip-width. As our main result, we show that the model checking problem for first-order logic is fixed-parameter tractable on graph classes of bounded merge-width, assuming the input includes a witnessing construction sequence. This unites and extends two previous model checking results: the result of Dvo\v{r}\'{a}k, Kr\'{a}l, and Thomas for classes of bounded expansion, and the result of Bonnet, Kim, Thomass\'e, and Watrigant for classes of bounded twin-width. Finally, we suggest future research directions that could impact the study of structural and algorithmic graph theory, in particular of monadically dependent graph classes, which we conjecture to coincide with classes of almost bounded merge-width.

The existence of a polynomial-time pivot rule for the simplex method is a fundamental open question in optimization. While many super-polynomial lower bounds exist for individual or very restricted classes of pivot rules, there currently is little hope for an unconditional lower bound that addresses all pivot rules. We approach this question by considering the active-set method as a natural generalization of the simplex method to non-linear objectives. This generalization allows us to prove the first unconditional lower bound for all pivot rules. More precisely, we construct a multivariate polynomial of degree linear in the number of dimensions such that the active-set method started in the origin visits all vertices of the hypercube. We hope that our framework serves as a starting point for a new angle of approach to understanding the complexity of the simplex method.

A segment (barrier) is specified on the plane, as well as depots, where the mobile devices (drones) can be placed. Each drone departs from its depot to the barrier, moves along the barrier and returns to its depot, traveling a path of a limited length. The part of the barrier along which the drone moved is \emph{covered} by this sensor. It is required to place a limited quantity of drones in the depots and determine the trajectory of each drone in such a way that the barrier is covered, and the total length of the paths traveled by the drones is minimal. Previously, this problem was considered for an unlimited number of drones. If each drone covers a segment of length at least 1, then the time complexity of the proposed algorithm was $O(mL^3)$, where $m$ is the number of depots and $L$ is the length of the barrier. In this paper, we generalize the problem by introducing an upper bound $n$ on the number of drones, and propose a new algorithm with time complexity equals $O(mnL^2)$. Since each drone covers a segment of length at least 1, then $n\leq L$ and $O(mnL^2)\leq O(mL^3)$. Assuming an unlimited number of drones, as investigated in our prior work, we present an $O(mL^2)$-time algorithm, achieving an $L$-fold reduction compared to previous methods. Here, the algorithm has a time complexity that equals $O(L^2)$, and the most time-consuming is preprocessing.

In recent years, much attention has been placed on the complexity of graph homomorphism problems when the input is restricted to ${\mathbb P}_k$-free and ${\mathbb P}_k$-subgraph-free graphs. We consider the directed version of this research line, by addressing the questions, is it true that digraph homomorphism problems CSP$({\mathbb H})$ have a P versus NP-complete dichotomy when the input is restricted to $\vec{\mathbb P}_k$-free (resp.\ $\vec{\mathbb P}_k$-subgraph-free) digraphs? Our main contribution in this direction shows that if CSP$({\mathbb H})$ is NP-complete, then there is a positive integer $N$ such that CSP$({\mathbb H})$ remains NP-hard even for $\vec{\mathbb P}_N$-subgraph-free digraphs. Moreover, it remains NP-hard for acyclic $\vec{\mathbb P}_N$-subgraph-free digraphs, and becomes polynomial-time solvable for $\vec{\mathbb P}_{N-1}$-subgraph-free acyclic digraphs. We then verify the questions above for digraphs on three vertices and a family of smooth tournaments. We prove these results by establishing a connection between $\mathbb F$-(subgraph)-free algorithmics and constraint satisfaction theory. On the way, we introduce restricted CSPs, i.e., problems of the form CSP$({\mathbb H})$ restricted to yes-instances of CSP$({\mathbb H}')$ -- these were called restricted homomorphism problems by Hell and Ne\v{s}et\v{r}il. Another main result of this paper presents a P versus NP-complete dichotomy for these problems. Moreover, this complexity dichotomy is accompanied by an algebraic dichotomy in the spirit of the finite domain CSP dichotomy.

The permanent is a function, defined for a square matrix, with applications in various domains including quantum computing, statistical physics, complexity theory, combinatorics, and graph theory. Its formula is similar to that of the determinant, however unlike the determinant, its exact computation is #P-complete, i.e., there is no algorithm to compute the permanent in polynomial time unless P=NP. For an $n \times n$ matrix, the fastest algorithm has a time complexity of $O(2^{n-1}n)$. Although supercomputers have been employed for permanent computation before, there is no work and more importantly, no publicly available software that leverages cutting-edge, yet widely accessible, High-Performance Computing accelerators such as GPUs. In this work, we designed, developed, and investigated the performance of SUperman, a complete software suite that can compute matrix permanents on multiple nodes/GPUs on a cluster while handling various matrix types, e.g., real/complex/binary and sparse/dense etc., with a unique treatment for each type. Compared to a state-of-the-art parallel algorithm on 44 cores, SUperman can be $86\times$ faster on a single Nvidia A100 GPU. Combining multiple GPUs, we also showed that SUperman can compute the permanent of a $56 \times 56$ matrix which is the largest reported in the literature., Comment: 34 pages, 7 figures, 5 tables, 4 algorithms, 31 references

Online learning is a model of machine learning where the learner is trained on sequential feedback. We investigate worst-case error for the online learning of real functions that have certain smoothness constraints. Suppose that $\mathcal{F}_q$ is the class of all absolutely continuous functions $f: [0, 1] \rightarrow \mathbb{R}$ such that $\|f'\|_q \le 1$, and $\operatorname{opt}_p(\mathcal{F}_q)$ is the best possible upper bound on the sum of the $p^{\text{th}}$ powers of absolute prediction errors for any number of trials guaranteed by any learner. We show that for any $\delta, \epsilon \in (0, 1)$, $\operatorname{opt}_{1+\delta} (\mathcal{F}_{1+\epsilon}) = O(\min(\delta, \epsilon)^{-1})$. Combined with the previous results of Kimber and Long (1995) and Geneson and Zhou (2023), we achieve a complete characterization of the values of $p, q \ge 1$ that result in $\operatorname{opt}_p(\mathcal{F}_q)$ being finite, a problem open for nearly 30 years. We study the learning scenarios of smooth functions that also belong to certain special families of functions, such as polynomials. We prove a conjecture by Geneson and Zhou (2023) that it is not any easier to learn a polynomial in $\mathcal{F}_q$ than it is to learn any general function in $\mathcal{F}_q$. We also define a noisy model for the online learning of smooth functions, where the learner may receive incorrect feedback up to $\eta \ge 1$ times, denoting the worst-case error bound as $\operatorname{opt}^{\text{nf}}_{p, \eta} (\mathcal{F}_q)$. We prove that $\operatorname{opt}^{\text{nf}}_{p, \eta} (\mathcal{F}_q)$ is finite if and only if $\operatorname{opt}_p(\mathcal{F}_q)$ is. Moreover, we prove for all $p, q \ge 2$ and $\eta \ge 1$ that $\operatorname{opt}^{\text{nf}}_{p, \eta} (\mathcal{F}_q) = \Theta (\eta)$.

We study monotonicity testing of high-dimensional distributions on $\{-1,1\}^n$ in the model of subcube conditioning, suggested and studied by Canonne, Ron, and Servedio~\cite{CRS15} and Bhattacharyya and Chakraborty~\cite{BC18}. Previous work shows that the \emph{sample complexity} of monotonicity testing must be exponential in $n$ (Rubinfeld, Vasilian~\cite{RV20}, and Aliakbarpour, Gouleakis, Peebles, Rubinfeld, Yodpinyanee~\cite{AGPRY19}). We show that the subcube \emph{query complexity} is $\tilde{\Theta}(n/\varepsilon^2)$, by proving nearly matching upper and lower bounds. Our work is the first to use directed isoperimetric inequalities (developed for function monotonicity testing) for analyzing a distribution testing algorithm. Along the way, we generalize an inequality of Khot, Minzer, and Safra~\cite{KMS18} to real-valued functions on $\{-1,1\}^n$. We also study uniformity testing of distributions that are promised to be monotone, a problem introduced by Rubinfeld, Servedio~\cite{RS09} , using subcube conditioning. We show that the query complexity is $\tilde{\Theta}(\sqrt{n}/\varepsilon^2)$. Our work proves the lower bound, which matches (up to poly-logarithmic factors) the uniformity testing upper bound for general distributions (Canonne, Chen, Kamath, Levi, Waingarten~\cite{CCKLW21}). Hence, we show that monotonicity does not help, beyond logarithmic factors, in testing uniformity of distributions with subcube conditional queries.

Benford's Law is an important instance of experimental mathematics that appears to constrain the information-theoretic behavior of numbers. Elias' encoding for integers is a remarkable approach to universality and optimality of codes. In the present analysis we seek to deduce a general law and its particular implications for these two cases from optimality and renormalization as applied to information-theoretical functionals. Both theoretical and experimental results corroborate our conclusions., Comment: 9 pages, 1 figure

Graph modification problems are computational tasks where the goal is to change an input graph $G$ using operations from a fixed set, in order to make the resulting graph satisfy a target property, which usually entails membership to a desired graph class $\mathcal{C}$. Some well-known examples of operations include vertex-deletion, edge-deletion, edge-addition and edge-contraction. In this paper we address an operation known as subgraph complement. Given a graph $G$ and a subset $S$ of its vertices, the subgraph complement $G \oplus S$ is the graph resulting of complementing the edge set of the subgraph induced by $S$ in $G$. We say that a graph $H$ is a subgraph complement of $G$ if there is an $S$ such that $H$ is isomorphic to $G \oplus S$. For a graph class $\mathcal{C}$, subgraph complementation to $\mathcal{C}$ is the problem of deciding, for a given graph $G$, whether $G$ has a subgraph complement in $\mathcal{C}$. This problem has been studied and its complexity has been settled for many classes $\mathcal{C}$ such as $\mathcal{H}$-free graphs, for various families $\mathcal{H}$, and for classes of bounded degeneracy. In this work, we focus on classes graphs of minimum/maximum degree upper/lower bounded by some value $k$. In particular, we answer an open question of Antony et al. [Information Processing Letters 188, 106530 (2025)], by showing that subgraph complementation to $\mathcal{C}$ is NP-complete when $\mathcal{C}$ is the class of graphs of minimum degree at least $k$, if $k$ is part of the input. We also show that subgraph complementation to $k$-regular parameterized by $k$ is fixed-parameter tractable.

Reed-Muller codes consist of evaluations of $n$-variate polynomials over a finite field $\mathbb{F}$ with degree at most $d$. Much like every linear code, Reed-Muller codes can be characterized by constraints, where a codeword is valid if and only if it satisfies all \emph{degree-$d$} constraints. For a subset $\tilde{X} \subseteq \mathbb{F}^n$, we introduce the notion of \emph{$\tilde{X}$-quotient} Reed-Muller code. A function $F : \tilde{X} \rightarrow \mathbb{F}$ is a valid codeword in the quotient code if it satisfies all the constraints of degree-$d$ polynomials \emph{lying in $\tilde{X}$}. This gives rise to a novel phenomenon: a quotient codeword may have \emph{many} extensions to original codewords. This weakens the connection between original codewords and quotient codewords which introduces a richer range of behaviors along with substantial new challenges. Our goal is to answer the following question: what properties of $\tilde{X}$ will imply that the quotient code inherits its distance and list-decoding radius from the original code? We address this question using techniques developed by Bhowmick and Lovett [BL14], identifying key properties of $\mathbb{F}^n$ used in their proof and extending them to general subsets $\tilde{X} \subseteq \mathbb{F}^n$. By introducing a new tool, we overcome the novel challenge in analyzing the quotient code that arises from the weak connection between original and quotient codewords. This enables us to apply known results from additive combinatorics and algebraic geometry [KZ18, KZ19, LZ21] to show that when $\tilde{X}$ is a \emph{high rank variety}, $\tilde{X}$-quotient Reed-Muller codes inherit the distance and list-decoding parameters from the original Reed-Muller codes.

The transmission of a vertex in a connected graph is the sum of distances from that vertex to all the other vertices. A connected graph is transmission irregular if any two distinct vertices have different transmissions. We present an efficient algorithm that generates all the transmission irregular trees up to a given order, up to isomorphism.

In this paper, we study different variants of the Cops-and-Robber game with respect to cop- and robber\-/monotonicity. We study a visible and invisible robber and variants where the robber is lazy, thus can only move when the cops announce to move on top of him. In all four combinations, we also vary the number $s$ of edges that the robber can traverse in a single round, called speed. We complete the study of the unbounded speed case by showing that, besides the active variants, also the visible lazy variant has both the cop- and robber\-/monotonicity property. Furthermore, we prove that the cop\-/monotone invisible lazy copwidth characterizes path-width, while the non\-/monotone and robber\-/monotone is known to characterize tree-width, thus these variants differ even in the unbounded speed case. We find that, even with speed restriction, the cop\-/monotone invisible copwidth and the robber\-/monotone invisible active copwidth all characterize path-width. On the other hand, we show that the path-width of a graph can be arbitrarily larger than the number of cops needed to win the non\-/monotone invisible active variant. To complete our study of cop\-/monotone variants, we show that also in the visible variants the cop\-/monotone copwidth can be arbitrarily larger than the non\-/monotone. Regarding robber\-/monotonicity, for all speeds $s\geq 4$, we give graphs where the non\-/monotone and robber\-/monotone copwidth differ. On the other hand, we prove that there is a function that bounds the robber\-/monotone copwidth in terms of the non\-/monotone copwidth and the speed, thus the gap between the variants is bounded. This proof also yields that a graph class has bounded expansion if and only if, for every speed $s$, the number of cops needed in any robber\-/monotone lazy variant is bounded by some constant $c(s)$.

The paper considers the NP-hard graph vertex coloring problem, which differs from traditional problems in which it is required to color vertices with a given (or minimal) number of colors so that adjacent vertices have different colors. In the problem under consideration, a simple edge-weighted graph is given. It is required to color its vertices in 3 colors to minimize the total weight of monochromatic (one-color) edges, i.e. edges with the same colors of their end vertices. This problem is poorly investigated. Previously, we developed graph decomposition algorithms that, in particular, allowed us to construct lower bounds for the optimum, as well as several greedy algorithms. In this paper, several new approximation algorithms are proposed. Among them are variable neighborhood search, simulated annealing, genetic algorithm and graph clustering with further finding the optimal coloring in each cluster. A numerical experiment was conducted on random graphs, as well as on real communication graphs. The characteristics of the algorithms are presented both in tables and graphically. The developed algorithms have shown high efficiency.

In this paper, we propose a novel parallel hierarchical Leiden-based algorithm for dynamic community detection. The algorithm, for a given batch update of edge insertions and deletions, partitions the network into communities using only a local neighborhood of the affected nodes. It also uses the inner hierarchical graph-based structure, which is updated incrementally in the process of optimizing the modularity of the partitioning. The algorithm has been extensively tested on various networks. The results demonstrate promising improvements in performance and scalability while maintaining the modularity of the partitioning., Comment: 45 pages, 9 figures, 30 tables

We show that for every graph $H$, there is a hereditary weakly sparse graph class $\mathcal C_H$ of unbounded treewidth such that the $H$-free (i.e., excluding $H$ as an induced subgraph) graphs of $\mathcal C_H$ have bounded treewidth. This refutes several conjectures and critically thwarts the quest for the unavoidable induced subgraphs in classes of unbounded treewidth, a wished-for counterpart of the Grid Minor theorem. We actually show a stronger result: For every positive integer $t$, there is a hereditary graph class $\mathcal C_t$ of unbounded treewidth such that for any graph $H$ of treewidth at most $t$, the $H$-free graphs of $\mathcal C_t$ have bounded treewidth. Our construction is a variant of so-called layered wheels. We also introduce a framework of abstract layered wheels, based on their most salient properties. In particular, we streamline and extend key lemmas previously shown on individual layered wheels. We believe that this should greatly help develop this topic, which appears to be a very strong yet underexploited source of counterexamples., Comment: 21 pages, 6 figures

Enumerating minimal dominating sets with polynomial delay in bipartite graphs is a long-standing open problem. To date, even the subcase of chordal bipartite graphs is open, with the best known algorithm due to Golovach, Heggernes, Kant\'e, Kratsch, Saether, and Villanger running in incremental-polynomial time. We improve on this result by providing a polynomial delay and space algorithm enumerating minimal dominating sets in chordal bipartite graphs. Additionally, we show that the total and connected variants admit polynomial and incremental-polynomial delay algorithms, respectively, within the same class. This provides an alternative proof of a result by Golovach et al. for total dominating sets, and answers an open question for the connected variant. Finally, we give evidence that the techniques used in this paper cannot be generalized to bipartite graphs for (total) minimal dominating sets, unless P = NP, and show that enumerating minimal connected dominating sets in bipartite graphs is harder than enumerating minimal transversals in general hypergraphs., Comment: 22 pages, 2 figures

Connectivity in temporal graphs relies on the notion of temporal paths, in which edges follow a chronological order (either strict or non-strict). In this work, we investigate the question of how to make a temporal graph connected. More precisely, we tackle the problem of finding, among a set of proposed temporal edges, the smallest subset such that its addition makes the graph temporally connected (TCA). We study the complexity of this problem and variants, under restricted lifespan of the graph, i.e. the maximum time step in the graph. Our main result on TCA is that for any fixed lifespan at least 2, it is NP-complete in both the strict and non-strict setting. We additionally provide a set of restrictions in the non-strict setting which makes the problem solvable in polynomial time and design an algorithm achieving this complexity. Interestingly, we prove that the source variant (making a given vertex a source in the augmented graph) is as difficult as TCA. On the opposite, we prove that the version where a list of connectivity demands has to be satisfied is solvable in polynomial time, when the size of the list is fixed. Finally, we highlight a variant of the previous case for which even with two pairs the problem is already NP-hard.

In this paper we show that a generalized version of the Nikoli puzzle Slant is NP-complete. We also give polynomial time algorithms for versions of the puzzle where some constraints are omitted. These problems correspond to simultaneously satisfying connectivity and vertex degree constraints in a grid graph and its dual., Comment: 9 pages, 12 figures. Appeared at the Canadian Conference on Computational Geometry 2024 (CCCG 2024)

Elementary cellular automata (ECA) are converted into multiplicative versions by using permuted n-dim Galois fields and octonion multiplication tables as binary pointers to each rule's Wolfram code truth table. This enables an extension of the binary ECA to complex numbers, identity solutions are found, produces a polynomial, and is implemented in Java.

Given a rectangular grid graph with a special vertex at a corner called base station, we study the problem of covering the vertices of the entire graph with tours that start and end at the base station and whose lengths do not exceed a given threshold, while minimizing a quality measure. We consider two objective functions: minimizing the number of tours and minimizing the sum of their lengths. We present an algorithm that computes the optimal solution for both objectives in linear time with respect to the grid size.

The paper proposes a novel machine learning-based approach to the pathfinding problem on extremely large graphs. This method leverages diffusion distance estimation via a neural network and uses beam search for pathfinding. We demonstrate its efficiency by finding solutions for 4x4x4 and 5x5x5 Rubik's cubes with unprecedentedly short solution lengths, outperforming all available solvers and introducing the first machine learning solver beyond the 3x3x3 case. In particular, it surpasses every single case of the combined best results in the Kaggle Santa 2023 challenge, which involved over 1,000 teams. For the 3x3x3 Rubik's cube, our approach achieves an optimality rate exceeding 98%, matching the performance of task-specific solvers and significantly outperforming prior solutions such as DeepCubeA (60.3%) and EfficientCube (69.6%). Additionally, our solution is more than 26 times faster in solving 3x3x3 Rubik's cubes while requiring up to 18.5 times less model training time than the most efficient state-of-the-art competitor., Comment: 12 pages, 3 tables, 3 figures

In this expository article, we study the relation between the boolean functions and the hypercontractivity theorems of Aline Bonami. We focus on the social choice theory, and present some of the most important results in the area, such as the Friedgut-Kalai-Naor (FKN) and the Kahn-Kalai-Linial (KKL) theorems, and the famous Fourier Entropy/Influence conjecture. -- En este art\'iculo expositivo estudiamos la relaci\'on entre las funciones booleanas y los teoremas de hipercontractividad de Aline Bonami. Nos concentramos en la teor\'ia de la elecci\'on social, y presentamos algunos de los resultados m\'as importantes en el \'area como los teoremas de Friedgut-Kalai-Naor (FKN) y de Kahn-Kalai-Linial (KKL), y la famosa conjetura Entrop\'i{\i}a de Fourier/Influencia., Comment: 37 pages, in Spanish, 2 photos

In 2010, Bre\v{s}ar, Klav\v{z}ar and Rall introduced the optimization variant of the graph domination game and the game domination number. In 2024, Leo Versteegen obtained the celebrated proof of the Conjecture $\frac{3}{5}$ on this variant of the domination game, proposed by Kinnersley, West and Zamani in 2013. In this paper, we investigate for the first time the normal play of the domination game, which we call \textsc{Normal Domination Game}, that is an impartial game where the last to play wins. We first prove that this game is PSPACE-complete. We also use the Sprague-Grundy theory to prove that Alice (the first player) wins in the path $P_n$ if and only if $n$ is not a multiple of $4$, and wins in the cycle $C_n$ if and only if $n=4k+3$ for some integer $k$. Finally, we obtain a polynomial time algorithm to decide the winner for any disjoint union of paths and cycles in the \textsc{Normal Domination Game} and its natural partizan variant.

Using the subdivision schemes theory, we develop a criterion to check if any natural number has at most one representation in the $n$-ary number system with a set of non-negative integer digits $A=\{a_1, a_2,\ldots, a_n\}$ that contains zero. This uniqueness property is shown to be equivalent to a certain restriction on the roots of the trigonometric polynomial $\sum_{k=1}^n e^{-2\pi i a_k t}$. From this criterion, under a natural condition of irreducibility for $A$, we deduce that in case of prime $n$ the uniqueness holds if and only if the digits of $A$ are distinct modulo $n$, whereas for any composite $n$ we show that the latter condition is not necessary. We also establish the connection of this uniqueness to the semigroup freeness problem for affine integer functions of equal integer slope; this together with the two criteria allows to fill the gap in the work of D$.$Klarner on Erd\"os question about densities of affine integer orbits and establish a simple algorithm to check the freeness and the positivity of density when the slope is a prime number., Comment: 11 pages

We consider a framework for clustering edge-colored hypergraphs, where the goal is to cluster (equivalently, to color) objects based on the primary type of multiway interactions they participate in. One well-studied objective is to color nodes to minimize the number of unsatisfied hyperedges -- those containing one or more nodes whose color does not match the hyperedge color. We motivate and present advances for several directions that extend beyond this minimization problem. We first provide new algorithms for maximizing satisfied edges, which is the same at optimality but is much more challenging to approximate, with all prior work restricted to graphs. We develop the first approximation algorithm for hypergraphs, and then refine it to improve the best-known approximation factor for graphs. We then introduce new objective functions that incorporate notions of balance and fairness, and provide new hardness results, approximations, and fixed-parameter tractability results.

Finding a minimum spanning tree (MST) for $n$ points in an arbitrary metric space is a fundamental primitive for hierarchical clustering and many other ML tasks, but this takes $\Omega(n^2)$ time to even approximate. We introduce a framework for metric MSTs that first (1) finds a forest of disconnected components using practical heuristics, and then (2) finds a small weight set of edges to connect disjoint components of the forest into a spanning tree. We prove that optimally solving the second step still takes $\Omega(n^2)$ time, but we provide a subquadratic 2.62-approximation algorithm. In the spirit of learning-augmented algorithms, we then show that if the forest found in step (1) overlaps with an optimal MST, we can approximate the original MST problem in subquadratic time, where the approximation factor depends on a measure of overlap. In practice, we find nearly optimal spanning trees for a wide range of metrics, while being orders of magnitude faster than exact algorithms.

We define generalized de Bruijn words, as those words having a Burrows--Wheeler transform that is a concatenation of permutations of the alphabet. We show how to interpret generalized de Bruijn words in terms of Hamiltonian cycles in the generalized de Bruijn graphs introduced in the early '80s in the context of network design. When the size of the alphabet is a prime, we give relations between generalized de Bruijn words, normal bases of finite fields, invertible circulant matrices, and Reutenauer groups. In particular, we highlight a correspondence between binary de Bruijn words of order $d+1$, binary necklaces of length $2^{d}$ having an odd number of $1$s, invertible BWT matrices of size $2^{d}\times 2^{d}$, and normal bases of the finite field $\mathbb{F}_{2^{2^{d}}}$., Comment: Submitted

Hofstadter's G function is recursively defined via $G(0)=0$ and then $G(n)=n-G(G(n-1))$. Following Hofstadter, a family $(F_k)$ of similar functions is obtained by varying the number $k$ of nested recursive calls in this equation. We study here some Fibonacci-like sequences that are deeply connected with these functions $F_k$. In particular, the Zeckendorf theorem can be adapted to provide digital expansions via sums of terms of these sequences. On these digital expansions, the functions $F_k$ are acting as right shifts of the digits. The Fibonacci-like sequences are then expressed in terms of zeros of the polynomial $X^k{-}X^{k-1}{-}1$. Thanks to that, we estimate the discrepancy of each function $F_k$, i.e., the maximal distance between $F_k$ and its linear equivalent. This discrepancy is finite exactly when $k \le 4$. This also solves some related questions, in particular two twenty-year-old OEIS conjectures stating how close the functions $F_3$ and $F_4$ are from the integer parts of their linear equivalents. Moreover $F_k$ can coincide exactly with such an integer part only when $k\le 2$, while $F_k$ is almost additive exactly when $k \le 4$. Finally, a nice fractal shape a la Rauzy has been encountered when investigating the discrepancy of $F_3$., Comment: (v2: add missing files for latex compilation)

Given a static vertex-selection problem (e.g. independent set, dominating set) on a graph, we can define a corresponding temporal reconfiguration problem on a temporal graph which asks for a sequence of solutions to the vertex-selection problem at each time such that we can reconfigure from one solution to the next. We can think of each solution in the sequence as a set of vertices with tokens placed on them; our reconfiguration model allows us to slide tokens along active edges of a temporal graph. We show that it is possible to efficiently check whether one solution can be reconfigured to another, and show that approximation results on the static vertex-selection problem can be adapted with a lifetime factor to the reconfiguration version. Our main contributions are fixed-parameter tractable algorithms with respect to: enumeration time of the related static problem; the combination of temporal neighbourhood diversity and lifetime of the input graph; and the combination of lifetime and treewidth of the footprint graph., Comment: 22 pages, 3 figures. Conference version with appendix

We obtain structure theorems for graphs excluding a fan (a path with a universal vertex) or a dipole ($K_{2,k}$) as a topological minor. The corresponding decompositions can be computed in FPT linear time. This is motivated by the study of a graph parameter we call treebandwidth which extends the graph parameter bandwidth by replacing the linear layout by a rooted tree such that neighbours in the graph are in ancestor-descendant relation in the tree. We deduce an approximation algorithm for treebandwidth running in FPT linear time from our structure theorems. We complement this result with a precise characterisation of the parameterised complexity of computing the parameter exactly.

A graph is \textit{rigid} if it only admits the identity endomorphism. We show that for every $d\ge 3$ there exist infinitely many mutually rigid $d$-regular graphs of arbitrary odd girth $g\geq 7$. Moreover, we determine the minimum order of a rigid $d$-regular graph for every $d\ge 3$. This provides strong positive answers to a question of van der Zypen [https://mathoverflow.net/q/296483, https://mathoverflow.net/q/321108]. Further, we use our construction to show that every finite monoid is isomorphic to the endomorphism monoid of a regular graph. This solves a problem of Babai and Pultr [J. Comb.~Theory, Ser.~B, 1980]., Comment: 32 pages, 10 figures

We present in this paper four new variants of the Tower of Hanoi problem, the optimal solution of each of these variants is related to one of the four known numbers Fibonacci, Lucas, Pell, and Jacobsthal. We give an optimal solution to each of these variants, and we present their associated graphs., Comment: 24 pages, 23 figures

There has recently been significant interest in fault tolerant spanners, which are spanners that still maintain their stretch guarantees after some nodes or edges fail. This work has culminated in an almost complete understanding of the three-way tradeoff between stretch, sparsity, and number of faults tolerated. However, despite some progress in metric settings, there have been no results to date on the tradeoff in general graphs between stretch, lightness, and number of faults tolerated. We initiate the study of light edge fault tolerant (EFT) graph spanners, obtaining the first such results. First, we observe that lightness can be unbounded if we use the traditional definition (normalizing by the MST). We then argue that a natural definition of fault-tolerant lightness is to instead normalize by a min-weight fault tolerant connectivity preserver; essentially, a fault-tolerant version of the MST. However, even with this, we show that it is still not generally possible to construct $f$-EFT spanners whose weight compares reasonably to the weight of a min-weight $f$-EFT connectivity preserver. In light of this lower bound, it is natural to then consider bicriteria notions of lightness, where we compare the weight of an $f$-EFT spanner to a min-weight $(f' > f)$-EFT connectivity preserver. The most interesting question is to determine the minimum value of $f'$ that allows for reasonable lightness upper bounds. Our main result is a precise answer to this question: $f' = 2f$. In particular, we show that the lightness can be untenably large (roughly $n/k$ for a $k$-spanner) if one normalizes by the min-weight $(2f-1)$-EFT connectivity preserver. But if one normalizes by the min-weight $2f$-EFT connectivity preserver, then we show that the lightness is bounded by just $O(f^{1/2})$ times the non-fault tolerant lightness (roughly $n^{1/k}$, for a $(1+\epsilon)(2k-1)$-spanner)., Comment: 25 pages, 3 figures