arXiv subjects

Saket Saurabh

Publications and source records attributed to Saket Saurabh.

At least 19 recordsLinked to original sources

arXiv2026

Vertex Deletion on $H$-Induced-Minor-Free Graphs: Subexponential FPT Algorithms and Approximation Schemes

We study vertex-deletion problems on string graphs and, more generally, on graphs excluding a fixed graph as an induced minor. For every fixed graph $H$, fixed nonnegative integer $t$, and fixed monadic second-order formula $ϕ$, we consider the problem of finding a minimum-cardinality set $S$ such that $G-S$ has treewidth at most $t$ and satisfies $ϕ$. This captures classic problems such as Vertex Cover, Feedback Vertex Set, and deletion to graphs of any fixed bounded treewidth. We give an algorithm with running time $2^{k^{2/3} \log^{O(1)} k}n^{O(1)}$ on $H$-induced-minor-free graphs, where $k$ is the optimum deletion-set size. This resolves in the affirmative the open problem highlighted by Bonnet and Rzążewski [Algorithmica 2019] and Berthe, Bougeret, Gonçalves, and Raymond [ICALP 2025] regarding the existence of a subexponential parameterized algorithm for Feedback Vertex Set on string graphs. Our algorithm significantly improves over existing parameterized algorithms for more restricted classes of problems on more restricted classes of graphs. When the property defined by $ϕ$ is hereditary, we also obtain a quasi-polynomial time approximation scheme, namely an algorithm that produces a $(1+\varepsilon)$-approximate solution in quasi-polynomial time. Both the sub-exponential algorithm and the quasi-polynomial time approximation scheme build on the same new structural and algorithmic insights.

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arXiv2026

FPTAS for Polygonal Multi-Robot Motion Planning

We study multi-robot motion planning in a polygonal environment with polygonal robots. The setting of the problem consists of a polygonal domain $\mathcal{W}$ with $n$ vertices and $k$ robots $R_1,\dots,R_k$ translating in $\mathcal{W}$ each of which is represented by a polygon of constant complexity. Given a source configuration $\mathbf{s}$ and a target configuration $\mathbf{t}$ of the robots in $\mathcal{W}$, the goal of the problem is to find a collision-free motion plan from $\mathbf{s}$ to $\mathbf{t}$ that is shortest with respect to some cost function. We present the first fully polynomial-time approximation schemes (FPTAS) for motion planning with any fixed number $k$ of robots, under the min-sum objective, the min-max objective, and the min-makespan objective. Here, the min-sum variant aims to minimize the total traveling distance of all robots, the min-max variant aims to minimize the maximum traveling distance of a single robot, and the min-makespan variant aims to minimize the total time cost under the assumption that all robots move with a maximum speed $1$. The algorithm directly generalizes to the problem in any fixed dimension $d$, with a polytope environment and polytope robots. Our result settles an open question of Agarwal, Halperin, Sharir, Steiger [SODA'24] and Agarwal, Holmgren, Steiger [SODA'26], who gave an FPTAS for $k=2$ unit-square robots and a bicriteria-approximation algorithm for a fixed number $k$ of unit-square or unit-disk robots, respectively. Compared to the previous results, our algorithm is substantially simpler.

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arXiv2026

How to Fix a Broken Metric: A Linear Kernel, Tight Bounds, and Tree Metrics

Given a complete graph whose edge weights represent dissimilarities, Metric Violation Distance asks whether at most $k$ weights can be changed to form a metric. Motivated by metric repair for noisy data, the problem admits a polynomial-time $O(\log n)$-approximation due to Cohen-Addad, Fan, Lee and de Mesmay [SIAM J. Comput., 2025]. In the context of parameterized complexity, Fan, Gilbert, Raichel, Sonthalia and Van Buskirk [SWAT 2020] gave a $k^{O(k)}n^{O(1)}$-time algorithm. Fomin, Golovach and More [IPEC 2026] obtained an $O(k^2)$ kernel and a single-exponential algorithm for the ultrametric case. They asked whether general metrics admit a single-exponential algorithm and a polynomial kernel, and whether the tree-metric analogue is fixed-parameter tractable. We answer all three questions. We give a $2^{O(k)}n^{O(1)}$-time algorithm and prove that, unless ETH fails, no $2^{o(k)}n^{O(1)}$-time algorithm exists, even when all input distances lie in $\{1,2,3\}$. We also give a kernel with at most $6k$ vertices. The kernel supports solution lifting and can precede any approximation algorithm. Combined with the $O(\log n)$-approximation of Cohen-Addad, Fan, Lee and de Mesmay, it yields an $O(\log \mathrm{OPT})$-approximation at no asymptotic cost in running time. Both results extend to an interval generalization in which each edge $e$ has an observed value $M_e$ and an admissible range $[A_e,B_e]$ within which it may be reassigned; the kernel then has $7k$ vertices. Finally, Tree Metric Violation Distance is fixed-parameter tractable and solvable in $k^{O(k)}n^{O(1)}$ time.

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arXiv2026

Connected Dominating Set on Semi-Ladder-Free Graphs

We study \textsc{Connected Dominating Set} on graphs whose closed-neighborhood set systems are $d$-semi-ladder-free. This structural condition strictly generalizes the biclique-free setting and provides a natural regime for connectivity-constrained domination. We obtain both a fixed-parameter algorithm and an approximate kernelization framework for the problem on this class. Our algorithmic result is based on a new compact representation theorem for inclusion-wise minimal set covers in $d$-semi-ladder-free set systems. Although the number of minimal set covers of size at most $k$ may be as large as $n^{Ω(k)}$, we show that all such set covers can nevertheless be encoded by a family of at most $k^{kd+1}$ tuples, and that this family can be enumerated in time $\Oh(k^{kd+2}\cdot nm)$. Combining this representation with a \textsc{Group Steiner Tree} subroutine, we obtain an algorithm for \textsc{Connected Set Cover}, which in turn yields an algorithm for \textsc{Connected Dominating Set} running in time $k^{kd+2}\cdot 2^k \cdot n^{\Oh(1)}$ and polynomial space. For the preprocessing result, we introduce grouped domination cores and dominator cores, and prove polynomial upper bounds on their sizes in $d$-semi-ladder-free graphs. Using these structures, we obtain, for every fixed $d$ and $\varepsilon>0$, a polynomial-time $(1+\varepsilon)$-lossy compression for \textsc{Connected Dominating Set} to an equivalent reduced instance of size $k^{\Oh(d^2/\varepsilon)}$. The reduced instance is a \textsc{Connected Dominating Set} instance on a $(d+2)$-semi-ladder-free graph.

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arXiv2026

Single-Exponential Algorithms for Directed Feedback Vertex Set on Planar Digraphs

We consider Directed Feedback Vertex Set on planar digraphs, parameterized by the solution size $k$. We give a randomized algorithm with one-sided error running in time $(2+\sqrt5)^k n^{O(1)}= 4.24^k n^{O(1)}$, and a deterministic algorithm running in time $8.04^k n^{O(1)}$. Both algorithms use polynomial space. To the best of our knowledge, these are the first single-exponential fixed-parameter algorithms for Directed Feedback Vertex Set on planar digraphs. This contrasts with general digraphs, where the best known algorithms run in time $2^{O(k\log k)}(n+m)$, and whether a $2^{o(k\log k)}n^{O(1)}$-time algorithm exists remains a major open problem. Our main tool is an exact Euler-type counting identity for plane digraphs. It shows that every small solution must carry a large share of the vertices whose in- and out-arcs alternate in the embedding, while solutions avoiding such vertices can be computed by reducing to Directed Feedback Arc Set, which is known to be solvable in polynomial time on planar digraphs via the Lucchesi-Younger theorem.

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arXiv2026

Counting Paths and Trees via Exterior Algebra

We give randomized approximation algorithms for counting k-paths and k-forests in a host graph. Here k denotes the number of pattern vertices, n and m denote the numbers of host vertices and edges or arcs, ε is the relative error, and δ is the failure probability. Our main results are: 1. Paths: We approximate the number of directed paths on $k$ vertices in $2^k k^{O(1)}(n+m)\varepsilon^{-2}\log(2/δ)$ arithmetic operations. 2. Trees and forests: For every fixed $η>0$, we approximate the number of non-induced copies of a given forest on $k$ vertices in $(2+η)^k n^{O_η(1)}\varepsilon^{-2}\log(2/δ)$ arithmetic operations. Our path algorithm resolves a conjecture of Koutis and Williams~[CACM 2016] and answers an open question of Lokshtanov, Saurabh, and Zehavi~[SODA 2021] by giving a $2^k poly(n,\varepsilon^{-1})$-time approximation scheme. Our algorithms combine exterior algebra with random matrix estimators, using the tensor-train moment bound of Rakhshan and Rabusseau~[AISTATS 2020]. For forests, we use a small-component separator to evaluate the estimator efficiently.

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arXiv2026

From One Solution to Many: An Oracle-Based FPT Framework for Diverse Solutions under Generalized Diversity Measures

The problem of computing \emph{diverse} solutions has recently emerged as an important area of study, motivated by applications in fairness, robustness, and security. Instead of returning a single feasible or optimal solution, the goal is to output a \emph{collection} of meaningfully different solutions, often measured by symmetric differences. Diverse variants have been studied using sparsification, network-flow reductions, and algebraic techniques. We investigate the fixed-parameter tractability of diverse variants of an implicit set-system problem. Given parameters $k$ and $r$ and a threshold $b$, the task is to compute $r$ feasible solutions, each of size at most $k$, whose diversity under a specified objective is at least $b$. Our main contribution is an oracle-based meta-theorem. We identify a broad class of objectives, called \emph{consistently diverse}, that includes several standard measures. Assuming an \emph{exact empty-extension oracle} given a forbidden set ${\sf Forb}$, which returns a feasible solution of a prescribed size avoiding ${\sf Forb}$ or reports that none exists, we obtain a fixed-parameter tractable algorithm parameterized by $k+r$. The algorithm makes at most $(2kr)^{kr} \cdot r$ oracle calls, and in each call the oracle parameter satisfies $s+|{\sf Forb}| \leq k+2kr$. Our framework unifies and strengthens previous oracle-based approaches. Compared with Kumabe's framework (ESA 2025), which gives a doubly exponential bound on the number of oracle calls, our approach achieves the single exponential bound $2^{O(kr\log(kr))}$ and directly constructs the desired tuple of solutions. We recover fixed-parameter tractable algorithms for all problems covered by that framework, with improved oracle complexity, and obtain strong bounds for diverse variants of classical graph and matroid problems.

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arXiv2026

Fixed Budget vs. Covering Target: The Partial Set Cover Boundary for Bounded VC-Dimension

Maximum Coverage and Partial Set Cover are fundamental parameterized covering problems. The former fixes a budget $k$ and maximizes coverage; the latter meets a target with as few sets as possible. Badanidiyuru, Kleinberg, and Lee (SoCG 2012) give an EPAS for the former on bounded-VC set systems, while Jain et al. (SODA 2023) show that on $K_{d,d}$-free incidence graphs, $k+1$ sets suffice whenever $k$ sets meet the target. We ask whether this guarantee extends to all bounded-VC set systems. Our first result is negative. Unless FPT = W[1], Partial Set Cover admits no parameterized $(2-δ)$-approximation even at VC-dimension seven. Under ETH, it has no parameterized approximation scheme there and no $2^{o(d)}$-approximation at VC-dimension $d$. On the positive side, bounded semi-ladder index restores this guarantee. It is stronger than bounded VC-dimension but strictly generalizes the $K_{d,d}$-free setting. For Weighted Partial Set Cover, if $k$ sets cover weight $W$, we find $k+1$ sets covering weight $W$ in $2^{O(Γk\log k)}N$ time, where $Γ$ is the downward intersection complexity and $N$ is the input size. The framework supports per-class targets and matroid independence, with applications to partial dominating set and geometric and bounded-size covering. Finally, we give a deterministic FPT reduction from Weighted CC-MaxSAT to a bounded family of Weighted Maximum Coverage instances, preserving incidence structure and approximation schemes with constant-factor accuracy loss. This gives an EPAS at bounded semi-ladder index. We improve the deterministic BKL bounded-VC implementation; combined with our reduction, it yields a $2^{\widetilde{O}(kd/\varepsilon)}N^{O(1)}$-time EPAS for bounded-VC Weighted CC-MaxSAT.

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arXiv2026

Covering Points with Rectangular Boundaries

Geometric covering problems ask for a small family of geometric objects whose union covers a given point set. We study the more restrictive \emph{boundary covering} variant, where every point must lie on the boundary of a chosen object. Motivated by the framework of Langerman and Morin\,[Discret.\ Comput.\ Geom., 2005] for hyperspheres, we initiate the study of boundary covering by axis-parallel rectangles. We first consider the \emph{discrete} setting, where rectangles must be selected from a given family. We define \bcdaprfull\ (\bcdaprshort): given a point set $P\subseteq\mathbb{R}^2$, a family $\mathcal{R}$ of axis-parallel rectangles, and an integer $k$, decide whether $P$ can be covered by the boundaries of at most $k$ rectangles from $\mathcal{R}$. We prove that \bcdaprshort\ is $\mathrm{W}[1]$-hard parameterized by $k$. We then study the \emph{continuous} variant, \prbcfull\ (\prbcshort), where rectangles may be placed freely. Given $P\subseteq\mathbb{R}^2$ and $k$, the goal is to 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 \prbcshort\ is fixed-parameter tractable, with running time $2^{\cO(k\log k)}\cdot n^{\cO(1)}$, where $n=|P|$. Our algorithm relies on a structural analysis of how $k$ rectangles interact with the point set, reducing \prbcshort\ to at most $2^{\cO(k\log k)}$ instances of \ddmtcsp, each solvable in polynomial time. On the hardness side, we prove NP-completeness for boundary covering by axis-aligned $L$-shapes and use this reduction to establish NP-completeness of \prbcshort.

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arXiv2026

Courcelle's Theorem in Truly Linear FPT

Recently, Bumpus, Downey, Eagling-Vose, Enright, Fellows, Kutner, Larios-Jones, Martin, Rosamond, and Yates defined Truly Linear FPT (TLFPT) to be the class of parameterized problems with algorithms running in time $O(n) + f(k)$, where $n$ is the input size and $k$ the parameter [arXiv:2606.02492]. They gave several algorithmic techniques for designing TLFPT algorithms, but left parameterization by treewidth open. In this paper, we give a general method for designing TLFPT algorithms parameterized by treewidth, solving three open problems posed by Bumpus et al. In particular, we give a TLFPT algorithm for Courcelle's theorem: We show that given an $n$-vertex $m$-edge graph $G$, an integer $k$, and a $\mathsf{CMSO}_2$-formula $φ$, we can in time $O(n+m) + f(k, φ)$ either conclude that the treewidth of $G$ is more than $k$, or check whether $G$ satisfies $φ$. As a part of our algorithm, we give an approximation algorithm for treewidth that runs in time $O(n+m)$ and returns a tree decomposition whose width is at most $2^{O(k)}$ times the optimum. Our result also implies a TLFPT algorithm for computing the value of treewidth exactly.

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arXiv2026

Fine-Grained Bounds for Courcelle's Theorem

Courcelle's theorem states that there exists an algorithm that takes as input a graph $G$ of treewidth at most $t$ and a MSO formula $ϕ$, and determines whether $G$ satisfies $ϕ$ in time $f(ϕ,t) \cdot n$. It is folklore that the the function $f$ contains a tower of exponentials whose height depends as a linear function of the number of quantifier alternations of the input formula $ϕ$. A classic reduction of Frick and Grohe shows that, assuming the Exponential Time Hypothesis (ETH), the linear growth of the height of the tower is unavoidable. Nevertheless, there is still a huge gap between existing upper and lower bounds -- after all, there is quite a difference between a single exponential and a double exponential running time. In addition, this only gives us a very coarse understanding in the time complexity of Courcelle's theorem. In this paper, we prove a fine-grained version of Courcelle's theorem with nearly ETH-tight dependence on the treewidth parameter $t$ and the quantifier structure of $ϕ$ (specifically, the number of first order and second order variables in each quantifier alternation block).

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arXiv2026

Polynomial Kernels for Spanning Tree with Diversity Requirements

Given a connected undirected graph $G$, a spanning tree is a subgraph $T$ of $G$ such that $V(T) = V(G)$ and $T$ is a tree. A collection of $\ell$ spanning trees $T_1,\ldots,T_\ell$ is pairwise $k$-diverse if for every $i \neq j$, $|E(T_i) \triangle E(T_j)| \geq k$. Given a connected undirected graph $G$ and integers $p, q, k, \ell$, Leaf & Internal-Constrained Diverse Spanning Trees asks whether there are $\ell$ distinct spanning trees $T_1,\ldots,T_{\ell}$ of $G$ that are pairwise $k$-diverse such that each tree has at least $p$ leaves and at least $q$ internal vertices. Similarly, Leaf & Non-terminal-Constrained Diverse Spanning Trees takes a connected undirected graph $G$, $V_{NT}\subseteq V(G)$, and three integers $p, k, \ell$, and asks if $G$ has $\ell$ spanning trees that are pairwise $k$-diverse, and each has at least $p$ leaves and conains the vertices of $V_{NT}$ as internal. We consider these two problems from the kernelization perspective and provide polynomial kernels for Leaf & Internal-Constrained Diverse Spanning Trees and Leaf & Non-terminal-Constrained Diverse Spanning Trees, when parameterized by $p + q + k + \ell$ and $p + |V_{\rm NT}| + k + \ell$, respectively.

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arXiv2026

Dominating Set with Quotas: Balancing Coverage and Constraints

We study a natural generalization of the classical \textsc{Dominating Set} problem, called \textsc{Dominating Set with Quotas} (DSQ). In this problem, we are given a graph $ G $, an integer $ k $, and for each vertex $ v \in V(G) $, a lower quota $ \mathrm{lo}_v $ and an upper quota $ \mathrm{up}_v $. The goal is to determine whether there exists a set $ S \subseteq V(G) $ of size at most $ k $ such that for every vertex $ v \in V(G) $, the number of vertices in its closed neighborhood that belong to $ S $, i.e., $ |N[v] \cap S| $, lies within the range $ [\mathrm{lo}_v, \mathrm{up}_v] $. This richer model captures a variety of practical settings where both under- and over-coverage must be avoided -- such as in fault-tolerant infrastructure, load-balanced facility placement, or constrained communication networks. While DS is already known to be computationally hard, we show that the added expressiveness of per-vertex quotas in DSQ introduces additional algorithmic challenges. In particular, we prove that DSQ becomes \W[1]-hard even on structurally sparse graphs -- such as those with degeneracy 2, or excluding $ K_{3,3} $ as a subgraph -- despite these classes admitting FPT algorithms for DS. On the positive side, we show that DSQ is fixed-parameter tractable when parameterized by solution size and treewidth, and more generally, on nowhere dense graph classes. Furthermore, we design a subexponential-time algorithm for DSQ on apex-minor-free graphs using the bidimensionality framework. These results collectively offer a refined view of the algorithmic landscape of DSQ, revealing a sharp contrast with the classical DS problem and identifying the key structural properties that govern tractability.

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arXiv2026

Algorithms for Euclidean Distance Matrix Completion: Exploiting Proximity to Triviality

In the d-Euclidean Distance Matrix Completion (d-EDMC) problem, one aims to determine whether a given partial matrix of pairwise distances can be extended to a full Euclidean distance matrix in d dimensions. This problem is a cornerstone of computational geometry with numerous applications. While classical work on this problem often focuses on exploiting connections to semidefinite programming typically leading to approximation algorithms, we focus on exact algorithms and propose a novel distance-from-triviality parameterization framework to obtain tractability results for d-EDMC. We identify key structural patterns in the input that capture entry density, including chordal substructures and coverability of specified entries by fully specified principal submatrices. We obtain: (1) The first fixed-parameter algorithm (FPT algorithm) for d-EDMC parameterized by d and the maximum number of unspecified entries per row/column. This is achieved through a novel compression algorithm that reduces a given instance to a submatrix on O(1) rows (for fixed values of the parameters). (2) The first FPT algorithm for d-EDMC parameterized by d and the minimum number of fully specified principal submatrices whose entries cover all specified entries of the given matrix. This result is also achieved through a compression algorithm. (3) A polynomial-time algorithm for d-EDMC when both d and the minimum fill-in of a natural graph representing the specified entries are fixed constants. This result is achieved by combining tools from distance geometry and algorithms from real algebraic geometry. Our work identifies interesting parallels between EDM completion and graph problems, with our algorithms exploiting techniques from both domains.

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arXiv2026

Maximum Partial List H-Coloring on P_5-free graphs in polynomial time

In this article we show that Maximum Partial List H-Coloring is polynomial-time solvable on P_5-free graphs for every fixed graph H. In particular, this implies that Maximum k-Colorable Subgraph is polynomial-time solvable on P_5-free graphs. This answers an open question from Agrawal, Lima, Lokshtanov, Saurabh & Sharma [SODA 2024]. This also improves the $n^{O(ω(G))}$-time algorithm for Maximum Partial H-Coloring by Chudnovsky, King, Pilipczuk, Rzążewski & Spirkl [SIDMA 2021] to polynomial-time algorithm.

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arXiv2026

FPT Approximations for Connected Maximum Coverage

We revisit connectivity-constrained coverage through a unifying model, Partial Connected Red-Blue Dominating Set. Given a red-blue bipartite graph $G$ and an auxiliary connectivity graph $G_{conn}$ on red vertices, and integers $k, t$, the task is to find a $k$-sized subset of red vertices that dominates at least $t$ blue vertices, and that induces a connected subgraph in $G_{conn}$. This formulation captures connected variants of Max Coverage, Partial Dominating Set, and Partial Vertex Cover studied in prior literature. After identifying (parameterized) inapproximability results inherited from known problems, we first show that the problem is fixed-parameter tractable by $t$. Furthermore, when the bipartite graph excludes $K_{d,d}$ as a subgraph, we design (resp. efficient) parameterized approximation schemes for approximating $t$ (resp. $k$). Notably, these FPT approximations do not impose any restrictions on $G_{conn}$. Together, these results chart the boundary between hardness and FPT-approximability for connectivity-constrained coverage.

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arXiv2025

Tight Parameterized (In)tractability of Layered Crossing Minimization: Subexponential Algorithms and Kernelization

The starting point of our work is a decade-old open question concerning the subexponential parameterized complexity of \textsc{2-Layer Crossing Minimization}. In this problem, the input is an $n$-vertex graph $G$ whose vertices are partitioned into two independent sets $V_1$ and $V_2$, and a non-negative integer $k$. The question is whether $G$ admits a 2-layered drawing with at most $k$ crossings, where each $V_i$ lies on a distinct line parallel to the $x$-axis, and all edges are straight lines. We resolve this open question by giving the first subexponential fixed-parameter algorithm for this problem, running in time $2^{O(\sqrt{k}\log k)} + n \cdot k^{O(1)}$. We then ask whether the subexponential phenomenon extends beyond two layers. In the general $h$-Layer Crossing Minimization problem, the vertex set is partitioned into $h$ independent sets $V_1, \ldots, V_h$, and the goal is to decide whether an $h$-layered drawing with at most $k$ crossings exists. We present a subexponential FPT algorithm for three layers with running time $2^{O(k^{2/3}\log k)} + n \cdot k^{O(1)}$ for $h = 3$ layers. In contrast, we show that for all $h \ge 5$, no algorithm with running time $2^{o(k/\log k)} \cdot n^{O(1)}$ exists unless the Exponential-Time Hypothesis fails. Finally, we address polynomial kernelization. While a polynomial kernel was already known for $h=2$, we design a new polynomial kernel for $h=3$. These kernels are essential ingredients in our subexponential algorithms. Finally, we rule out polynomial kernels for all $h \ge 4$ unless the polynomial hierarchy collapses.

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arXiv2025

More Efforts Towards Fixed-Parameter Approximability of Multiwinner Rules

Multiwinner Elections have emerged as a prominent area of research with numerous practical applications. We contribute to this area by designing parameterized approximation algorithms and also resolving an open question by Yang and Wang [AAMAS'18]. More formally, given a set of candidates, \mathcal{C}, a set of voters,\mathcal{V}, approving a subset of candidates (called approval set of a voter), and an integer $k$, we consider the problem of selecting a ``good'' committee using Thiele rules. This problem is computationally challenging for most Thiele rules with monotone submodular satisfaction functions, as there is no (1-\frac{1}{e}-ε)\footnote{Here, $e$ denotes the base of the natural logarithm.}-approximation algorithm in f(k)(|\mathcal{C}| + |\mathcal{V}|)^{o(k)} time for any fixed $ε> 0$ and any computable function $f$, and no {\sf PTAS} even when the length of approval set is two. Skowron [WINE'16] designed an approximation scheme running in FPT time parameterized by the combined parameter, size of the approval set and $k$. In this paper, we consider a parameter $d+k$ (no $d$ voters approve the same set of $d$ candidates), where $d$ is upper bounded by the size of the approval set (thus, can be much smaller). With respect to this parameter, we design parameterized approximation schemes, a lossy polynomial-time preprocessing method, and show that an extra committee member suffices to achieve the desired score (i.e., $1$-additive approximation). Additionally, we resolve an open question by Yang and Wang~[AAMAS'18] regarding the fixed-parameter tractability of the problem under the PAV rule with the total score as the parameter, demonstrating that it admits an FPT algorithm.

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