arXiv subjects

Saket Saurabh

Publications and source records attributed to Saket Saurabh.

At least 19 recordsLinked to original sources

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.

cs.DS↗

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.

cs.DS↗

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.

cs.DS↗

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.

cs.CG↗

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.

cs.DS↗

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).

cs.DS↗

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.

cs.DS↗

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.

cs.DS↗

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.

cs.DS↗

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.

cs.DS↗

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.

cs.DS↗

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.

cs.DS↗

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.

cs.GT↗

arXiv2025

Path Contraction Faster than $2^n$

A graph $G$ is contractible to a graph $H$ if there is a set $X \subseteq E(G)$, such that $G/X$ is isomorphic to $H$. Here, $G/X$ is the graph obtained from $G$ by contracting all the edges in $X$. For a family of graphs $\cal F$, the $\mathcal{F}$-\textsc{Contraction} problem takes as input a graph $G$ on $n$ vertices, and the objective is to output the largest integer $t$, such that $G$ is contractible to a graph $H \in {\cal F}$, where $|V(H)|=t$. When $\cal F$ is the family of paths, then the corresponding $\mathcal{F}$-\textsc{Contraction} problem is called \textsc{Path Contraction}. The problem \textsc{Path Contraction} admits a simple algorithm running in time $2^{n}\cdot n^{\mathcal{O}(1)}$. In spite of the deceptive simplicity of the problem, beating the $2^{n}\cdot n^{\mathcal{O}(1)}$ bound for \textsc{Path Contraction} seems quite challenging. In this paper, we design an exact exponential time algorithm for \textsc{Path Contraction} that runs in time $1.99987^n\cdot n^{\mathcal{O}(1)}$. We also define a problem called \textsc{$3$-Disjoint Connected Subgraphs}, and design an algorithm for it that runs in time $1.88^n\cdot n^{\mathcal{O}(1)}$. The above algorithm is used as a sub-routine in our algorithm for {\sc Path Contraction}

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arXiv2025

When Distances Lie: Euclidean Embeddings in the Presence of Outliers and Distance Violations

Distance geometry explores the properties of distance spaces that can be exactly represented as the pairwise Euclidean distances between points in $\mathbb{R}^d$ ($d \geq 1$), or equivalently, distance spaces that can be isometrically embedded in $\mathbb{R}^d$. In this work, we investigate whether a distance space can be isometrically embedded in $\mathbb{R}^d$ after applying a limited number of modifications. Specifically, we focus on two types of modifications: outlier deletion (removing points) and distance modification (adjusting distances between points). The central problem, Euclidean Embedding Editing (EEE), asks whether an input distance space on $n$ points can be transformed, using at most $k$ modifications, into a space that is isometrically embeddable in $\mathbb{R}^d$. We present several fixed-parameter tractable (FPT) and approximation algorithms for this problem. Our first result is an algorithm that solves EEE in time $(dk)^{\mathcal{O}(d+k)} + n^{\mathcal{O}(1)}$. The core subroutine of this algorithm, which is of independent interest, is a polynomial-time method for compressing the input distance space into an equivalent instance of EEE with $\mathcal{O}((dk)^2)$ points. For the special but important case of EEE where only outlier deletions are allowed, we improve the parameter dependence of the FPT algorithm and obtain a running time of $\min\{(d+3)^k, 2^{d+k}\} \cdot n^{\mathcal{O}(1)}$. Additionally, we provide an FPT-approximation algorithm for this problem, which outputs a set of at most $2 \cdot {\rm OPT}$ outliers in time $2^d \cdot n^{\mathcal{O}(1)}$. This 2-approximation algorithm improves upon the previous $(3+\varepsilon)$-approximation algorithm by Sidiropoulos, Wang, and Wang [SODA '17]. Furthermore, we complement our algorithms with hardness results motivating our choice of parameterizations.

cs.CG↗

arXiv2025

A Quadratic Vertex Kernel and a Subexponential Algorithm for Subset-FAST

In the Subset Feedback Arc Set in Tournaments, Subset-FAST problem we are given as input a tournament $T$ with a vertex set $V(T)$ and an arc set $A(T)$, along with a terminal set $S \subseteq V(T)$, and an integer $ k$. The objective is to determine whether there exists a set $ F \subseteq A(T) $ with $|F| \leq k$ such that the resulting graph $T-F $ contains no cycle that includes any vertex of $S$. When $S=V(T)$ this is the classic Feedback Arc Set in Tournaments (FAST) problem. We obtain the first polynomial kernel for this problem parameterized by the solution size. More precisely, we obtain an algorithm that, given an input instance $(T, S, k)$, produces an equivalent instance $(T',S',k')$ with $k'\leq k$ and $V(T')=O(k^2)$. It was known that FAST admits a simple quadratic vertex kernel and a non-trivial linear vertex kernel. However, no such kernel was previously known for Subset-FAST. Our kernel employs variants of the most well-known reduction rules for FAST and introduces two new reduction rules to identify irrelevant vertices. As a result of our kernelization, we also obtain the first sub-exponential time FPT algorithm for Subset-FAST.

cs.DM↗

arXiv2025

Subexponential Parameterized Algorithms for Hitting Subgraphs

For a finite set $\mathcal{F}$ of graphs, the $\mathcal{F}$-Hitting problem aims to compute, for a given graph $G$ (taken from some graph class $\mathcal{G}$) of $n$ vertices (and $m$ edges) and a parameter $k\in\mathbb{N}$, a set $S$ of vertices in $G$ such that $|S|\leq k$ and $G-S$ does not contain any subgraph isomorphic to a graph in $\mathcal{F}$. As a generic problem, $\mathcal{F}$-Hitting subsumes many fundamental vertex-deletion problems that are well-studied in the literature. The $\mathcal{F}$-Hitting problem admits a simple branching algorithm with running time $2^{O(k)}\cdot n^{O(1)}$, while it cannot be solved in $2^{o(k)}\cdot n^{O(1)}$ time on general graphs assuming the ETH. In this paper, we establish a general framework to design subexponential parameterized algorithms for the $\mathcal{F}$-Hitting problem on a broad family of graph classes. Specifically, our framework yields algorithms that solve $\mathcal{F}$-Hitting with running time $2^{O(k^c)}\cdot n+O(m)$ for a constant $c<1$ on any graph class $\mathcal{G}$ that admits balanced separators whose size is (strongly) sublinear in the number of vertices and polynomial in the size of a maximum clique. Examples include all graph classes of polynomial expansion and many important classes of geometric intersection graphs. Our algorithms also apply to the \textit{weighted} version of $\mathcal{F}$-Hitting, where each vertex of $G$ has a weight and the goal is to compute the set $S$ with a minimum weight that satisfies the desired conditions. The core of our framework is an intricate subexponential branching algorithm that reduces an instance of $\mathcal{F}$-Hitting (on the aforementioned graph classes) to $2^{O(k^c)}$ general hitting-set instances, where the Gaifman graph of each instance has treewidth $O(k^c)$, for some constant $c<1$ depending on $\mathcal{F}$ and the graph class.

cs.DS↗

arXiv2025

Exponential-Time Approximation (Schemes) for Vertex-Ordering Problems

In this paper, we begin the exploration of vertex-ordering problems through the lens of exponential-time approximation algorithms. In particular, we ask the following question: Can we simultaneously beat the running times of the fastest known (exponential-time) exact algorithms and the best known approximation factors that can be achieved in polynomial time? Following the recent research initiated by Esmer et al. (ESA 2022, IPEC 2023, SODA 2024) on vertex-subset problems, and by Inamdar et al. (ITCS 2024) on graph-partitioning problems, we focus on vertex-ordering problems. In particular, we give positive results for Feedback Arc Set, Optimal Linear Arrangement, Cutwidth, and Pathwidth. Most of our algorithms build upon a novel ``balanced-cut'' approach, which is our main conceptual contribution. This allows us to solve various problems in very general settings allowing for directed and arc-weighted input graphs. Our main technical contribution is a (1+ε)-approximation for any ε > 0 for (weighted) Feedback Arc Set in O*((2-δ)^n) time, where δ > 0 is a constant only depending on ε.

cs.DS↗