arXiv subjects

Ignasi Sau

Publications and source records attributed to Ignasi Sau.

At least 19 recordsLinked to original sources

arXiv2026

Model Checking for Low Monodimensionality Fragments of CMSO on Topological-Minor-Free Graph Classes

Algorithmic meta-theorems explain the tractability of large classes of computational problems by linking logical expressibility with structural graph properties. While extensions of first-order logic such as FO+dp admit efficient model checking on graph classes excluding a fixed topological minor, comparable results for richer fragments of CMSO were previously unknown. We further develop the framework of Sau, Stamoulis, and Thilikos [SODA 2025] for fragmenting CMSO via annotated graph parameters, which restrict set quantification to vertex sets satisfying bounded structural conditions. Following this approach, we identify a fragment of CMSO, namely the one defined by allowing quantification only over sets having what we call low monodimensionality, that generalizes several previously-known logics and we show that model checking for this fragment, enhanced with the disjoint-paths predicate, is fixed-parameter tractable on topological-minor-free graph classes. Such classes essentially delimit the tractability for this logic on subgraph-closed classes. As a consequence, our results lift several known algorithmic meta-theorems beyond first-order logic to the topological-minor-free setting.

cs.LO↗

arXiv2026

A more versatile model for enumerative kernelization: a case study for Vertex Cover

Enumerative kernelization is a recent promising at the intersection of parameterized complexity and enumeration algorithms, with two proposed models. The first, known as enum-kernels and due to Creignou et al., was too permissive, leading to constant-sized kernels for every problem solvable with FPT-delay. To remedy this, Golovach et al. proposed the polynomial-delay enumeration kernelization model that, while addressing the shortcoming of the previous one, appears to be too strict, which we believe is a central reason for the slow development of the area. In this paper, we propose a new model for enumeration kernels, which we have called polynomial-delay (PD) kernels. It is more flexible than Golovach et al.'s kernels while still preserving their qualities; informally, it allows us to ignore ``bad'' solutions of the compressed instance when producing the solution set of the input instance, but still requires that the ``good'' solutions are lifted with polynomial-delay. After discussing the main properties of our model, we design a generic framework for vertex-subset problems to adapt decision kernels into PD kernels of the same size. We showcase our model's versatility and the framework's expressive power on the \textsc{Enum Vertex Cover} problem, where we want to list all vertex covers of size at most $k$ of a given graph. We generalize the kernelization dichotomy by Bougeret et al. about the existence of polynomial kernels for \textsc{Vertex Cover} parameterized by the vertex deletion distance to a minor-closed graph class, as well as by the solution size or feedback vertex number. The second one, in particular, is significantly simpler than the known kernel, requiring only a few lines for its lifting algorithm. Beyond our framework, we also show how to generalize to the enumeration setting the kernel of Bougeret et al. for the vertex-deletion distance to $c$-treedepth.

cs.DS↗

arXiv2025

Computing parameters that generalize interval graphs using restricted modular partitions

Recently, Lafond and Luo [MFCS 2023] defined the $\mathcal{G}$-modular cardinality of a graph $G$ as the minimum size of a partition of $V(G)$ into modules that belong to a graph class $\mathcal{G}$. We analyze the complexity of calculating parameters that generalize interval graphs when parameterized by the $\mathcal{G}$-modular cardinality, where $\mathcal{G}$ corresponds either to the class of interval graphs or to the union of complete graphs. Namely, we analyze the complexity of computing the thinness and the simultaneous interval number of a graph. We present a linear kernel for the Thinness problem parameterized by the interval-modular cardinality and an FPT algorithm for Simultaneous Interval Number when parameterized by the cluster-modular cardinality plus the solution size. The interval-modular cardinality of a graph is not greater than the cluster-modular cardinality, which in turn generalizes the neighborhood diversity and the twin-cover number. Thus, our results imply a linear kernel for Thinness when parameterized by the neighborhood diversity of the input graph, FPT algorithms for Thinness when parameterized by the twin-cover number and vertex cover number, and FPT algorithms for Simultaneous Interval Number when parameterized by the neighborhood diversity plus the solution size, twin-cover number, and vertex cover number. To the best of our knowledge, prior to our work no parameterized algorithms (FPT or XP) for computing the thinness or the simultaneous interval number were known. On the negative side, we observe that Thinness and Simultaneous Interval Number parameterized by treewidth, pathwidth, bandwidth, (linear) mim-width, clique-width, modular-width, or even the thinness or simultaneous interval number themselves, admit no polynomial kernels assuming NP $\not\subseteq$ coNP/poly.

cs.DS↗

arXiv2025

Kernelization dichotomies for hitting minors under structural parameterizations

For a finite collection of connected graphs $\mathcal{F}$, the $\mathcal{F}$-MINOR-DELETION problem consists in, given a graph $G$ and an integer $\ell$, deciding whether $G$ contains a vertex set of size at most $\ell$ whose removal results in an $\mathcal{F}$-minor-free graph. We lift the existence of (approximate) polynomial kernels for $\mathcal{F}$-MINOR-DELETION by the solution size to (approximate) polynomial kernels parameterized by the vertex-deletion distance to graphs of bounded elimination distance to $\mathcal{F}$-minor-free graphs. This results in exact polynomial kernels for every family $\mathcal{F}$ that contains a planar graph, and an approximate polynomial kernel for PLANAR VERTEX DELETION. Moreover, combining our result with a previous lower bound, we obtain the following infinite set of dichotomies, assuming $NP \not\subseteq coNP/poly$: for any finite set $\mathcal{F}$ of biconnected graphs on at least three vertices containing a planar graph, and any minor-closed class of graphs $\mathcal{C}$, $\mathcal{F}$-MINOR-DELETION admits a polynomial kernel parameterized by the vertex-deletion distance to $\mathcal{C}$ if and only if $\mathcal{C}$ has bounded elimination distance to $\mathcal{F}$-minor-free graphs. For instance, this yields dichotomies for CACTUS VERTEX DELETION, OUTERPLANAR VERTEX DELETION, and TREEWIDTH-$t$ VERTEX DELETION for every integer $t \geq 0$. Prior to our work, such dichotomies were only known for the particular cases of VERTEX COVER and FEEDBACK VERTEX SET. We also provide lower bounds on the size of the kernels.

cs.DS↗

arXiv2025

Enumeration kernels for Vertex Cover and Feedback Vertex Set

Enumerative kernelization is a recent and promising area sitting at the intersection of parameterized complexity and enumeration algorithms. Its study began with the paper of Creignou et al. [Theory Comput. Syst., 2017], and development in the area has started to accelerate with the work of Golovach et al. [J. Comput. Syst. Sci., 2022]. The latter introduced polynomial-delay enumeration kernels and applied them in the study of structural parameterizations of the \textsc{Matching Cut} problem and some variants. Few other results, mostly on \textsc{Longest Path} and some generalizations of \textsc{Matching Cut}, have also been developed. However, little success has been seen in enumeration versions of \textsc{Vertex Cover} and \textsc{Feedback Vertex Set}, some of the most studied problems in kernelization. In this paper, we address this shortcoming. Our first result is a polynomial-delay enumeration kernel with $2k$ vertices for \textsc{Enum Vertex Cover}, where we wish to list all solutions with at most $k$ vertices. This is obtained by developing a non-trivial lifting algorithm for the classical crown decomposition reduction rule, and directly improves upon the kernel with $\mathcal{O}(k^2)$ vertices derived from the work of Creignou et al. Our other result is a polynomial-delay enumeration kernel with $\mathcal{O}(k^3)$ vertices and edges for \textsc{Enum Feedback Vertex Set}; the proof is inspired by some ideas of Thomass\'e [TALG, 2010], but with a weaker bound on the kernel size due to difficulties in applying the $q$-expansion technique.

cs.DS↗

arXiv2025

Finding subdigraphs in digraphs of bounded directed treewidth

It is well known that directed treewidth does not enjoy the nice algorithmic properties of its undirected counterpart. There exist, however, some positive results that, essentially, present XP algorithms for the problem of finding, in a given digraph $D$, a subdigraph isomorphic to a digraph $H$ that can be formed by the union of $k$ directed paths (with some extra properties), parameterized by $k$ and the directed treewidth of $D$. Our motivation is to tackle the following question: Are there subdigraphs, other than the directed paths, that can be found efficiently in digraphs of bounded directed treewidth? In a nutshell, the main message of this article is that, other than the directed paths, the only digraphs that seem to behave well with respect to directed treewidth are the stars. For this, we present a number of positive and negative results, generalizing several results in the literature, as well as some directions for further research.

cs.DS↗

arXiv2025

A Parameterized Perspective on Uniquely Restricted Matchings

Given a graph G, a matching is a subset of edges of G that do not share an endpoint. A matching M is uniquely restricted if the subgraph induced by the endpoints of the edges of M has exactly one perfect matching. Given a graph G and a positive integer \ell, Uniquely Restricted Matching asks whether G has a uniquely restricted matching of size at least \ell. In this paper, we study the parameterized complexity of Uniquely Restricted Matching under various parameters. Specifically, we show that Uniquely Restricted Matching admits a fixed-parameter tractable (FPT) algorithm on line graphs when parameterized by the solution size. We also establish that the problem is FPT when parameterized by the treewidth of the input graph. Furthermore, we show that Uniquely Restricted Matching does not admit a polynomial kernel with respect to the vertex cover number plus the size of the matching unless NP \subseteq coNP/poly.

cs.DS↗

arXiv2025

Revisiting Directed Disjoint Paths on tournaments (and relatives)

In the Directed Disjoint Paths problem ($k$-DDP), we are given a digraph $k$ pairs of terminals, and the goal is to find $k$ pairwise vertex-disjoint paths connecting each pair of terminals. Bang-Jensen and Thomassen [SIAM J. Discrete Math. 1992] claimed that $k$-DDP is NP-complete on tournaments, and this result triggered a very active line of research about the complexity of the problem on tournaments and natural superclasses. We identify a flaw in their proof, which has been acknowledged by the authors, and provide a new NP-completeness proof. From an algorithmic point of view, Fomin and Pilipczuk [J. Comb. Theory B 2019] provided an FPT algorithm for the edge-disjoint version of the problem on semicomplete digraphs, and showed that their technique cannot work for the vertex-disjoint version. We overcome this obstacle by showing that the version of $k$-DDP where we allow congestion $c$ on the vertices is FPT on semicomplete digraphs provided that $c$ is greater than $k/2$. This is based on a quite elaborate irrelevant vertex argument inspired by the edge-disjoint version, and we show that our choice of $c$ is best possible for this technique, with a counterexample with no irrelevant vertices when $c \leq k/2$. We also prove that $k$-DDP on digraphs that can be partitioned into $h$ semicomplete digraphs is $W[1]$-hard parameterized by $k+h$, which shows that the XP algorithm presented by Chudnovsky, Scott, and Seymour [J. Comb. Theory B 2019] is essentially optimal.

cs.DS↗

arXiv2025

Computing distances is FPT on graph associahedra and W[2]-hard on hypergraphic polytopes

An elimination tree of a connected graph $G$ is a rooted tree on the vertices of $G$ obtained by choosing a root $v$ and recursing on the connected components of $G-v$ to obtain the subtrees of $v$. The graph associahedron of $G$ is a polytope whose vertices correspond to elimination trees of $G$ and whose edges correspond to tree rotations, a natural operation between elimination trees. These objects generalize associahedra, which correspond to the case where $G$ is a path. Ito et al. [ICALP 2023] recently proved that the problem of computing distances on graph associahedra is NP-hard. In this paper we prove that the problem, for a general graph $G$, is fixed-parameter tractable parameterized by the distance $k$. Prior to our work, only the case where $G$ is a path was known to be fixed-parameter tractable. To prove our result, we use a novel approach based on a marking scheme that restricts the search to a set of vertices whose size is bounded by a (large) function of $k$. On the negative side, we show that it is unlikely that FPT algorithms exist on a natural generalization of graph associahedra, namely hypergraphic polytopes, by proving that computing distances on them is W[2]-hard parameterized by the distance. We also prove that, on hypergraphic polytopes, the distance cannot be approximated in polynomial time within a factor $c \cdot \log(|V|+|\mathcal{E}|)$ for some constant $c > 0$ unless P = NP, where $H=(V, \mathcal{E})$ is the input hypergraph. This result strengthens the hardness result of Cardinal and Steiner [Combin. Theory 2025], who proved that the problem cannot be approximated within a factor $(1 + \varepsilon)$ for some absolute constant $\varepsilon > 0$ unless P = NP. Finally, we rule out the existence of polynomial kernels parameterized by the number of vertices of the input hypergraph, a parameter for which the problem is easily seen to be FPT.

cs.DS↗

arXiv2025

Graph modification of bounded size to minor-closed classes as fast as vertex deletion

A replacement action is a function $\mathcal{L}$ that maps each graph $H$ to a collection of graphs of size at most $|V(H)|$. Given a graph class $\mathcal{H}$, we consider a general family of graph modification problems, called $\mathcal{L}$-Replacement to $\mathcal{H}$, where the input is a graph $G$ and the question is whether it is possible to replace some induced subgraph $H_1$ of $G$ on at most $k$ vertices by a graph $H_2$ in $\mathcal{L}(H_1)$ so that the resulting graph belongs to $\mathcal{H}$. $\mathcal{L}$-Replacement to $\mathcal{H}$ can simulate many graph modification problems including vertex deletion, edge deletion/addition/edition/contraction, vertex identification, subgraph complementation, independent set deletion, (induced) matching deletion/contraction, etc. We present two algorithms. The first one solves $\mathcal{L}$-Replacement to $\mathcal{H}$ in time $2^{{\rm poly}(k)}\cdot |V(G)|^2$ for every minor-closed graph class $\mathcal{H}$, where {\rm poly} is a polynomial whose degree depends on $\mathcal{H}$, under a mild technical condition on $\mathcal{L}$. This generalizes the results of Morelle, Sau, Stamoulis, and Thilikos [ICALP 2020, ICALP 2023] for the particular case of Vertex Deletion to $\mathcal{H}$ within the same running time. Our second algorithm is an improvement of the first one when $\mathcal{H}$ is the class of graphs embeddable in a surface of Euler genus at most $g$ and runs in time $2^{\mathcal{O}(k^{9})}\cdot |V(G)|^2$, where the $\mathcal{O}(\cdot)$ notation depends on $g$. To the best of our knowledge, these are the first parameterized algorithms with a reasonable parametric dependence for such a general family of graph modification problems to minor-closed classes.

cs.DS↗

arXiv2024

Vertex identification to a forest

Let $\mathcal{H}$ be a graph class and $k\in\mathbb{N}$. We say a graph $G$ admits a \emph{$k$-identification to $\mathcal{H}$} if there is a partition $\mathcal{P}$ of some set $X\subseteq V(G)$ of size at most $k$ such that after identifying each part in $\mathcal{P}$ to a single vertex, the resulting graph belongs to $\mathcal{H}$. The graph parameter ${\sf id}_{\mathcal{H}}$ is defined so that ${\sf id}_{\mathcal{H}}(G)$ is the minimum $k$ such that $G$ admits a $k$-identification to $\mathcal{H}$, and the problem of \textsc{Identification to $\mathcal{H}$} asks, given a graph $G$ and $k\in\mathbb{N}$, whether ${\sf id}_{\mathcal{H}}(G)\le k$. If we set $\mathcal{H}$ to be the class $\mathcal{F}$ of acyclic graphs, we generate the problem \textsc{Identification to Forest}, which we show to be {\sf NP}-complete. We prove that, when parameterized by the size $k$ of the identification set, it admits a kernel of size $2k+1$. For our kernel we reveal a close relation of \textsc{Identification to Forest} with the \textsc{Vertex Cover} problem. We also study the combinatorics of the \textsf{yes}-instances of \textsc{Identification to $\mathcal{H}$}, i.e., the class $\mathcal{H}^{(k)}:=\{G\mid {\sf id}_{\mathcal{H}}(G)\le k\}$, {which we show to be minor-closed for every $k$} when $\mathcal{H}$ is minor-closed. We prove that the minor-obstructions of $\mathcal{F}^{(k)}$ are of size at most $2k+4$. We also prove that every graph $G$ such that ${\sf id}_{\mathcal{F}}(G)$ is sufficiently big contains as a minor either a cycle on $k$ vertices, or $k$ disjoint triangles, or the \emph{$k$-marguerite} graph, that is the graph obtained by $k$ disjoint triangles by identifying one vertex of each of them into the same vertex.

cs.DS↗

arXiv2024

Constant congestion linkages in polynomially strong digraphs in polynomial time

Given integers $k,c > 0$, we say that a digraph $D$ is $(k,c)$-linked if for every pair of ordered sets $\{s_1, \ldots, s_k\}$ and $\{t_1, \ldots, t_k\}$ of vertices of $D$, there are $P_1, \ldots, P_k$ such that for $i \in [k]$ each $P_i$ is a path from $s_i$ to $t_i$ and every vertex of $D$ appears in at most $c$ of those paths. Thomassen [Combinatorica, 1991] showed that for every fixed $k \geq 2$ there is no integer $p$ such that every $p$-strong digraph is $(k,1)$-linked. Edwards et al. [ESA, 2017] showed that every digraph $D$ with directed treewidth at least some function $f(k)$ contains a large bramble of congestion $2$ and that every $(36k^3 + 2k)$-strong digraph containing a bramble of congestion $2$ and size roughly $188k^3$ is $(k,2)$-linked. Since the directed treewidth of a digraph has to be at least its strong connectivity, this implies that there is a function $L(k)$ such that every $L(k)$-strong digraph is $(k,2)$-linked. This result was improved by Campos et al. [ESA, 2023], who showed that any $k$-strong digraph containing a bramble of size at least $2k(c\cdot k -c + 2) + c(k-1)$ and congestion $c$ is $(k,c)$-linked. Regarding the bramble, although the given bound on $f(k)$ is very large, Masa\v{r}\'ik et al. [SIDMA, 2022] showed that directed treewidth $\mathcal{O}(k^{48}\log^{13} k)$ suffices if the congestion is relaxed to $8$. We first show how to drop the dependence on $c$, for even $c$, on the size of the bramble that is needed in the work of Campos et al. [ESA, 2023]. Then, by making two local changes in the proof of Masa\v{r}\'ik et al. [SIDMA, 2022] we show how to build in polynomial time a bramble of size $k$ and congestion $8$ assuming that a large obstruction to directed treewidth (namely, a path system) is given. Applying these results, we show that there is a polynomial function $g(k)$ such that every $g(k)$-strong digraph is $(k,8)$-linked.

cs.DS↗

arXiv2024

On the parameterized complexity of computing good edge-labelings

A good edge-labeling (gel for short) of a graph $G$ is a function $\lambda: E(G) \to \mathbb{R}$ such that, for any ordered pair of vertices $(x, y)$ of $G$, there do not exist two distinct increasing paths from $x$ to $y$, where ``increasing'' means that the sequence of labels is non-decreasing. This notion was introduced by Bermond et al. [Theor. Comput. Sci. 2013] motivated by practical applications arising from routing and wavelength assignment problems in optical networks. Prompted by the lack of algorithmic results about the problem of deciding whether an input graph admits a gel, called GEL, we initiate its study from the viewpoint of parameterized complexity. We first introduce the natural version of GEL where one wants to use at most $c$ distinct labels, which we call $c$-GEL, and we prove that it is NP-complete for every $c \geq 2$ on very restricted instances. We then provide several positive results, starting with simple polynomial kernels for GEL and $c$-\GEL parameterized by neighborhood diversity or vertex cover. As one of our main technical contributions, we present an FPT algorithm for GEL parameterized by the size of a modulator to a forest of stars, based on a novel approach via a 2-SAT formulation which we believe to be of independent interest. We also present FPT algorithms based on dynamic programming for $c$-GEL parameterized by treewidth and $c$, and for GEL parameterized by treewidth and the maximum degree. Finally, we answer positively a question of Bermond et al. [Theor. Comput. Sci. 2013] by proving the NP-completeness of a problem strongly related to GEL, namely that of deciding whether an input graph admits a so-called UPP-orientation.

cs.DS↗

arXiv2024

Parameterizing the quantification of CMSO: model checking on minor-closed graph classes

Given a graph $G$ and a vertex set $X$, the annotated treewidth tw$(G,X)$ of $X$ in $G$ is the maximum treewidth of an $X$-rooted minor of $G$, i.e., a minor $H$ where the model of each vertex of $H$ contains some vertex of $X$. That way, tw$(G,X)$ can be seen as a measure of the contribution of $X$ to the tree-decomposability of $G$. We introduce the logic CMSO/tw as the fragment of monadic second-order logic on graphs obtained by restricting set quantification to sets of bounded annotated treewidth. We prove the following Algorithmic Meta-Theorem (AMT): for every non-trivial minor-closed graph class, model checking for CMSO/tw formulas can be done in quadratic time. Our proof works for the more general CMSO/tw+dp logic, that is CMSO/tw enhanced by disjoint-path predicates. Our AMT can be seen as an extension of Courcelle's theorem to minor-closed graph classes where the bounded-treewidth condition in the input graph is replaced by the bounded-treewidth quantification in the formulas. Our results yield, as special cases, all known AMTs whose combinatorial restriction is non-trivial minor-closedness.

cs.LO↗

arXiv2024

Redicolouring digraphs: directed treewidth and cycle-degeneracy

Given a digraph $D=(V,A)$ on $n$ vertices and a vertex $v\in V$, the cycle-degree of $v$ is the minimum size of a set $S \subseteq V(D) \setminus \{v\}$ intersecting every directed cycle of $D$ containing $v$. From this definition of cycle-degree, we define the $c$-degeneracy (or cycle-degeneracy) of $D$, which we denote by $δ^*_c(D)$. It appears to be a nice generalisation of the undirected degeneracy. In this work, using this new definition of cycle-degeneracy, we extend several evidences for Cereceda's conjecture to digraphs. The $k$-dicolouring graph of $D$, denoted by $\mathcal{D}_k(D)$, is the undirected graph whose vertices are the $k$-dicolourings of $D$ and in which two $k$-dicolourings are adjacent if they differ on the colour of exactly one vertex. We show that $\mathcal{D}_k(D)$ has diameter at most $O_{δ^*_c(D)}(n^{δ^*_c(D) + 1})$ (respectively $O(n^2)$ and $(δ^*_c(D)+1)n$) when $k$ is at least $δ^*_c(D)+2$ (respectively $\frac{3}{2}(δ^*_c(D)+1)$ and $2(δ^*_c(D)+1)$). This improves known results on digraph redicolouring (Bousquet et al.). Next, we extend a result due to Feghali to digraphs, showing that $\mathcal{D}_{d+1}(D)$ has diameter at most $O_{d,ε}(n(\log n)^{d-1})$ when $D$ has maximum average cycle-degree at most $d-ε$. We then show that two proofs of Bonamy and Bousquet for undirected graphs can be extended to digraphs. The first one uses the digrundy number of a digraph and the second one uses the $\mathscr{D}$-width. Finally, we give a general theorem which makes a connection between the recolourability of a digraph $D$ and the recolourability of its underlying graph $UG(D)$. This result directly extends a number of results on planar graph recolouring to planar digraph redicolouring.

math.CO↗

arXiv2024

Kernelization Dichotomies for Hitting Subgraphs under Structural Parameterizations

For a fixed graph $H$, the $H$-SUBGRAPH HITTING problem consists in deleting the minimum number of vertices from an input graph to obtain a graph without any occurrence of $H$ as a subgraph. This problem can be seen as a generalization of VERTEX COVER, which corresponds to the case $H = K_2$. We initiate a study of $H$-SUBGRAPH HITTING from the point of view of characterizing structural parameterizations that allow for polynomial kernels, within the recently active framework of taking as the parameter the number of vertex deletions to obtain a graph in a "simple" class $C$. Our main contribution is to identify graph parameters that, when $H$-SUBGRAPH HITTING is parameterized by the vertex-deletion distance to a class $C$ where any of these parameters is bounded, and assuming standard complexity assumptions and that $H$ is biconnected, allow us to prove the following sharp dichotomy: the problem admits a polynomial kernel if and only if $H$ is a clique. These new graph parameters are inspired by the notion of $C$-elimination distance introduced by Bulian and Dawar [Algorithmica 2016], and generalize it in two directions. Our results also apply to the version of the problem where one wants to hit $H$ as an induced subgraph, and imply in particular, that the problems of hitting minors and hitting (induced) subgraphs have a substantially different behavior with respect to the existence of polynomial kernels under structural parameterizations.

cs.DS↗

arXiv2023

On the Complexity of the Median and Closest Permutation Problems

Genome rearrangements are events where large blocks of DNA exchange places during evolution. The analysis of these events is a promising tool for understanding evolutionary genomics, providing data for phylogenetic reconstruction based on genome rearrangement measures. Many pairwise rearrangement distances have been proposed, based on finding the minimum number of rearrangement events to transform one genome into the other, using some predefined operation. When more than two genomes are considered, we have the more challenging problem of rearrangement-based phylogeny reconstruction. Given a set of genomes and a distance notion, there are at least two natural ways to define the "target" genome. On the one hand, finding a genome that minimizes the sum of the distances from this to any other, called the median genome. Finding a genome that minimizes the maximum distance to any other, called the closest genome. Considering genomes as permutations, some distance metrics have been extensively studied. We investigate median and closest problems on permutations over the metrics: breakpoint, swap, block-interchange, short-block-move, and transposition. In biological matters some values are usually small, such as the solution value d or the number k of input permutations. For each of these metrics and parameters d or k, we analyze the closest and the median problems from the viewpoint of parameterized complexity. We obtain the following results: NP-hardness for finding the median/closest permutation for some metrics, even for k = 3; Polynomial kernels for the problems of finding the median permutation of all studied metrics, considering the target distance d as parameter; NP-hardness result for finding the closest permutation by short-block-moves; FPT algorithms and infeasibility of polynomial kernels for finding the closest permutation for some metrics parameterized by the target distance d.

cs.CC↗

arXiv2023

Dynamic programming on bipartite tree decompositions

We revisit a graph width parameter that we dub bipartite treewidth (btw). Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number, and is closely related to odd-minors. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one "bipartite" vertex, while the width of such decomposition measures the number of "non-bipartite" vertices in a bag. We provide para-NP-completeness results and develop dynamic programming techniques to solve problems on graphs of small btw. In particular, we show that $K_t$-Subgraph-Cover, Weighted Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are $FPT$ parameterized by btw. We also provide the following dichotomy when $H$ is a 2-connected graph: if $H$ is bipartite, then $H$-Subgraph/Induced-Subgraph/Odd-Minor/Scattered-Packing is para-NP-complete parameterized by btw while, if $H$ is non-bipartite, then the problem is solvable in XP-time.

cs.DS↗