Harmender Gahlawat

Dibyayan Chakraborty, Harmender Gahlawat, Bodhayan Roy

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.

Parinya Chalermsook, Harmender Gahlawat, Felix Klingelhoefer et al.

The dichromatic number $\vecχ(D)$ 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 $\vecα(D)$ 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 $\vecχ$ and $\vecα$ remain as challenging as their undirected counterparts on general graphs. Specifically, we establish that for every $ε>0$, it is hard to approximate both $\vecα$ and $\vecχ$ 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 $\ell$-dicolorable digraphs using at most $\ell \cdot n^{1-\frac{1}{\ell}}$ colors in time $O(n^{2\ell})$; in particular, we can color $2$-dicolorable digraphs with $2\sqrt{n}$ 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 $\vecχ(D)\leq 2$ and digraphs that do not contain any directed triangle. For these cases, we present algorithms which generalize and improve existing tools and results.

Harmender Gahlawat, Meirav Zehavi

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

Foivos Fioravantes, Harmender Gahlawat, Nikolaos Melissinos

Imagine we want to split a group of agents into teams in the most \emph{efficient} way, considering that each agent has their own preferences about their teammates. This scenario is modeled by the extensively studied \textsc{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 \emph{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 \emph{vertex cover number}).

Florent Foucaud, Harmender Gahlawat, Fionn Mc Inerney et al.

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 $2^{\mathcal{O}(\rm{tw}\cdot \log \rm{tw})}\cdot |V|$-time algorithm for any graph $G=(V,E)$ of treewidth $\rm{tw}$ (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).