arXiv subjects

Fahad Panolan

Publications and source records attributed to Fahad Panolan.

At least 19 recordsLinked to original sources

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 $\phi$, and determines whether $G$ satisfies $\phi$ in time $f(\phi,t) \cdot n$. 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 $\phi$. 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 $\phi$ (specifically, the number of first order and second order variables in each quantifier alternation block).

cs.DS↗

arXiv2026

Clausal Deletion Backdoors for QBF: a Parameterized Complexity Approach

Determining the validity of a quantified Boolean formula (QBF) is a PSPACE-complete problem with rich expressive power. Despite interest in efficient solvers, there is, compared to problems in NP, a lack of positive theoretical results, and in the parameterized complexity setting one often has to restrict the quantifier prefix (e.g., bounding alternations) to obtain fixed parameter tractability (FPT). We propose a new parameter: the number of variables in clauses that has to be removed before reaching a tractable class (a clause covering (CC) backdoor). We are then interested in solving QBF in FPT time given a CC-backdoor of size $k$. We consider the three classical, tractable cases of QBF as base classes: Horn, 2-CNF, and linear equations. We establish W[1]-hardness for Horn but prove FPT for the others, and prove that in a precise, algebraic sense, we are only missing one important case for a full dichotomy. Our algorithms are non-trivial and depend on propagation, and Gaussian elimination, respectively, and are comparably unexplored for QBF.

cs.CC↗

arXiv2026

Backdoors for Quantified Boolean Formulas

The quantified Boolean formula problem (QBF) is a well-known PSpace-complete problem with rich expressive power, and is generally viewed as the SAT analogue for PSpace. Given that many problems today are solved in practice by reducing to SAT, and then using highly optimized SAT solvers, it is natural to ask whether problems in PSpace are amenable to this approach. While SAT solvers exploit hidden structural properties, such as backdoors to tractability, backdoor analysis for QBF is comparatively very limited. We present a comprehensive study of the (parameterized) complexity of QBF parameterized by backdoor size to the largest tractable syntactic classes: HORN, 2-SAT, and AFFINE. While SAT is in FPT under this parameterization, we prove that QBF remains PSpace-hard even on formulas with backdoors of constant size. Parameterizing additionally by the quantifier depth, we design FPT-algorithms for the classes 2-SAT and AFFINE, and show that 3-HORN is W[1]-hard. As our next contribution, we vastly extend the applicability of QBF backdoors not only for the syntactic classes defined above but also for tractable classes defined via structural restrictions, such as formulas with bounded incidence treewidth and quantifier depth. To this end, we introduce enhanced backdoors: these are separators S of size at most k in the primal graph such that S together with all variables contained in any purely universal component of the primal graph minus S is a backdoor. We design FPT-algorithms with respect to k for both evaluation and detection of enhanced backdoors to all tractable classes of QBF listed above and more.

cs.DS↗

arXiv2026

The Parameterized Complexity of Vertex-Coloring Edge-Weighting

Motivated by the landmark resolution of the 1-2-3 Conjecture, we initiate the study of the parameterized complexity of the Vertex-Coloring {0,1}-Edge-Weighting problem and its generalization, Vertex-Coloring Pre-edge-Weighting, under various structural parameters. The base problem, Vertex-Coloring {0,1}-Edge-Weighting, asks whether we can assign a weight from {0,1} to each edge of a graph. The goal is to ensure that for every pair of adjacent vertices, the sums of their incident edge weights are distinct. In the Vertex-Coloring Pre-edge-Weighting variant, we are given a graph where a subset of edges is already assigned fixed weights from {0,1}. The goal is to determine if this partial weighting can be extended to all remaining edges such that the final, complete assignment satisfies the proper vertex coloring property. While the existence of such weightings is well-understood for specific graph classes, their algorithmic complexity under structural parameterization has remained unexplored. We prove both hardness and tractability for the problem, across a hierarchy of structural parameters. We show that both the base problem and the Pre-edge-Weighting variant are W[1]-hard when parameterized by the size of a feedback vertex set of the input graph. On the positive side, we establish that the base problem and a restricted Pre-edge-Weighting variant where the pre-assigned weights are all 1, become FPT when parameterized by the size of a vertex cover of the input graph. Further, we show that both the base problem and the Pre-edge-Weighting variant have XP algorithms when parameterized by the treewidth of the input graph.

cs.DS↗

arXiv2024

Fast FPT Algorithms for Grundy Number on Dense Graphs

In this paper, we investigate the \textsc{Grundy Coloring} problem for graphs with a cluster modulator, a structure commonly found in dense graphs. The Grundy chromatic number, representing the maximum number of colors needed for the first-fit coloring of a graph in the worst-case vertex ordering, is known to be $W[1]$-hard when parameterized by the number of colors required by the most adversarial ordering. We focus on fixed-parameter tractable (FPT) algorithms for solving this problem on graph classes characterized by dense substructures, specifically those with a cluster modulator. A cluster modulator is a vertex subset whose removal results in a cluster graph (a disjoint union of cliques). We present FPT algorithms for graphs where the cluster graph consists of one, two, or $k$ cliques, leveraging the cluster modulator's properties to achieve the best-known FPT runtimes, parameterized by both the modulator's size and the number of cliques.

cs.DS↗

arXiv2024

Parameterized Saga of First-Fit and Last-Fit Coloring

The classic greedy coloring (first-fit) algorithm considers the vertices of an input graph $G$ in a given order and assigns the first available color to each vertex $v$ in $G$. In the {\sc Grundy Coloring} problem, the task is to find an ordering of the vertices that will force the greedy algorithm to use as many colors as possible. In the {\sc Partial Grundy Coloring}, the task is also to color the graph using as many colors as possible. This time, however, we may select both the ordering in which the vertices are considered and which color to assign the vertex. The only constraint is that the color assigned to a vertex $v$ is a color previously used for another vertex if such a color is available. Whether {\sc Grundy Coloring} and {\sc Partial Grundy Coloring} admit fixed-parameter tractable (FPT) algorithms, algorithms with running time $f(k)n^{\OO(1)}$, where $k$ is the number of colors, was posed as an open problem by Zaker and by Effantin et al., respectively. Recently, Aboulker et al. (STACS 2020 and Algorithmica 2022) resolved the question for \Grundycol\ in the negative by showing that the problem is W[1]-hard. For {\sc Partial Grundy Coloring}, they obtain an FPT algorithm on graphs that do not contain $K_{i,j}$ as a subgraph (a.k.a. $K_{i,j}$-free graphs). Aboulker et al.~re-iterate the question of whether there exists an FPT algorithm for {\sc Partial Grundy Coloring} on general graphs and also asks whether {\sc Grundy Coloring} admits an FPT algorithm on $K_{i,j}$-free graphs. We give FPT algorithms for {\sc Partial Grundy Coloring} on general graphs and for {\sc Grundy Coloring} on $K_{i,j}$-free graphs, resolving both the questions in the affirmative. We believe that our new structural theorems for partial Grundy coloring and ``representative-family'' like sets for $K_{i,j}$-free graphs that we use in obtaining our results may have wider algorithmic applications.

cs.DM↗

arXiv2024

Packing Short Cycles

Cycle packing is a fundamental problem in optimization, graph theory, and algorithms. Motivated by recent advancements in finding vertex-disjoint paths between a specified set of vertices that either minimize the total length of the paths [Bj\"orklund, Husfeldt, ICALP 2014; Mari, Mukherjee, Pilipczuk, and Sankowski, SODA 2024] or request the paths to be shortest [Lochet, SODA 2021], we consider the following cycle packing problems: Min-Sum Cycle Packing and Shortest Cycle Packing. In Min-Sum Cycle Packing, we try to find, in a weighted undirected graph, $k$ vertex-disjoint cycles of minimum total weight. Our first main result is an algorithm that, for any fixed $k$, solves the problem in polynomial time. We complement this result by establishing the W[1]-hardness of Min-Sum Cycle Packing parameterized by $k$. The same results hold for the version of the problem where the task is to find $k$ edge-disjoint cycles. Our second main result concerns Shortest Cycle Packing, which is a special case of Min-Sum Cycle Packing that asks to find a packing of $k$ shortest cycles in a graph. We prove this problem to be fixed-parameter tractable (FPT) when parameterized by $k$ on weighted planar graphs. We also obtain a polynomial kernel for the edge-disjoint variant of the problem on planar graphs. Deciding whether Min-Sum Cycle Packing is FPT on planar graphs and whether Shortest Cycle Packing is FPT on general graphs remain challenging open questions.

cs.DS↗

arXiv2024

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↗

arXiv2024

Bipartizing (Pseudo-)Disk Graphs: Approximation with a Ratio Better than 3

In a disk graph, every vertex corresponds to a disk in $\mathbb{R}^2$ and two vertices are connected by an edge whenever the two corresponding disks intersect. Disk graphs form an important class of geometric intersection graphs, which generalizes both planar graphs and unit-disk graphs. We study a fundamental optimization problem in algorithmic graph theory, Bipartization (also known as Odd Cycle Transversal), on the class of disk graphs. The goal of Bipartization is to delete a minimum number of vertices from the input graph such that the resulting graph is bipartite. A folklore (polynomial-time) $3$-approximation algorithm for Bipartization on disk graphs follows from the classical framework of Goemans and Williamson [Combinatorica'98] for cycle-hitting problems. For over two decades, this result has remained the best known approximation for the problem (in fact, even for Bipartization on unit-disk graphs). In this paper, we achieve the first improvement upon this result, by giving a $(3-\alpha)$-approximation algorithm for Bipartization on disk graphs, for some constant $\alpha>0$. Our algorithm directly generalizes to the broader class of pseudo-disk graphs. Furthermore, our algorithm is robust in the sense that it does not require a geometric realization of the input graph to be given.

cs.DS↗

arXiv2024

When far is better: The Chamberlin-Courant approach to obnoxious committee selection

Classical work on metric space based committee selection problem interprets distance as ``near is better''. In this work, motivated by real-life situations, we interpret distance as ``far is better''. Formally stated, we initiate the study of ``obnoxious'' committee scoring rules when the voters' preferences are expressed via a metric space. To this end, we propose a model where large distances imply high satisfaction and study the egalitarian avatar of the well-known Chamberlin-Courant voting rule and some of its generalizations. For a given integer value $1 \le λ\le k$, the committee size k, a voter derives satisfaction from only the $λ$-th favorite committee member; the goal is to maximize the satisfaction of the least satisfied voter. For the special case of $λ= 1$, this yields the egalitarian Chamberlin-Courant rule. In this paper, we consider general metric space and the special case of a $d$-dimensional Euclidean space. We show that when $λ$ is $1$ and $k$, the problem is polynomial-time solvable in $\mathbb{R}^2$ and general metric space, respectively. However, for $λ= k-1$, it is NP-hard even in $\mathbb{R}^2$. Thus, we have ``double-dichotomy'' in $\mathbb{R}^2$ with respect to the value of λ, where the extreme cases are solvable in polynomial time but an intermediate case is NP-hard. Furthermore, this phenomenon appears to be ``tight'' for $\mathbb{R}^2$ because the problem is NP-hard for general metric space, even for $λ=1$. Consequently, we are motivated to explore the problem in the realm of (parameterized) approximation algorithms and obtain positive results. Interestingly, we note that this generalization of Chamberlin-Courant rules encodes practical constraints that are relevant to solutions for certain facility locations.

cs.DS↗

arXiv2024

Solving Quantified Boolean Formulas with Few Existential Variables

The quantified Boolean formula (QBF) problem is an important decision problem generally viewed as the archetype for PSPACE-completeness. Many problems of central interest in AI are in general not included in NP, e.g., planning, model checking, and non-monotonic reasoning, and for such problems QBF has successfully been used as a modelling tool. However, solvers for QBF are not as advanced as state of the art SAT solvers, which has prevented QBF from becoming a universal modelling language for PSPACE-complete problems. A theoretical explanation is that QBF (as well as many other PSPACE-complete problems) lacks natural parameters} guaranteeing fixed-parameter tractability (FPT). In this paper we tackle this problem and consider a simple but overlooked parameter: the number of existentially quantified variables. This natural parameter is virtually unexplored in the literature which one might find surprising given the general scarcity of FPT algorithms for QBF. Via this parameterization we then develop a novel FPT algorithm applicable to QBF instances in conjunctive normal form (CNF) of bounded clause length. We complement this by a W[1]-hardness result for QBF in CNF of unbounded clause length as well as sharper lower bounds for the bounded arity case under the (strong) exponential-time hypothesis.

cs.CC↗

arXiv2024

Parameterized Algorithms for Minimum Sum Vertex Cover

Minimum sum vertex cover of an $n$-vertex graph $G$ is a bijection $ϕ: V(G) \to [n]$ that minimizes the cost $\sum_{\{u,v\} \in E(G)} \min \{ϕ(u), ϕ(v) \}$. Finding a minimum sum vertex cover of a graph (the MSVC problem) is NP-hard. MSVC is studied well in the realm of approximation algorithms. The best-known approximation factor in polynomial time for the problem is $16/9$ [Bansal, Batra, Farhadi, and Tetali, SODA 2021]. Recently, Stankovic [APPROX/RANDOM 2022] proved that achieving an approximation ratio better than $1.014$ for MSVC is NP-hard, assuming the Unique Games Conjecture. We study the MSVC problem from the perspective of parameterized algorithms. The parameters we consider are the size of a minimum vertex cover and the size of a minimum clique modulator of the input graph. We obtain the following results. 1. MSVC can be solved in $2^{2^{O(k)}} n^{O(1)}$ time, where $k$ is the size of a minimum vertex cover. 2. MSVC can be solved in $f(k)\cdot n^{O(1)}$ time for some computable function $f$, where $k$ is the size of a minimum clique modulator.

cs.DS↗

arXiv2023

Efficient Approximation for Subgraph-Hitting Problems in Sparse Graphs and Geometric Intersection Graphs

We investigate a fundamental vertex-deletion problem called (Induced) Subgraph Hitting: given a graph $G$ and a set $\mathcal{F}$ of forbidden graphs, the goal is to compute a minimum-sized set $S$ of vertices of $G$ such that $G-S$ does not contain any graph in $\mathcal{F}$ as an (induced) subgraph. This is a generic problem that encompasses many well-known problems that were extensively studied on their own, particularly (but not only) from the perspectives of both approximation and parameterization. In this paper, we study the approximability of the problem on a large variety of graph classes. Our first result is a linear-time $(1+\varepsilon)$-approximation reduction from (Induced) Subgraph Hitting on any graph class $\mathcal{G}$ of bounded expansion to the same problem on bounded degree graphs within $\mathcal{G}$. This directly yields linear-size $(1+\varepsilon)$-approximation lossy kernels for the problems on any bounded-expansion graph classes. Our second result is a linear-time approximation scheme for (Induced) Subgraph Hitting on any graph class $\mathcal{G}$ of polynomial expansion, based on the local-search framework of Har-Peled and Quanrud [SICOMP 2017]. This approximation scheme can be applied to a more general family of problems that aim to hit all subgraphs satisfying a certain property $π$ that is efficiently testable and has bounded diameter. Both of our results have applications to Subgraph Hitting (not induced) on wide classes of geometric intersection graphs, resulting in linear-size lossy kernels and (near-)linear time approximation schemes for the problem.

cs.DS↗

arXiv2023

On the Polynomial Kernelizations of Finding a Shortest Path with Positive Disjunctive Constraints

We study the SHORTEST PATH problem with positive disjunctive constraints from the perspective of parameterized complexity. For positive disjunctive constraints, there are certain pair of edges such that any feasible solution must contain at least one edge from every such pair. In this paper, we initiate the study of SHORTEST PATH problem subject to some positive disjunctive constraints the classical version is known to be NP-Complete. Formally, given an undirected graph G = (V, E) with a forcing graph H = (E, F) such that the vertex set of H is same as the edge set of G. The goal is to find a set S of at most k edges from G such that S forms a vertex cover in H and there is a path from s to t in the subgraph of G induced by the edge set S. In this paper, we consider two natural parameterizations for this problem. One natural parameter is the solution size, i.e. k for which we provide a kernel with O(k^5) vertices when both G and H are general graphs. Additionally, when either G or H (but not both) belongs to some special graph classes, we provied kernelization results with O(k^3) vertices . The other natural parameter we consider is structural properties of H, i.e. the size of a vertex deletion set of H to some special graph classes. We provide some fixed-parameter tractability results for those structural parameterizations.

cs.DM↗

arXiv2023

Meta-theorems for Parameterized Streaming Algorithms

The streaming model was introduced to parameterized complexity independently by Fafianie and Kratsch [MFCS14] and by Chitnis, Cormode, Hajiaghayi and Monemizadeh [SODA15]. Subsequently, it was broadened by Chitnis, Cormode, Esfandiari, Hajiaghayi and Monemizadeh [SPAA15] and by Chitnis, Cormode, Esfandiari, Hajiaghayi, McGregor, Monemizadeh and Vorotnikova [SODA16]. Despite its strong motivation, the applicability of the streaming model to central problems in parameterized complexity has remained, for almost a decade, quite limited. Indeed, due to simple $Ω(n)$-space lower bounds for many of these problems, the $k^{O(1)}\cdot {\rm polylog}(n)$-space requirement in the model is too strict. Thus, we explore {\em semi-streaming} algorithms for parameterized graph problems, and present the first systematic study of this topic. Crucially, we aim to construct succinct representations of the input on which optimal post-processing time complexity can be achieved. - We devise meta-theorems specifically designed for parameterized streaming and demonstrate their applicability by obtaining the first $k^{O(1)}\cdot n\cdot {\rm polylog}(n)$-space streaming algorithms for well-studied problems such as Feedback Vertex Set on Tournaments, Cluster Vertex Deletion, Proper Interval Vertex Deletion and Block Vertex Deletion. In the process, we demonstrate a fundamental connection between semi-streaming algorithms for recognizing graphs in a graph class H and semi-streaming algorithms for the problem of vertex deletion into H. - We present an algorithmic machinery for obtaining streaming algorithms for cut problems and exemplify this by giving the first $k^{O(1)}\cdot n\cdot {\rm polylog}(n)$-space streaming algorithms for Graph Bipartitization, Multiway Cut and Subset Feedback Vertex Set.

cs.DS↗

arXiv2022

A Framework for Approximation Schemes on Disk Graphs

We initiate a systematic study of approximation schemes for fundamental optimization problems on disk graphs, a common generalization of both planar graphs and unit-disk graphs. Our main contribution is a general framework for designing efficient polynomial-time approximation schemes (EPTASes) for vertex-deletion problems on disk graphs, which results in EPTASes for many problems including Vertex Cover, Feedback Vertex Set, Small Cycle Hitting (in particular, Triangle Hitting), $P_k$-Hitting for $k\in\{3,4,5\}$, Path Deletion, Pathwidth $1$-Deletion, Component Order Connectivity, Bounded Degree Deletion, Pseudoforest Deletion, Finite-Type Component Deletion, etc. All EPTASes obtained using our framework are robust in the sense that they do not require a realization of the input graph. To the best of our knowledge, prior to this work, the only problems known to admit (E)PTASes on disk graphs are Maximum Clique, Independent Set, Dominating set, and Vertex Cover, among which the existing PTAS [Erlebach et al., SICOMP'05] and EPTAS [Leeuwen, SWAT'06] for Vertex Cover require a realization of the input disk graph (while ours does not). The core of our framework is a reduction for a broad class of (approximation) vertex-deletion problems from (general) disk graphs to disk graphs of bounded local radius, which is a new invariant of disk graphs introduced in this work. Disk graphs of bounded local radius can be viewed as a mild generalization of planar graphs, which preserves certain nice properties of planar graphs. Specifically, we prove that disk graphs of bounded local radius admit the Excluded Grid Minor property and have locally bounded treewidth. This allows existing techniques for designing approximation schemes on planar graphs (e.g., bidimensionality and Baker's technique) to be directly applied to disk graphs of bounded local radius.

cs.CG↗

arXiv2022

Boolean and $\mathbb{F}_p$-Matrix Factorization: From Theory to Practice

Boolean Matrix Factorization (BMF) aims to find an approximation of a given binary matrix as the Boolean product of two low-rank binary matrices. Binary data is ubiquitous in many fields, and representing data by binary matrices is common in medicine, natural language processing, bioinformatics, computer graphics, among many others. Unfortunately, BMF is computationally hard and heuristic algorithms are used to compute Boolean factorizations. Very recently, the theoretical breakthrough was obtained independently by two research groups. Ban et al. (SODA 2019) and Fomin et al. (Trans. Algorithms 2020) show that BMF admits an efficient polynomial-time approximation scheme (EPTAS). However, despite the theoretical importance, the high double-exponential dependence of the running times from the rank makes these algorithms unimplementable in practice. The primary research question motivating our work is whether the theoretical advances on BMF could lead to practical algorithms. The main conceptional contribution of our work is the following. While EPTAS for BMF is a purely theoretical advance, the general approach behind these algorithms could serve as the basis in designing better heuristics. We also use this strategy to develop new algorithms for related $\mathbb{F}_p$-Matrix Factorization. Here, given a matrix $A$ over a finite field GF($p$) where $p$ is a prime, and an integer $r$, our objective is to find a matrix $B$ over the same field with GF($p$)-rank at most $r$ minimizing some norm of $A-B$. Our empirical research on synthetic and real-world data demonstrates the advantage of the new algorithms over previous works on BMF and $\mathbb{F}_p$-Matrix Factorization.

cs.LG↗

arXiv2022

Partial Vertex Cover on Graphs of Bounded Degeneracy

In the Partial Vertex Cover (PVC) problem, we are given an $n$-vertex graph $G$ and a positive integer $k$, and the objective is to find a vertex subset $S$ of size $k$ maximizing the number of edges with at least one end-point in $S$. This problem is W[1]-hard on general graphs, but admits a parameterized subexponential time algorithm with running time $2^{O(\sqrt{k})}n^{O(1)}$ on planar and apex-minor free graphs [Fomin et al. (FSTTCS 2009, IPL 2011)], and a $k^{O(k)}n^{O(1)}$ time algorithm on bounded degeneracy graphs [Amini et al. (FSTTCS 2009, JCSS 2011)]. Graphs of bounded degeneracy contain many sparse graph classes like planar graphs, $H$-minor free graphs, and bounded tree-width graphs. In this work, we prove the following results: 1) There is an algorithm for PVC with running time $2^{O(k)}n^{O(1)}$ on graphs of bounded degeneracy which is an improvement on the previous $k^{O(k)}n^{O(1)}$ time algorithm by Amini et al. 2) PVC admits a polynomial compression on graphs of bounded degeneracy, resolving an open problem posed by Amini et al.

cs.DS↗