XP Algorithm Research Articles

XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure

AbstractIn this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna (ACM Trans Comput Theory 9:1–36, 2018), that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, Capacitated Vertex Cover and Bipartite Bandwidth.

On the Parameterized Complexity of Bend-Minimum Orthogonal Planarity

Computing planar orthogonal drawings with the minimum number of bends is one of the most studied topics in Graph Drawing. The problem is known to be NP-hard, even when we want to test the existence of a rectilinear planar drawing, i.e., an orthogonal drawing without bends (Garg and Tamassia in SIAM J Comput 31(2):601–625, 2001). From the parameterized complexity perspective, the problem is fixed-parameter tractable when parameterized by the sum of three parameters: the number b of bends, the number k of vertices of degree at most two, and the treewidth tw\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{tw}$$\\end{document} of the input graph (Di Giacomo et al. in J Comput Syst Sci 125:129–148, 2022). We improve this last result by showing that the problem remains fixed-parameter tractable when parameterized only by b+k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b+k$$\\end{document}. As a consequence, rectilinear planarity testing lies in FPT parameterized by the number of vertices of degree at most two. We also prove that our choice of parameters is minimal, as deciding if an orthogonal drawing with at most b bends exists is already NP-hard when k is zero (i.e., the problem is para-NP-hard parameterized in k); hence, there is neither an FPT nor an XP algorithm parameterized only by the parameter k (unless P = NP). In addition, we prove that the problem is W[1]-hard parameterized by k+tw\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k+\ extsf{tw}$$\\end{document}, complementing a recent result (Jansen et al. in Upward and orthogonal planarity are W[1]-hard parameterized by treewidth. CoRR, abs/2309.01264, 2023; in: Bekos MA, Chimani M (eds) Graph Drawing and Network Visualization, vol 14466, Springer, Cham, pp 203–217, 2023) that shows W[1]-hardness for the parameterization b+tw\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b+\ extsf{tw}$$\\end{document}. As a consequence, we are able to trace a clear parameterized tractability landscape for the bend-minimum orthogonal planarity problem with respect to the three parameters b, k, and tw\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{tw}$$\\end{document}.

Structure-based multitargeted docking screening, pharmacokinetics, DFT, and dynamics simulation studies reveal mitoglitazone as a potent inhibitor of cellular survival and stress response proteins of lung cancer.

Lung cancer is a disease in which lung cells grow abnormally and uncontrollably, and the cause of it is direct smoking, secondhand smoke, radon, asbestos, and certain chemicals. The worldwide leading cause of death is lung cancer, which is responsible for more than 1.8 million deaths yearly and is expected to rise to 2.2 million by 2030. The most common type of lung cancer is non-small cell lung cancer (NSCLC), which accounts for about 80% and small cell lung cancer (SCLC), which is more aggressive than NSCLC and is often diagnosed later and accounts for 20% of cases. The global concern for lung cancer demands efficient drugs with the slightest chance of developing resistance, and the idea of multitargeted drug designing came up with the solution. In this study, we have performed multitargeted molecular docking studies of Drug Bank compounds with HTVS, SP and XP algorithms followed by MM\GBSA against the four proteins of lung cancer cellular survival and stress responses, which revealed Mitoglitazone as a multitargeted inhibitor with a docking and MM\GBSA score ranging from - 5.784 to - 7.739 kcal/mol and - 25.81 to - 47.65kcal/mol, respectively. Moreover, we performed pharmacokinetics studies and QM-based DFT analysis, showing suitable candidate and interaction pattern analysis revealed the most count of interacting residues was 4GLY, 5PHE, 6ASP, 6GLU, 6LYS, and 6THR. Further, the results were validated with SPC water model-based MD simulation for 100ns in neutralised condition, showing the cumulative deviation and fluctuation < 2Å with many intermolecular interactions. The whole analysis has suggested that Mitoglitazone can be used as a multitargeted inhibitor against lung cancer-however, experimental studies are needed before human use.

Mitoxantrone 2HCl’s adroit activity against cervical cancer replication and maintenance proteins: a multitargeted approach

Cervical cancer poses a significant global health challenge, ranking as the fourth most common cancer among women worldwide and resulting in approximately 300,000 deaths yearly, predominantly caused by high-risk human papillomavirus strains (HPV), mainly types 16 and 18. The scenario poses the urgent need of the hour to develop effective treatment strategies that can address the complexity of cervical cancer and multitargeted inhibitor designing that holds promise as it can simultaneously target multiple proteins and pathways involved in its progression and have the potential to enhance treatment efficacy, reduce the likelihood of drug resistance. In this study, we have performed multitargeted molecular docking of FDA-approved drugs against cervical cancer replication and maintenance proteins- Xenopus kinesin-like protein-2 (3KND), cell division cycle protein-20 (4N14), MCM2-histone complex (4UUZ) and MCM6 Minichromosome maintenance (2KLQ) with HTVS, SP and XP algorithms and have obtained the docking and MM\\GBSA score ranging from −8.492 to −5.189 Kcal/mol and −58.16 to −39.07 Kcal/mol. Further, the molecular interaction fingerprints identified ALA, THR, SER, ASN, LEU, and ILE were among the most interacted residues, leaning towards hydrophobic and polar amino acids. The pharmacokinetics and DFT of the compound have shown promising results. The complexes were simulated for 100 ns to study the stability by computing the deviation, fluctuations, and intermolecular interactions formed during the simulation. This study produced promising results, satisfying the criteria that Mitoxantrone 2HCl can be a multitargeted inhibitor against cervical cancer proteins—however, experimental validation is a must before human use. Communicated by Ramaswamy H. Sarma

Unveiling Novel Urease Inhibitors for Helicobacter pylori: A Multi-Methodological Approach from Virtual Screening and ADME to Molecular Dynamics Simulations.

Helicobacter pylori (Hp) infections pose a global health challenge demanding innovative therapeutic strategies by which to eradicate them. Urease, a key Hp virulence factor hydrolyzes urea, facilitating bacterial survival in the acidic gastric environment. In this study, a multi-methodological approach combining pharmacophore- and structure-based virtual screening, molecular dynamics simulations, and MM-GBSA calculations was employed to identify novel inhibitors for Hp urease (HpU). A refined dataset of 8,271,505 small molecules from the ZINC15 database underwent pharmacokinetic and physicochemical filtering, resulting in 16% of compounds for pharmacophore-based virtual screening. Molecular docking simulations were performed in successive stages, utilizing HTVS, SP, and XP algorithms. Subsequent energetic re-scoring with MM-GBSA identified promising candidates interacting with distinct urease variants. Lys219, a residue critical for urea catalysis at the urease binding site, can manifest in two forms, neutral (LYN) or carbamylated (KCX). Notably, the evaluated molecules demonstrated different interaction and energetic patterns in both protein variants. Further evaluation through ADMET predictions highlighted compounds with favorable pharmacological profiles, leading to the identification of 15 candidates. Molecular dynamics simulations revealed comparable structural stability to the control DJM, with candidates 5, 8 and 12 (CA5, CA8, and CA12, respectively) exhibiting the lowest binding free energies. These inhibitors suggest a chelating capacity that is crucial for urease inhibition. The analysis underscores the potential of CA5, CA8, and CA12 as novel HpU inhibitors. Finally, we compare our candidates with the chemical space of urease inhibitors finding physicochemical similarities with potent agents such as thiourea.

Delineated 3-1-BenCarMethInYlPro-Phosphonic Acid's Adroit Activity against Lung Cancer through Multitargeted Docking, MM\GBSA, QM-DFT and Multiscale Simulations.

Lung cancer is a pervasive and challenging disease with limited treatment options, with global health challenges often present with complex molecular profiles necessitating the exploration of innovative therapeutic strategies. Single-target drugs have shown limited success due to the heterogeneity of this disease. Multitargeted drug designing is imperative to combat this complexity by simultaneously targeting multiple target proteins and pathways, which can enhance treatment efficacy and overcome resistance by addressing the dynamic nature of the disease and stopping tumour growth and spread. In this study, we performed the molecular docking studies of Drug Bank compounds with a multitargeted approach against crucial proteins of lung cancer such as heat shock protein 5 (BIP/GRP78) ATPase, myosin 9B RhoGAP, EYA2 phosphatase inhibitor, RSK4 N-terminal kinase, and collapsin response mediator protein-1 (CRMP-1) using HTVS, SP with XP algorithms, and poses were filtered using MM\GBSA which identified [3-(1-Benzyl-3-Carbamoylmethyl-2-Methyl-1h-Indol-5-Yloxy)-Propyl-]-Phosphonic Acid (3-1-BenCarMethIn YlPro-Phosphonic Acid) (DB02504) as multitargeted drug candidate with docking and MM\GBSA score ranges from -5.83 to -10.66 and -7.56 to -50.14 Kcal/mol, respectively. Further, the pharmacokinetic and QM-based DFT studies have shown complete acceptance results, and interaction fingerprinting reveals that ILE, GLY, VAL, TYR, LEU, and GLN were among the most interacting residues. The 100 ns MD simulation in the SPC water model with NPT ensemble showed stable performance with deviation and fluctuations <2 Å with huge interactions, making it a promising multitargeted drug candidate; however, experimental studies are needed before use.

Unrevealing the multitargeted potency of 3-1-BCMIYPPA against lung cancer structural maintenance and suppression proteins through pharmacokinetics, QM-DFT, and multiscale MD simulation studies.

Lung cancer, a relentless and challenging disease, demands unwavering attention in drug design research. Single-target drugs have yielded limited success, unable to effectively address this malignancy's profound heterogeneity and often developed resistance. Consequently, the clarion call for lung cancer drug design echoes louder than ever, and multitargeted drug design emerges as an imperative approach in this landscape, which is done by concurrently targeting multiple proteins and pathways and offering a beacon of hope. This study is focused on the multitargeted drug designing approach by identifying drug candidates against human cyclin-dependent kinase-2, SRC-2 domains of C-ABL, epidermal growth factor and receptor extracellular domains, and insulin-like growth factor-1 receptor kinase. We performed the multitargeted molecular docking studies of Drug Bank compounds using HTVS, SP and XP algorithms and poses filter with MM\GBSA against all proteins and identified DB02504, namely [3-(1-Benzyl-3-Carbamoylmethyl-2-Methyl-1h-Indol-5-Yloxy)-Propyl-]-Phosphonic Acid (3-1-BCMIYPPA) as multitargeted lead with docking and MM\GBSA score range from -8.242 to -6.274 and -28.2 and -44.29 Kcal/mol, respectively. Further, the QikProp-based pharmacokinetic computations and QM-based DFT showed acceptance results against standard values, and interaction fingerprinting reveals that THR, MET, GLY, VAL, LEU, GLU and ASP were among the most interacting residues. The NPT ensemble-based 100ns MD simulation in a neutralised state with an SPC water model has also shown a stable performance and produced deviation and fluctuations <2Å with huge interactions, making it a promising multitargeted drug candidate-however, experimental studies are suggested.

FPT algorithms for packing [formula omitted]-safe spanning rooted sub(di)graphs

We study three problems introduced by Bang-Jensen and Yeo (2015) and by Bang-Jensen et al. (2016) about finding disjoint “balanced” spanning rooted substructures in graphs and digraphs, which generalize classic packing problems such as detecting the existence of multiple arc-disjoint spanning arborescences. Namely, given a positive integer k, a digraph D=(V,A), and a root r∈V, we first consider the problem of finding two arc-disjoint k-safe spanning r-arborescences, meaning arborescences rooted at a vertex r such that deleting any arc rv and every vertex in the sub-arborescence rooted at v leaves at least k vertices. Then, we consider the problem of finding two arc-disjoint (r,k)-flow branchings meaning arc sets admitting a flow that distributes one unit from r to every other vertex while respecting a capacity limit of n−k on every arc. We show that both these problems are FPT with parameter k, improving on existing XP algorithms. The latter of these results answers a question of Bang-Jensen et al. (2016). Further, given a positive integer k, a graph G=(V,E), and r∈V, we consider the problem of finding two edge-disjoint (r,k)-safe spanning trees meaning spanning trees such that the component containing r has size at least k when deleting any vertex different from r. We show that this problem is also FPT with parameter k, again improving on a previous XP algorithm. Our main technical contribution is to prove that the existence of such spanning substructures is equivalent to the existence of substructures with size and maximum (out-)degree both bounded by a (linear or quadratic) function of k, which may be of independent interest.

Multitargeted inhibitory effect of Mitoxantrone 2HCl on cervical cancer cell cycle regulatory proteins: a multitargeted docking-based MM\GBSA and MD simulation study.

Cervical cancer remains a significant global health concern that starts in the cervix, the lower part of the uterus that connects to the vagina and is caused by the human papillomavirus (HPV), necessitating the development of effective multitargeted effective and resistance-proof therapies. In early-stage cervical cancer may not show any symptoms, however, as the cancer progresses, some people may experience- abnormal vaginal bleeding, watery or bloody vaginal discharge, pain in the pelvis or lower back, pain during sex, and frequent and painful urination. In this study, we screened the complete FDA-approved drug library using a multitargeted inhibitory approach against four cervical cancer proteins, namely mitotic arrest deficient -2, DNA polymerase epsilon B-subunit, benzimidazole-related -1, and threonine-protein kinase-1 which crucially plays its role for the in its development process. We employed the HTVS, SP and XP algorithms for efficient filtering and screening that helped to identify Mitoxantrone 2HCl against all of them with docking and MM\GBSA scores ranging from -11.63 to -7.802kcal/mol and -74.38 to -47.73kcal/mol, respectively. We also evaluated the interaction patterns of each complex and the pharmacokinetics properties that helped gain insight into interactions. Subsequently, we performed multiscale MD simulations for 100ns to understand the dynamic behaviour and stability of the Mitoxantrone 2HCl -protein complexes that revealed the formation of stable drug-protein complexes and provided insights into the molecular interactions that contribute to Mitoxantrone's inhibitory effects on these proteins and can be a better drug for cervical cancer. However, experimental studies of these findings could pave the way for therapies to combat cervical cancer effectively.

Unveiling the multitargeted repurposing potential of taxifolin (dihydroquercetin) in cervical cancer: an extensive MM\GBSA-based screening, and MD simulation study.

Cervical cancer is a significant cause of morbidity and mortality in women worldwide. Despite the availability of effective therapies, the development of drug resistance and adverse side effects remain significant challenges in cervical cancer treatment. Thus, repurposing existing drugs as multitargeted therapies for cervical cancer is an attractive approach. In this study, we extensively screened the complete prepared FDA-approved drugs and identified the repurposing potential of taxifolin, a flavonoid with known antioxidant and anti-inflammatory properties, as a multitargeted therapy for cervical cancer. We performed a computational analysis using molecular docking with various sampling algorithms, namely HTVS, SP, and XP algorithms, for robust sampling pose and filtered with MM/GBSA analysis to determine the binding affinity of taxifolin with potential targets involved in cervical cancer, such as Symmetric Mad2 Dimer, replication initiation factor MCM10-ID, TPX2, DNA polymerase epsilon B-subunit, human TBK1, and alpha-v beta-8. We then conducted MD simulations to investigate the stability and conformational changes of the complex formed between taxifolin and the mentioned proteins. Our results suggest that taxifolin has a high binding affinity ranging from -6.094 to -9.558kcal/mol, indicating its potential as a multitargeted therapy for cervical cancer. Furthermore, interaction fingerprints, pharmacokinetics and MD simulations revealed that the Taxifolin-target complexes remained stable over the simulation period, indicating that taxifolin may bind to the targets for an extended period. Our study suggests that taxifolin has the potential as a multitargeted therapy for cervical cancer, and further experimental studies are necessary to validate our findings.

Hardness of the generalized coloring numbers

The generalized coloring numbers of Kierstead and Yang (Order 2003) offer an algorithmically-useful characterization of graph classes with bounded expansion. In this work, we consider the hardness and approximability of these parameters. First, we complete the work of Grohe et al. (WG 2015) by showing that computing the weak 2-coloring number is NP-hard. Our approach further establishes that determining if a graph has weak r-coloring number at most k is para-NP-hard when parameterized by k for all r≥2. We adapt this to determining if a graph has r-coloring number at most k as well, proving para-NP-hardness for all r≥2. Para-NP-hardness implies that no XP algorithm (runtime O(nf(k))) exists for testing if a generalized coloring number is at most k. Moreover, there exists a constant c such that it is NP-hard to approximate the generalized coloring numbers within a factor of c. To complement these results, we give an approximation algorithm for the generalized coloring numbers, improving both the runtime and approximation factor of the existing approach of Dvořák (EuJC 2013). We prove that greedily ordering vertices with small estimated backconnectivity achieves a (k−1)r−1-approximation for the r-coloring number and an O(kr−1)-approximation for the weak r-coloring number.

Reducing the vertex cover number via edge contractions

Given a graph G on n vertices and two integers k and d, the Contraction(vc) problem asks whether one can contract at most k edges to reduce the vertex cover number of G by at least d. Recently, Lima et al. [JCSS 2021] proved that Contraction(vc) admits an XP algorithm running in time f(d)⋅nO(d). They asked whether this problem is FPT under this parameterization. In this article, we prove that: (i) Contraction(vc) is W[1]-hard parameterized by k+d. Moreover, unless the ETH fails, the problem does not admit an algorithm running in time f(k+d)⋅no(k+d) for any function f. This answers negatively the open question stated in Lima et al. [JCSS 2021]. (ii) Contraction(vc) is NP-hard even when k=d. (iii) Contraction(vc) can be solved in time 2O(d)⋅nk−d+O(1). This improves the algorithm of Lima et al. [JCSS 2021], and shows that when k=d, Contraction(vc) is FPT parameterized by d (or by k).

Disconnected matchings

In 2005, Goddard et al. (2005) [12] asked the computational complexity of determining the maximum cardinality of a matching whose vertex set induces a disconnected graph. In this paper, we answer this question. In fact, we consider the generalized problem of finding c-disconnected matchings; such matchings are ones whose vertex sets induce subgraphs with at least c connected components. We show that, for every fixed c≥2, this problem is NP-complete even if we restrict the input to bounded diameter bipartite graphs, while can be solved in polynomial time if c=1. For the case when c is part of the input, we show that the problem is NP-complete for chordal graphs, while being solvable in polynomial time for interval graphs. Finally, we explore the parameterized complexity of the problem. We present an FPT algorithm under the treewidth parameterization, and an XP algorithm for graphs with a polynomial number of minimal separators when parameterized by c. We complement these results by showing that, unless NP⊆coNP/poly, the related Induced Matching problem does not admit a polynomial kernel when parameterized by vertex cover and size of the matching nor when parameterized by vertex deletion distance to clique and size of the matching. As for Connected Matching, we show how to obtain a maximum connected matching in linear time given an arbitrary maximum matching in the input.

Group Activity Selection with Few Agent Types

In this paper we establish the complexity map for the Group Activity Selection Problem (GASP), along with two of its prominent variants called sGASP and gGASP, focusing on the case when the number of types of agents is the parameter. In all these problems, one is given a set of agents (each with their own preferences) and a set of activities, and the aim is to assign agents to activities in a way which satisfies certain global as well as preference-based conditions. Our positive results, consisting of one fixed-parameter algorithm and one XP algorithm, rely on a combination of novel Subset Sum machinery (which may be of general interest) and identifying certain compression steps that allow us to focus on solutions with a simpler, well-defined structure (in particular, they are “acyclic”). These algorithms are complemented by matching lower bounds, which among others close a gap to a recently obtained tractability result of Gupta et al. (in: Algorithmic game theory—10th international symposium, SAGT 2017, vol 10504 of lecture notes in computer science, Springer, 2017). In this direction, the techniques used to establish W[1]-hardness of sGASP are of particular interest: as an intermediate step, we use Sidon sequences to show the W[1]-hardness of a highly restricted variant of multi-dimensional Subset Sum, which may find applications in other settings as well.

Adapting the Directed Grid Theorem into an FPT Algorithm

The Grid Theorem of Robertson and Seymour [JCTB, 1986], is one of the most important tools in the field of structural graph theory, finding numerous applications in the design of algorithms for undirected graphs. An analogous version of the Grid Theorem in digraphs was conjectured by Johnson et al. [JCTB, 2001], and proved by Kawarabayashi and Kreutzer [STOC, 2015]. Namely, they showed that there is a function $f(k)$ such that every digraph of directed tree-width at least $f(k)$ contains a cylindrical grid of size $k$ as a butterfly minor and stated that their proof can be turned into an XP algorithm, with parameter $k$, that either constructs a decomposition of the appropriate width, or finds the claimed large cylindrical grid as a butterfly minor. In this paper, we adapt some of the steps of the proof of Kawarabayashi and Kreutzer to improve this XP algorithm into an FPT algorithm. Towards this, our main technical contributions are two FPT algorithms with parameter $k$. The first one either produces an arboreal decomposition of width $3k-2$ or finds a haven of order $k$ in a digraph $D$, improving on the original result for arboreal decompositions by Johnson et al. The second algorithm finds a well-linked set of order $k$ in a digraph $D$ of large directed tree-width. As tools to prove these results, we show how to solve a generalized version of the problem of finding balanced separators for a given set of vertices $T$ in FPT time with parameter $|T|$, a result that we consider to be of its own interest.

Controlling the Spread of Two Secrets in Diverse Social Networks (Student Abstract)

Information diffusion in social networks is a well-studied concept in social choice theory. We propose the study of the diffusion of two secrets in a heterogeneous environment from the complexity perspective, that is, there are two different networks with the same set of agents (e.g., the structure of the set of followers might be different in two distinct social networks). Formally, our model combines two group identification processes for which we do have independent desiderata---either constructive, where we would like a given group of agents to be exposed to a secret, or destructive, where a given group of agents should not be exposed to a secret. To be able to reach these targets, we can either delete an agent or introduce a previously latent agent. Our results are mostly negative---all of the problems are NP-hard. Therefore, we propose a parameterized study with respect to the natural parameters, the number of influenced agents, the size of the required/protected agent sets, and the duration of the diffusion process. Most of the studied problems remain W[1]-hard even for a combination of these parameters. We complement these results with nearly optimal XP algorithms.

Spy game: FPT-algorithm, hardness and graph products

In the (s,d)-spy game over a graph G, k guards and one spy occupy some vertices of G and, at each turn, the spy may move with speed s (along at most s edges) and each guard may move along one edge. The spy and the guards may occupy the same vertices. The spy wins if she reaches a vertex at distance more than the surveilling distance d from every guard. This game was introduced by Cohen et al. (2016) [15] and is related to two well-studied games: Cops and Robber game and Eternal Dominating game. The guard number gns,d(G) is the minimum number of guards such that the guards have a winning strategy (of controlling the spy) in the graph G. In 2018, it was proved that deciding if the spy has a winning strategy is NP-hard for every speed s≥2 and distance d≥0. In this paper, we initiate the investigation of the guard number in grids and in graph products. We obtain a strict upper bound on the strong product of two general graphs and obtain examples with King grids that match this bound and other examples for which the guard number is smaller. We also obtain the exact value of the guard number in the lexicographical product of two general graphs for any distance d≥2. From the algorithmic point of view, we prove a positive result: if the number k of guards is fixed, the spy game is solvable in polynomial XP time O(n3k+2) for every speed s≥2 and distance d≥0. In other words, the spy game is XP when parameterized by the number of guards. This XP algorithm is used to obtain an FPT algorithm on the P4-fewness of the graph. As a negative result, we prove that the spy game is W[2]-hard even in bipartite graphs when parameterized by the number of guards, for every speed s≥2 and distance d≥0, extending the hardness result of Cohen et al. in 2018.

Node Multiway Cut and Subset Feedback Vertex Set on Graphs of Bounded Mim-Width

The two weighted graph problems Node Multiway Cut (NMC) and Subset Feedback Vertex Set (SFVS) both ask for a vertex set of minimum total weight, that for NMC disconnects a given set of terminals, and for SFVS intersects all cycles containing a vertex of a given set. We design a meta-algorithm that allows to solve both problems in time 2^{O(rw^3)}cdot n^{4}, 2^{O(q^2log (q))}cdot n^{4}, and n^{O(k^2)} where rw is the rank-width, q the {mathbb {Q}}-rank-width, and k the mim-width of a given decomposition. This answers in the affirmative an open question raised by Jaffke et al. (Algorithmica 82(1):118–145, 2020) concerning an XP algorithm for SFVS parameterized by mim-width. By a unified algorithm, this solves both problems in polynomial-time on the following graph classes: Interval, Permutation, and Bi-Interval graphs, Circular Arc and Circular Permutation graphs, Convex graphs, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k; and also on Leaf Power graphs if a leaf root is given as input, on H-Graphs for fixed H if an H-representation is given as input, and on arbitrary powers of graphs in all the above classes. Prior to our results, only SFVS was known to be tractable restricted only on Interval and Permutation graphs, whereas all other results are new.

Length-bounded cuts: Proper interval graphs and structural parameters

We study the Length-Bounded Cut problem for special graph classes and from a parameterized complexity viewpoint. Here, we are given a graph G, two vertices s and t, and positive integers β and λ. The task is to find a set F of at most β edges such that each s-t-path of length at most λ in G contains some edge in F. Bazgan et al. [20] conjectured that Length-Bounded Cut admits a polynomial-time algorithm if the input graph is a proper interval graph. We confirm this conjecture by providing a dynamic-programming-based polynomial-time algorithm. Moreover, we strengthen the W[1]-hardness result of Dvořák and Knop [15] for Length-Bounded Cut parameterized by pathwidth by showing W[1]-hardness for the combined parameter pathwidth and maximum degree of the input graph. Finally, we prove that Length-Bounded Cut is W[1]-hard for the feedback vertex number. Both our hardness results complement known XP algorithms.

C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that (1) the subgraph induced by each cluster is drawn in the interior of the corresponding disk, (2) each edge intersects any disk at most once, and (3) the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA’95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA’20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT (resp., XP) algorithm for embedded flat (resp., non-flat) clustered graphs, when parameterized by the carving-width of the dual graph of the input. These are the first FPT and XP algorithms for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. In particular, our algorithm runs in quadratic time for flat instances of bounded treewidth and bounded face size. To further strengthen the relevance of this result, we show that an algorithm with running time O(r(n)) for flat instances whose underlying graph has pathwidth 1 would result in an algorithm with running time O(r(n)) for flat instances and with running time O(r(n^2) + n^2) for general, possibly non-flat, instances.