CyberSec Research

The Local Structure Theorem (LST) for graph minors roughly states that every $H$-minor free graph $G$ that contains a sufficiently large wall $W$, there is a set of few vertices $A$ such that, upon removing $A$, the resulting graph $G':=G - A$ admits an "almost embedding" $\delta$ into a surface $\Sigma$ in which $H$ does not embed. By almost embedding, we mean that there exists a hypergraph $\mathcal{H}$ whose vertex set is a subset of the vertex set of $G$ and an embedding of $\mathcal{H}$ in $\Sigma$ such that 1) the drawing of each hyperedge of $\mathcal{H}$ corresponds to a cell of $\delta$, 2) the boundary of each cell intersects only the vertices of the corresponding hyperedge, and 3) all remaining vertices and edges of $G'$ are drawn in the interior of cells. The cells corresponding to hyperedges of arity at least $4$, called vortices, are few in number and have small "depth", while a "large" part of the wall $W$ is drawn outside the vortices and is "grounded" in the embedding $\delta$. Now suppose that the subgraphs drawn inside each of the non-vortex cells are equipped with some finite index, i.e., each such cell is assigned a color from a finite set. We prove a version of the LST in which the set $C$ of colors assigned to the non-vortex cells exhibits "large" bidimensionality: The graph $G'$ contains a minor model of a large grid $\Gamma$ where each bag corresponding to a vertex $v$ of $\Gamma$, contains the subgraph drawn within a cell carrying color $\alpha$, for every color $\alpha \in C$. Moreover, the grid $\Gamma$ can be chosen in a way that is "well-connected" to the original wall $W$.

Mutually Unbiased Bases (MUBs) are closely connected with quantum physics, and the structure has a rich mathematical background. We provide equivalent criteria for extending a set of MUBs for $C^n$ by studying real points of a certain affine algebraic variety. This variety comes from the relations that determine the extendability of a system of MUBs. Finally, we show that some part of this variety gives rise to complete intersection domains. Further, we show that there is a one-to-one correspondence between MUBs and the maximal commuting classes (bases) of orthogonal normal matrices in $\mathcal M_n({\mathbb{C}})$. It means that for $m$ MUBs in $C^n$, there are $m$ commuting classes, each consisting of $n$ commuting orthogonal normal matrices and the existence of maximal commuting basis for $\mathcal M_n({\mathbb{C}})$ ensures the complete set of MUBs in $\mathcal M_n({\mathbb{C}})$.

This paper addresses the problem of scheduling jobs on identical machines with conflict constraints, where certain jobs cannot be scheduled simultaneously on different machines. We focus on the case where conflicts can be represented by a simple undirected graph, and the objective is to minimize the mean flow time. We show that the problem is NP-hard even on two machines and two distinct processing times. For unit-time jobs, the problem becomes NP-hard when the number of machines increases to three. We also identify polynomial-time solvable cases for specific classes of conflict graphs. For the general problem, we propose mathematical models, lower bounds, and a genetic algorithm. We evaluate their performance through computational experiments on a wide range of instances derived from well-known benchmark instances in the literature.

For an undirected graph $G$, a dominating broadcast on $G$ is a function $f : V(G) \rightarrow \mathbb{N}$ such that for any vertex $u \in V(G)$, there exists a vertex $v \in V(G)$ with $f(v) \geqslant 1$ and $d(u,v) \leqslant f(v)$. The cost of $f$ is $\sum_{v \in V} f(v)$. The minimum cost over all the dominating broadcasts on $G$ is defined as the broadcast domination number $\gamma_b(G)$ of $G$. A multipacking in $G$ is a subset $M \subseteq V(G)$ such that, for every vertex $v \in V(G)$ and every positive integer $r$, the number of vertices in $M$ within distance $r$ of $v$ is at most $r$. The multipacking number of $G$, denoted $\operatorname{mp}(G)$, is the maximum cardinality of a multipacking in $G$. These two optimisation problems are duals of each other, and it easily follows that $\operatorname{mp}(G) \leqslant \gamma_b(G)$. It is known that $\gamma_b(G) \leqslant 2\operatorname{mp}(G)+3$ and conjectured that $\gamma_b(G) \leqslant 2\operatorname{mp}(G)$. In this paper, we show that for the $n$-dimensional hypercube $Q_n$ $$ \left\lfloor\frac{n}{2} \right\rfloor \leqslant \operatorname{mp}(Q_n) \leqslant \frac{n}{2} + 6\sqrt{2n}. $$ Since $\gamma_b(Q_n) = n-1$ for all $n \geqslant 3$, this verifies the above conjecture on hypercubes and, more interestingly, gives a sequence of connected graphs for which the ratio $\frac{\gamma_b(G)}{\operatorname{mp}(G)}$ approaches $2$, a search for which was initiated by Beaudou, Brewster and Foucaud in 2018. It follows that, for connected graphs $G$ $$ \limsup_{\operatorname{mp}(G) \rightarrow \infty} \left\{\frac{\gamma_b(G)}{\operatorname{mp}(G)}\right\} = 2.$$ The lower bound on $\operatorname{mp}(Q_n)$ is established by a recursive construction, and the upper bound is established using a classic result from discrepancy theory.

The CFI-graphs, named after Cai, F\"urer, and Immerman, are central to the study of the graph isomorphism testing and of first-order logic with counting. They are colored graphs, and the coloring plays a role in many of their applications. As usual, it is not hard to remove the coloring by some extra graph gadgets, but at the cost of blowing up the size of the graphs and changing some parameters of them as well. This might lead to suboptimal combinatorial bounds important to their applications. Since then for some uncolored variants of the CFI-graphs it has been shown that they serve the same purposes. We show that this already applies to the graphs obtained from the original CFI-graphs by forgetting the colors. Moreover, we will see that there is a first-order formula $\varphi(x,y)$ expressing in almost all uncolored CFI-graphs that $x$ and $y$ have the same color in the corresponding colored graphs.

We consider the model of limited visibility Cops and Robbers, where the cops can only see within their $l$-neighbourhood. We prove that the number of cops needed to see the robber can be arbitrarily smaller than the number needed to capture the robber, answering an open question from the literature. We then consider how close we can get to seeing the robber when we do not have enough cops, along with a probabilistic interpretation.

In the Inverse Matroid problem, we are given a matroid, a fixed basis $B$, and an initial weight function, and the goal is to minimally modify the weights -- measured by some function -- so that $B$ becomes a maximum-weight basis. The problem arises naturally in settings where one wishes to explain or enforce a given solution by minimally perturbing the input. We extend this classical problem by replacing the fixed basis with a subset $S_0$ of the ground set and imposing various structural constraints on the set of maximum-weight bases relative to $S_0$. Specifically, we study six variants: (A) Inverse Matroid Exists, where $S_0$ must contain at least one maximum-weight basis; (B) Inverse Matroid All, where all bases contained in $S_0$ are maximum-weight; and (C) Inverse Matroid Only, where $S_0$ contains exactly the maximum-weight bases, along with their natural negated counterparts. For all variants, we develop combinatorial polynomial-time algorithms under the $\ell_\infty$-norm. A key ingredient is a refined min-max theorem for Inverse Matroid under the $\ell_\infty$-norm, which enables simpler and faster algorithms than previous approaches and may be of independent combinatorial interest. Our work significantly broadens the range of inverse optimization problems on matroids that can be solved efficiently, especially those that constrain the structure of optimal solutions through subset inclusion or exclusion.

The problem Orienteering asks whether there exists a walk which visits a number of sites without exceeding some fuel budget. In the variant of the problem we consider, the cost of each edge in the walk is dependent on the time we depart one endpoint and the time we arrive at the other endpoint. This mirrors applications such as travel between orbiting objects where fuel costs are dependent on both the departure time and the length of time spent travelling. In defining this problem, we introduce a natural generalisation of the standard notion of temporal graphs: the pair consisting of the graph of the sites and a cost function, in which costs as well as shortest travel times between pairs of objects change over time. We believe this model is likely to be of independent interest. The problem of deciding whether a stated goal is feasible is easily seen to be NP-complete; we investigate three different ways to restrict the input which lead to efficient algorithms. These include the number of times an edge can be used, an analogue of vertex-interval-membership width, and the number of sites to be visited.

A graph covering projection, also referred to as a locally bijective homomorphism, is a mapping between the vertices and edges of two graphs that preserves incidences and is a local bijection. This concept originates in topological graph theory but has also found applications in combinatorics and theoretical computer science. In this paper we consider undirected graphs in the most general setting -- graphs may contain multiple edges, loops, and semi-edges. This is in line with recent trends in topological graph theory and mathematical physics. We advance the study of the computational complexity of the {\sc $H$-Cover} problem, which asks whether an input graph allows a covering projection onto a parameter graph $H$. The quest for a complete characterization started in 1990's. Several results for simple graphs or graphs without semi-edges have been known, the role of semi-edges in the complexity setting has started to be investigated only recently. One of the most general known NP-hardness results states that {\sc $H$}-Cover is NP-complete for every simple connected regular graph of valency greater than two. We complement this result by considering regular graphs $H$ arising from connected acyclic graphs by adding semi-edges. Namely, we prove that any graph obtained by adding semi-edges to the vertices of a tree making it a $d$-regular graph with $d \geq 3$, defines an NP-complete graph covering problem. In line with the so called Strong Dichotomy Conjecture, we prove that the NP-hardness holds even for simple graphs on input.

We initiate the study of extremal problems about faces in convex rectilinear drawings of~$K_n$, that is, drawings where vertices are represented by points in the plane in convex position and edges by line segments between the points representing the end-vertices. We show that if a convex rectilinear drawing of $K_n$ does not contain a common interior point of at least three edges, then there is always a face forming a convex 5-gon while there are such drawings without any face forming a convex $k$-gon with $k \geq 6$. A convex rectilinear drawing of $K_n$ is \emph{regular} if its vertices correspond to vertices of a regular convex $n$-gon. We characterize positive integers $n$ for which regular drawings of $K_n$ contain a face forming a convex 5-gon. To our knowledge, this type of problems has not been considered in the literature before and so we also pose several new natural open problems.

We consider the existence and construction of \textit{biclique covers} of graphs, consisting of coverings of their edge sets by complete bipartite graphs. The \textit{size} of such a cover is the sum of the sizes of the bicliques. Small-size biclique covers of graphs are ubiquitous in computational geometry, and have been shown to be useful compact representations of graphs. We give a brief survey of classical and recent results on biclique covers and their applications, and give new families of graphs having biclique covers of near-linear size. In particular, we show that semilinear graphs, whose edges are defined by linear relations in bounded dimensional space, always have biclique covers of size $O(n\polylog n)$. This generalizes many previously known results on special classes of graphs including interval graphs, permutation graphs, and graphs of bounded boxicity, but also new classes such as intersection graphs of L-shapes in the plane. It also directly implies the bounds for Zarankiewicz's problem derived by Basit, Chernikov, Starchenko, Tao, and Tran (\textit{Forum Math. Sigma}, 2021). We also consider capped graphs, also known as terrain-like graphs, defined as ordered graphs forbidding a certain ordered pattern on four vertices. Terrain-like graphs contain the induced subgraphs of terrain visibility graphs. We give an elementary proof that these graphs admit biclique partitions of size $O(n\log^3 n)$. This provides a simple combinatorial analogue of a classical result from Agarwal, Alon, Aronov, and Suri on polygon visibility graphs (\textit{Discrete Comput. Geom.} 1994). Finally, we prove that there exists families of unit disk graphs on $n$ vertices that do not admit biclique coverings of size $o(n^{4/3})$, showing that we are unlikely to improve on Szemer\'edi-Trotter type incidence bounds for higher-degree semialgebraic graphs.

We consider the problem of enumerating the irreducible closed sets of a closure system given by an implicational base. In the context of Horn logic, these correspond to Horn expressions and characteristic models, respectively. To date, the complexity status of this problem is widely open, and it is further known to generalize the notorious hypergraph dualization problem, even in the context of acyclic convex geometries, i.e., closure systems admitting an acyclic implicational base. This paper studies this later class with a focus on the degree, which corresponds to the maximal number of implications in which an element occurs. We show that the problem is tractable for bounded values of this parameter, even when relaxed to the notions of premise- and conclusion-degree. Our algorithms rely on structural properties of acyclic convex geometries and involve various techniques from algorithmic enumeration such as solution graph traversal, saturation techniques, and a sequential approach leveraging from acyclicity. They are shown to perform in incremental-polynomial time for the computation of irreducible closed sets, and in polynomial time for the construction of an implicational base. Finally, we argue that our running times cannot be improved to polynomial delay using the standard framework of flashlight search.

We prove that Boolean matrices with bounded $\gamma_2$-norm or bounded normalized trace norm must contain a linear-sized all-ones or all-zeros submatrix, verifying a conjecture of Hambardzumyan, Hatami, and Hatami. We also present further structural results about Boolean matrices of bounded $\gamma_2$-norm and discuss applications in communication complexity, operator theory, spectral graph theory, and extremal combinatorics. As a key application, we establish an inverse theorem for MaxCut. A celebrated result of Edwards states that every graph $G$ with $m$ edges has a cut of size at least $\frac{m}{2}+\frac{\sqrt{8m+1}-1}{8}$, with equality achieved by complete graphs with an odd number of vertices. To contrast this, we prove that if the MaxCut of $G$ is at most $\frac{m}{2}+O(\sqrt{m})$, then $G$ must contain a clique of size $\Omega(\sqrt{m})$.

A linear layout of a graph consists of a linear ordering of its vertices and a partition of its edges into pages such that the edges assigned to the same page obey some constraint. The two most prominent and widely studied types of linear layouts are stack and queue layouts, in which any two edges assigned to the same page are forbidden to cross and nest, respectively. The names of these two layouts derive from the fact that, when parsing the graph according to the linear vertex ordering, the edges in a single page can be stored using a single stack or queue, respectively. Recently, the concepts of stack and queue layouts have been extended by using a double-ended queue or a restricted-input queue for storing the edges of a page. We extend this line of study to edge-weighted graphs by introducing priority queue layouts, that is, the edges on each page are stored in a priority queue whose keys are the edge weights. First, we show that there are edge-weighted graphs that require a linear number of priority queues. Second, we characterize the graphs that admit a priority queue layout with a single queue, regardless of the edge-weight function, and we provide an efficient recognition algorithm. Third, we show that the number of priority queues required independently of the edge-weight function is bounded by the pathwidth of the graph, but can be arbitrarily large already for graphs of treewidth two. Finally, we prove that determining the minimum number of priority queues is NP-complete if the linear ordering of the vertices is fixed.

We consider the problem of finding a Hamiltonian path or a Hamiltonian cycle with precedence constraints in the form of a partial order on the vertex set. We show that the path problem is $\mathsf{NP}$-complete for graphs of pathwidth 4 while the cycle problem is $\mathsf{NP}$-complete on graphs of pathwidth 5. We complement these results by giving polynomial-time algorithms for graphs of pathwidth 3 and treewidth 2 for Hamiltonian paths as well as pathwidth 4 and treewidth 3 for Hamiltonian cycles. Furthermore, we study the complexity of the path and cycle problems on rectangular grid graphs of bounded height. For these, we show that the path and cycle problems are $\mathsf{NP}$-complete when the height of the grid is greater or equal to 7 and 9, respectively. In the variant where we look for minimum edge-weighted Hamiltonian paths and cycles, the problems are $\mathsf{NP}$-hard for heights 5 and 6, respectively.

Maximal Ancestral Graphs (MAGs) provide an abstract representation of Directed Acyclic Graphs (DAGs) with latent (selection) variables. These graphical objects encode information about ancestral relations and d-separations of the DAGs they represent. This abstract representation has been used amongst others to prove the soundness and completeness of the FCI algorithm for causal discovery, and to derive a do-calculus for its output. One significant inherent limitation of MAGs is that they rule out the possibility of cyclic causal relationships. In this work, we address that limitation. We introduce and study a class of graphical objects that we coin ''$\sigma$-Maximal Ancestral Graphs'' (''$\sigma$-MAGs''). We show how these graphs provide an abstract representation of (possibly cyclic) Directed Graphs (DGs) with latent (selection) variables, analogously to how MAGs represent DAGs. We study the properties of these objects and provide a characterization of their Markov equivalence classes.

Consider a graph with n nodes and m edges, independent edge weights and lengths, and arbitrary distance demands for node pairs. The spanner problem asks for a minimum-weight subgraph that satisfies these demands via sufficiently short paths w.r.t. the edge lengths. For multiplicative alpha-spanners (where demands equal alpha times the original distances) and assuming that each edge's weight equals its length, the simple Greedy heuristic by Alth\"ofer et al. (1993) is known to yield strong solutions, both in theory and practice. To obtain guarantees in more general settings, recent approximations typically abandon this simplicity and practicality. Still, so far, there is no known non-trivial approximation algorithm for the spanner problem in its most general form. We provide two surprisingly simple approximations algorithms. In general, our Augmented Greedy achieves the first unconditional approximation ratio of m, which is non-trivial due to the independence of weights and lengths. Crucially, it maintains all size and weight guarantees Greedy is known for, i.e., in the aforementioned multiplicative alpha-spanner scenario and even for additive +beta-spanners. Further, it generalizes some of these size guarantees to derive new weight guarantees. Our second approach, Randomized Rounding, establishes a graph transformation that allows a simple rounding scheme over a standard multicommodity flow LP. It yields an O(n log n)-approximation, assuming integer lengths and polynomially bounded distance demands. The only other known approximation guarantee in this general setting requires several complex subalgorithms and analyses, yet we match it up to a factor of O(n^{1/5-eps}) using standard tools. Further, on bounded-degree graphs, we yield the first O(log n) approximation ratio for constant-bounded distance demands (beyond multiplicative 2-spanners in unit-length graphs).

Abstract notions of convexity over the vertices of a graph, and corresponding notions of halfspaces, have recently gained attention from the machine learning community. In this work we study monophonic halfspaces, a notion of graph halfspaces defined through closure under induced paths. Our main result is a $2$-satisfiability based decomposition theorem, which allows one to represent monophonic halfspaces as a disjoint union of certain vertex subsets. Using this decomposition, we achieve efficient and (nearly) optimal algorithms for various learning problems, such as teaching, active, and online learning. Most notably, we obtain a polynomial-time algorithm for empirical risk minimization. Independently of the decomposition theorem, we obtain an efficient, stable, and proper sample compression scheme. This makes monophonic halfspaces efficiently learnable with proper learners and linear error rate $1/\varepsilon$ in the realizable PAC setting. Our results answer open questions from the literature, and show a stark contrast with geodesic halfspaces, for which most of the said learning problems are NP-hard.