Geometric covering problems typically ask for a small family of geometric objects whose union contains all input points. In this paper we study a more rigid variant, boundary covering, where every point must lie on the boundary of at least one chosen object. Motivated by the framework of Langerman and Morin [Discret. Comput. Geom., 2005] for boundary covering by hyperspheres, we initiate a systematic study of boundary covering by axis-parallel rectangles in the plane. We first consider the discrete setting, where the rectangles must be chosen from a given family. We define Boundary Covering with Discrete Axis-Parallel Rectangles (BCDAPR) as follows: given a point set P ⊆ ℝ², a collection ℛ of axis-parallel rectangles, and an integer k, decide whether P can be covered by the boundaries of at most k rectangles from ℛ. We prove that this discrete boundary-covering problem is W[1]-hard when parameterized by k. This motivates the continuous variant, where we are allowed to place rectangles freely. We define Boundary Covering with Continuous Axis-Parallel Rectangles (BCCAPR) as follows: given a point set P ⊆ ℝ² and an integer k, decide whether P can be covered by the boundaries of at most k axis-parallel rectangles. In contrast to the discrete case, we show that BCCAPR is fixed-parameter tractable parameterized by k, with running time 2^𝒪(k log k) ⋅ n^𝒪(1), where n = |P|. Our results does a fine-grained structural analysis of how k rectangles can interact with the point set. On the hardness side, we show that moving from lines to slightly richer shapes already incurs intractability: we prove NP-completeness for boundary covering by axis-aligned L-shapes, and then lift it to NP-completeness of BCCAPR. For the algorithm we reduce BCCAPR to at most 2^𝒪(k log k) instances of Distinct Domain Monotone ,$-CSP, each solvable in polynomial time.

We revisit connectivity-constrained coverage through a unifying model, Partial Connected Red-Blue Dominating Set (PartialConRBDS). Given a bipartite graph G = (R∪ B,E) with red vertices R and blue vertices B, an auxiliary connectivity graph G_{conn} on R, and integers k,t, the task is to find a set S ⊆ R with |S| ≤ k such that G_{conn}[S] is connected and S dominates at least t blue vertices. This formulation captures connected variants of Maximum Coverage [Hochbaum-Rao, Inf. Proc. Lett., 2020; D'Angelo-Delfaraz, AAMAS 2025], Partial Vertex Cover, and Partial Dominating Set [Khuller et al., SODA 2014; Lamprou et al., TCS 2021] via standard encodings. Limits to parameterized tractability. PartialConRBDS is W[1]-hard parameterized by k even under strong restrictions: it remains hard when G_{conn} is a clique or a star and the incidence graph G is 3-degenerate, or when G is K_{2,2}-free. Inapproximability. For every ε > 0, there is no polynomial-time (1, 1-1/e+ε)-approximation unless 𝖯 = NP. Moreover, under ETH, no algorithm running in f(k)⋅ n^{o(k)} time achieves an g(k)-approximation for k for any computable function g(⋅), or for any ε > 0, a (1-1/e+ε)-approximation for t. Graphical special cases. Partial Connected Dominating Set is W[2]-hard parameterized by k and inherits the same ETH-based f(k)⋅ n^{o(k)} inapproximability bound as above; Partial Connected Vertex Cover is W[1]-hard parameterized by k. These hardness boundaries delineate a natural "sweet spot" for study: within appropriate structural restrictions on the incidence graph, one can still aim for fine-grained (FPT) approximations. Our algorithms. We solve PartialConRBDS exactly by reducing it to Relaxed Directed Steiner Out-Tree in time (2e)^t ⋅ n^{𝒪(1)}. For biclique-free incidences (i.e., when G excludes K_{d,d} as an induced subgraph), we obtain two complementary parameterized schemes: - An Efficient Parameterized Approximation Scheme (EPAS) running in time 2^{𝒪(k² d/ε)}⋅ n^{𝒪(1)} that either returns a connected solution of size at most k covering at least (1-ε)t blue vertices, or correctly reports that no connected size-k solution covers t; and - A Parameterized Approximation Scheme (PAS) running in time 2^{𝒪(kd(k²+log d))}⋅ n^{𝒪(1/ε)} that either returns a connected solution of size at most (1+ε)k covering at least t blue vertices, or correctly reports that no connected size-k solution covers t. Together, these results chart the boundary between hardness and FPT-approximability for connectivity-constrained coverage.

We study Maximum Span, motivated by the recent Maximum Span Hypothesis of Karthik and Khot [SODA 2025], which suggests strong parameterized intractability for finding large structured subsets in vector spaces. Formally, given a matrix M and integers k and t, the task is to decide whether there exists a linearly independent set S of at most k columns such that at least t additional columns of M lie in span(S). Equivalently, the goal is to identify a low-rank witness whose span covers many input columns. We initiate a systematic study of the parameterized complexity of Maximum Span on natural matroid classes, revealing a diverse complexity landscape. We first show that the problem is polynomial-time solvable on laminar matroids, via a dynamic program over the laminar tree. In sharp contrast, on graphic matroids the problem is W[1]-hard parameterized by k+t, and, assuming Gap-ETH, admits no f(k)⋅ n^𝒪(1)-time k^o(1)-approximation. On cographic matroids, we show that the problem is equivalent to deleting at most k+t edges so as to create at least t+1 connected components; this yields fixed-parameter tractability parameterized by k+t, and W[1]-hardness parameterized by t. On transversal matroids, using a Hall-type interpretation, we prove W[1]-hardness parameterized by k+t. For strict gammoids, we develop a separator-based formulation. We prove W[1]-hardness parameterized by k+t, give an XP algorithm parameterized by t, and obtain FPT 2^k-approximation algorithms in both the directed and undirected settings. For general gammoids, we establish W[1]-hardness parameterized by k+t, NP-hardness already for t = 1, and an XP algorithm parameterized by k. Together, these results give a detailed parameterized complexity map for Maximum Span across fundamental matroid classes, ranging from polynomial-time solvability to fixed-parameter algorithms, XP algorithms, approximation algorithms, and strong hardness.

In the last decade, FPT approximation has witnessed tremendous growth, with the development of several powerful upper- and lower-bound techniques. Within this framework, a newly emerging direction focuses on problems that admit algorithms with running time of the form ck · nO(1) for some constant c. This line of inquiry naturally leads to the notion of time–approximation ratio trade-offs (or time-ratio trade-offs): by relaxing the approximation guarantee in a controlled manner, one can improve the exponential dependence on the parameter in the running time. The contribution of this paper is threefold: (i) a formal language for parameterized randomized branching algorithms (called Oracle Subset Problems); (ii) a meta-algorithm applicable to all problems expressible in this language; and (iii) new time–ratio trade-offs obtained by instantiating the framework on fundamental problems, including Above-Guarantee Vertex Cover (parameterized by excess over the LP lower bound), Odd Cycle Transversal, Node Multiway Cut, Subset/Group Feedback Vertex Set, Min-Weight d-SAT, and Matroid-Rank d-Hitting Set (where solution is measured by the rank in a matroid accessible via an independence oracle), among others. Our applications demonstrate substantially broader applicability. For the first time, they apply to cut problems, problems with parity constraints (Odd Cycle Transversal), “complex” cycle hitting problems (hitting all cycles whose length mod73 is non-zero), and even a generalization where the user specifies the subset of vertices such that only the cycles passing through that subset of vertices should be hit. These results are obtained by developing time–ratio trade-offs for two meta-algorithms, expressed in our language: (i) the biased-graph framework [Wahlström, SODA 2017; Lee and Wahlström, arXiv 2020], and (ii) the Vertex Cover above LP framework [Lokshtanov et al., TALG 2014].

Learning the structure of a Bayesian network from data is one of the key problems in probabilistic graphical models. Unfortunately, the problem is NP-hard and this has motivated recent works where the problem has been studied from the perspective of algorithmic paradigms meant for coping with hardness, such as parameterized complexity. We contribute to this area by designing fixed parameter tractable algorithms (FPT) to learn the Bayesian network structure when only a few variables are important. In particular, we study score-based structure learning where each graph is given with a score, based on how well it fits to the data, and the goal is to select the acyclic directed graph (DAG) that maximizes the score. Typically, one uses decomposable scores, where the score of a DAG is the sum of local scores for node-parent set pairs. We study a variant of this problem in which our objective is to find a k-heavy DAG, which is a DAG whose k most scoring nodes have a total score of at least some target value ℓ. We show that 1. if there is a k-heavy DAG with a maximum degree of d, then we can learn it in time f(k,d)nO(d) and 2. if there is a k-heavy DAG whose moralized graph has a treewidth of t and a maximum degree of t, then we can learn it in time f(k,t)nO(t). These algorithms leverage the color-coding technique from the field of Parameterized Complexity in a non-trivial manner.

For numerous graph problems in the realm of parameterized algorithms, using the size of a smallest deletion set (called a modulator) into well-understood graph families as parameterization has led to a long and successful line of research. Recently, however, there has been an extensive study of structural parameters that are potentially much smaller than the modulator size. In particular, recent papers [Jansen et al. STOC 2021; Agrawal et al. SODA 2022] have studied parameterization by the size of the modulator to a graph family $\mathcal{H}$ ($\textbf{mod}_{\mathcal{H}}$), elimination distance to $\mathcal{H}$ ($\textbf{ed}_{\mathcal{H}}$), and $\mathcal{H}$-treewidth ($\textbf{tw}_{\mathcal{H}}$). While these new parameters have been successfully exploited to design fast exact algorithms their utility (especially that of latter two) in the context of approximation algorithms is mostly unexplored. The conceptual contribution of this paper is to present novel algorithmic meta-theorems that expand the impact of these structural parameters to the area of FPT Approximation, mirroring their utility in the design of exact FPT algorithms. Precisely, we show that if a covering or packing problem is definable in Monadic Second Order Logic and has a property called Finite Integer Index, then the existence of an FPT Approximation Scheme (FPT-AS, i.e., ($1\pm ε$)-approximation) parameterized these three parameters is in fact equivalent. As concrete exemplifications of our meta-theorems, we obtain FPT-ASes for well-studied graph problems such as Vertex Cover, Feedback Vertex Set, Cycle Packing and Dominating Set, parameterized by these three parameters.

Selection of a group of representatives satisfying certain fairness constraints, is a commonly occurring scenario. Motivated by this, we initiate a systematic algorithmic study of a \emph{fair} version of \textsc{Hitting Set}. In the classical \textsc{Hitting Set} problem, the input is a universe $\mathcal{U}$, a family $\mathcal{F}$ of subsets of $\mathcal{U}$, and a non-negative integer $k$. The goal is to determine whether there exists a subset $S \subseteq \mathcal{U}$ of size $k$ that \emph{hits} (i.e., intersects) every set in $\mathcal{F}$. Inspired by several recent works, we formulate a fair version of this problem, as follows. The input additionally contains a family $\mathcal{B}$ of subsets of $\mathcal{U}$, where each subset in $\mathcal{B}$ can be thought of as the group of elements of the same \emph{type}. We want to find a set $S \subseteq \mathcal{U}$ of size $k$ that (i) hits all sets of $\mathcal{F}$, and (ii) does not contain \emph{too many} elements of each type. We call this problem \textsc{Fair Hitting Set}, and chart out its tractability boundary from both classical as well as multivariate perspective. Our results use a multitude of techniques from parameterized complexity including classical to advanced tools, such as, methods of representative sets for matroids, FO model checking, and a generalization of best known kernels for \textsc{Hitting Set}.