The dichromatic number χ(D) of a digraph is the minimum number k such that V(D) can be partitioned into k subsets, each inducing an acyclic digraph. The acyclic number α(D) is the cardinality of a largest induced acyclic subdigraph of D. We study these problems from an approximation point of view. We begin with establishing that even when restricted to tournaments, approximating χ and α remain as challenging as their undirected counterparts on general graphs. Specifically, we establish that for every ε > 0, it is hard to approximate both α and χ up to a factor of n^{1-ε} even when restricted to tournaments. We next consider approximate coloring of digraphs in special cases. We begin with establishing that we can color 𝓁-dicolorable digraphs using at most 𝓁 ⋅ n^{1-1/(𝓁)} colors in time O(n^{2𝓁}); in particular, we can color 2-dicolorable digraphs with 2√n colors in polynomial time. We then focus on bounding the dichromatic number of dense digraphs as a function of the independence number α of the underlying graph. We consider two special cases in this regard: digraphs with χ(D) ≤ 2 and digraphs that do not contain any directed triangle. For these cases, we present algorithms which generalize and improve existing tools and results.

Abstract Fair division of resources among competing agents is a fundamental problem in computational social choice and game theory. It has been intensively studied for various types of items ( divisible and indivisible ) and under various notions of fairness . We focus on Connected Fair Division ( $$\textsf{CFD}$$ ), the variant of fair division on graphs, where the resources are modeled as an item graph . Here, each agent has to be assigned a connected subgraph of the item graph, and each item has to be assigned to some agent. We introduce a generalization of $$\textsf{CFD}$$ , termed Incomplete $$\textsf{CFD}$$ ( $$\textsf{ICFD}$$ ), where exactly p vertices of the item graph should be assigned to the agents. This might be useful, in particular when the allocations are intended to be “economical” as well as fair. We consider four well-known notions of fairness: $$\textsf{PROP}$$ , $$\textsf{EF}$$ , $$\textsf{EF1}$$ , $$\textsf{EFX}$$ . First, we prove that $$\textsf{EF}$$ - $$\textsf{ICFD}$$ , $$\textsf{EF1}$$ - $$\textsf{ICFD}$$ , and $$\textsf{EFX}$$ - $$\textsf{ICFD}$$ are W[1]-hard parameterized by p plus the number of agents, even for graphs having constant vertex cover number ( $$\textsf{vcn}$$ ). In contrast, we present a randomized $$\textsf{FPT}$$ algorithm for $$\textsf{PROP}$$ - $$\textsf{ICFD}$$ parameterized only by p . Additionally, we prove both positive and negative results concerning the kernelization complexity of $$\textsf{ICFD}$$ under all four fairness notions, parameterized by p , $$\textsf{vcn}$$ , and the total number of different valuations in the item graph ( $$\textsf{val}$$ ).

We study the computational complexity of finding the geodetic number of a graph on chordal graphs and interval graphs. A set $S$ of vertices of a graph $G$ is a \textit{geodetic set} if every vertex of $G$ lies in a shortest path between some pair of vertices of $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality of a given graph. We show that \textsc{Minimum Geodetic Set} is fixed parameter tractable for chordal graphs when parameterized by its \emph{tree-width} (which equals its clique number). This implies a polynomial-time algorithm for $k$-trees, for fixed $k$. Then, we show that \textsc{Minimum Geodetic Set} is NP-hard on interval graphs, thereby answering a question of Ekim et al. (LATIN, 2012), who showed that \textsc{Minimum Geodetic Set} is polynomial-time solvable on proper interval graphs. As interval graphs are very constrained, to prove the latter result, we design a rather sophisticated reduction technique to work around their inherent linear structure.

Pursuit-evasion games have been intensively studied for several decades due to their numerous applications in artificial intelligence, robot motion planning, database theory, distributed computing, and algorithmic theory. Cops and Robber ( CnR ) is one of the most well-known pursuit-evasion games played on graphs, where multiple cops pursue a single robber . The aim is to compute the cop number of a graph, k , which is the minimum number of cops that ensures the capture of the robber. From the viewpoint of parameterized complexity, CnR is W[2]-hard parameterized by k [Fomin et al., TCS, 2010]. Thus, we study structural parameters of the input graph. We begin with the vertex cover number ( vcn ). First, we establish that k ≤ vcn 3 + 1 . Second, we prove that CnR parameterized by vcn is FPT by designing an exponential kernel. We complement this result by showing that it is unlikely for CnR parameterized by vcn to admit a polynomial compression. We extend our exponential kernels to the parameters cluster vertex deletion number and deletion to stars number , and design a linear vertex kernel for neighborhood diversity . Additionally, we extend all of our results to several well-studied variations of CnR .

Imagine we want to split a group of agents into teams in the most efficient way, considering that each agent has their own preferences about their teammates. This scenario is modeled by the extensively studied Coalition Formation problem. Here, we study a version of this problem where each team must additionally be of bounded size. We conduct a systematic algorithmic study, providing several intractability results as well as multiple exact algorithms that scale well as the input grows (FPT), which could prove useful in practice. Our main contribution is an algorithm that deals efficiently with tree-like structures (bounded treewidth) for ``small'' teams. We complement this result by proving that our algorithm is asymptotically optimal. Particularly, there can be no algorithm that vastly outperforms the one we present, under reasonable theoretical assumptions, even when considering star-like structures (bounded vertex cover number).

Research of cycles through specific vertices is a central topic in graph theory. In this context, we focus on a well-studied computational problem, T-Cycle: given an undirected n-vertex graph G and a set of k vertices T ⊆ V(G) termed terminals, the objective is to determine whether G contains a simple cycle C through all the terminals. Our contribution is twofold: (i) We provide a 2^{O(√klog k)}⋅ n-time fixed-parameter deterministic algorithm for T-Cycle on planar graphs; (ii) We provide a k^{O(1)}⋅ n-time deterministic kernelization algorithm for T-Cycle on planar graphs where the produced instance is of size klog^{O(1)}k. Both of our algorithms are optimal in terms of both k and n up to (poly)logarithmic factors in k under the ETH. In fact, our algorithms are the first subexponential-time fixed-parameter algorithm for T-Cycle on planar graphs, as well as the first polynomial kernel for T-Cycle on planar graphs. This substantially improves upon/expands the known literature on the parameterized complexity of the problem.

The VC-dimension is a well-studied and fundamental complexity measure of a set system (or hypergraph) that is central to many areas of machine learning. We establish several new results on the complexity of computing the VC-dimension. In particular, given a hypergraph $\mathcal{H}=(\mathcal{V},\mathcal{E})$, we prove that the naive $2^{\mathcal{O}(|\mathcal{V}|)}$-time algorithm is asymptotically tight under the Exponential Time Hypothesis (ETH). We then prove that the problem admits a $1$-additive fixed-parameter approximation algorithm when parameterized by the maximum degree of $\mathcal{H}$ and a fixed-parameter algorithm when parameterized by its dimension, and that these are essentially the only such exploitable structural parameters. Lastly, we consider a generalization of the problem, formulated using graphs, which captures the VC-dimension of both set systems and graphs. We design a $2^{\mathcal{O}(\texttt{tw}\cdot \log \texttt{tw})}\cdot |V|$-time algorithm for any graph $G=(V,E)$ of treewidth $\texttt{tw}$ (which, for a set system, applies to the treewidth of its incidence graph). This is in contrast with closely related problems that require a double-exponential dependency on the treewidth (assuming the ETH).

Decision trees is a fundamental tool in machine learning for representing, classifying, and generalizing data. It is desirable to construct ``small'' decision trees, by minimizing either the size (s) or the depth (d) of the decision tree (DT). Recently, the parameterized complexity of Decision Tree Learning has attracted a lot of attention. We consider a generalization of Decision Tree Learning where given a classification instance E and an integer t, the task is to find a ``small'' DT that disagrees with E in at most t examples. We consider two problems: DTSO and DTDO, where the goal is to construct a DT minimizing s and d, respectively. We first establish that both DTSO and DTDO are W[1]-hard when parameterized by s+y and d+y, respectively, where y is the maximum number of features in which two differently labeled examples can differ. We complement this result by showing that these problems become FPT if we include the parameter t. We also consider the kernelization complexity of these problems and establish several positive and negative results for both DTSO and DTDO.

Cops and Robber is one of the most studied two-player pursuit-evasion games played on graphs, where multiple cops, controlled by one player, pursue a single robber. The cop number of a graph is the minimum number of cops that can ensure the capture of the robber. In directed graphs, two kinds of moves are defined for players: strong move, where a player can move both along and against the orientation of an arc to an adjacent vertex; and weak move, where a player can only move along the orientation of an arc to an out-neighbor. We study three variants of Cops and Robber on oriented graphs: strong cop model, where the cops can make strong moves while the robber can only make weak moves; normal cop model, where both cops and the robber can only make weak moves; and weak cop model, where the cops can make weak moves while the robber can make strong moves. We study the cop number of these models with respect to several variants of retracts on oriented graphs and establish that the strong and normal cop number of an oriented graph remains invariant in their strong and distributed retracts, respectively. Next, we go on to study all three variants with respect to the subdivisions of graphs and oriented graphs. Finally, we establish that all these variants remain computationally difficult even when restricted to the class of 2-degenerate bipartite graphs.

The Hunters and Rabbit game is played on a graph G where the Hunter player shoots at k vertices in every round while the Rabbit player occupies an unknown vertex and, if it is not shot, must move to a neighbouring vertex after each round. The Rabbit player wins if it can ensure that its position is never shot. The Hunter player wins otherwise. The hunter number h(G) of a graph G is the minimum integer k such that the Hunter player has a winning strategy (i.e., allowing him to win whatever be the strategy of the Rabbit player). This game has been studied in several graph classes, in particular in bipartite graphs (grids, trees, hypercubes...), but the computational complexity of computing h(G) remains open in general graphs and even in more restricted graph classes such as trees. To progress further in this study, we propose a notion of monotonicity (a well-studied and useful property in classical pursuit-evasion games such as Graph Searching games) for the Hunters and Rabbit game imposing that, roughly, a vertex that has already been shot "must not host the rabbit anymore". This allows us to obtain new results in various graph classes. More precisely, let the monotone hunter number mh(G) of a graph G be the minimum integer k such that the Hunter player has a monotone winning strategy. We show that pw(G) ≤ mh(G) ≤ pw(G)+1 for any graph G with pathwidth pw(G), which implies that computing mh(G), or even approximating mh(G) up to an additive constant, is NP-hard. Then, we show that mh(G) can be computed in polynomial time in split graphs, interval graphs, cographs and trees. These results go through structural characterisations which allow us to relate the monotone hunter number with the pathwidth in some of these graph classes. In all cases, this allows us to specify the hunter number or to show that there may be an arbitrary gap between h and mh, i.e., that monotonicity does not help. In particular, we show that, for every k ≥ 3, there exists a tree T with h(T) = 2 and mh(T) = k. We conclude by proving that computing h (resp., mh) is FPT parameterised by the minimum size of a vertex cover.

A set $S$ of vertices of a graph $G$ is a \emph{geodetic set} if every vertex of $G$ lies in a shortest path between some pair of vertices of $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality of a given graph. A \emph{grid embedding} of a graph is a set of points in two dimensions with integer coordinates such that each point in the set represents a vertex of the graph and, for each edge, the points corresponding to its endpoints are at Euclidean distance~$1$. A graph is a \emph{partial grid} if it has a grid embedding. In this paper, we first prove that \textsc{Minimum Geodetic Set} remains NP-hard even for subcubic partial grids of arbitrary girth. This jointly strengthens three existing hardness results: for bipartite graphs (Dourado et al., Discrete. Math, 2010), subcubic graphs (Bueno et al., Inf. Process. Lett., 2018)~\cite{bueno2018}, and planar graphs (Chakraborty et al., CALDAM, 2020). The \emph{area} of an internal face is the number of integer points lying on the boundary or interior of the face. A graph is a \emph{solid grid} if it has a grid embedding such that all interior faces have area exactly four. To complement the above hardness result, we design a linear-time algorithm for \textsc{Minimum Geodetic Set} on solid grids, improving on a $3$-approximation algorithm by Chakraborty et al. (CALDAM, 2020). Our results hold for \textsc{Edge Geodetic Set} as well. A set $S$ of vertices of a graph $G$ is a \emph{geodetic set} if every edge of $G$ lies in a shortest path between some pair of vertices of $S$. The \textsc{Minimum Edge Geodetic Set (MEGS)} problem is to find an edge geodetic set with minimum cardinality of a given graph. As corollaries, we obtain that \textsc{MEGS} remains NP-hard on partial grids and is linear-time solvable on solid grids.

A path is isometric if it is a shortest path between its endpoints. In this article, we consider the graph covering problem Isometric Path Cover, where we want to cover all the vertices of the graph using a minimum-size set of isometric paths. Although this problem has been considered from a structural point of view (in particular, regarding applications to pursuit-evasion games), it is little studied from the algorithmic perspective. We consider Isometric Path Cover on chordal graphs, and show that the problem is NP-hard for this class. On the positive side, for chordal graphs, we design a 4-approximation algorithm and an FPT algorithm for the parameter solution size. The approximation algorithm is based on a reduction to the classic path covering problem on a suitable directed acyclic graph obtained from a breadth first search traversal of the graph. The approximation ratio of our algorithm is 3 for interval graphs and 2 for proper interval graphs. Moreover, we extend the analysis of our approximation algorithm to k-chordal graphs (graphs whose induced cycles have length at most k) by showing that it has an approximation ratio of k+7 for such graphs, and to graphs of treelength at most 𝓁, where the approximation ratio is at most 6𝓁+2.

A set $S$ of vertices of a graph $G$ is a \emph{geodetic set} if every vertex of $G$ lies in a shortest path between some pair of vertices of $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality of a given graph. A \emph{grid embedding} of a graph is a set of points in two dimensions with integer coordinates such that each point in the set represents a vertex of the graph and, for each edge, the points corresponding to its endpoints are at Euclidean distance~$1$. A graph is a \emph{partial grid} if it has a grid embedding. In this paper, we first prove that \textsc{Minimum Geodetic Set} remains NP-hard even for subcubic partial grids of arbitrary girth. This jointly strengthens three existing hardness results: for bipartite graphs (Dourado et al., Discrete. Math, 2010), subcubic graphs (Bueno et al., Inf. Process. Lett., 2018)~\cite{bueno2018}, and planar graphs (Chakraborty et al., CALDAM, 2020). The \emph{area} of an internal face is the number of integer points lying on the boundary or interior of the face. A graph is a \emph{solid grid} if it has a grid embedding such that all interior faces have area exactly four. To complement the above hardness result, we design a linear-time algorithm for \textsc{Minimum Geodetic Set} on solid grids, improving on a $3$-approximation algorithm by Chakraborty et al. (CALDAM, 2020). Our results hold for \textsc{Edge Geodetic Set} as well. A set $S$ of vertices of a graph $G$ is a \emph{geodetic set} if every edge of $G$ lies in a shortest path between some pair of vertices of $S$. The \textsc{Minimum Edge Geodetic Set (MEGS)} problem is to find an edge geodetic set with minimum cardinality of a given graph. As corollaries, we obtain that \textsc{MEGS} remains NP-hard on partial grids and is linear-time solvable on solid grids.