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Paloma T. Lima

Publications and source records attributed to Paloma T. Lima.

At least 19 recordsLinked to original sources
arXiv2025

Maximum list $r$-colorable induced subgraphs in $kP_3$-free graphs

We show that, for every fixed positive integers $r$ and $k$, \textsc{Max-Weight List $r$-Colorable Induced Subgraph} admits a polynomial-time algorithm on $kP_3$-free graphs. This problem is a common generalization of \textsc{Max-Weight Independent Set}, \textsc{Odd Cycle Transversal} and \textsc{List $r$-Coloring}, among others. Our result has several consequences. First, it implies that, for every fixed $r \geq 5$, assuming $\mathsf{P}\neq \mathsf{NP}$, \textsc{Max-Weight List $r$-Colorable Induced Subgraph} is polynomial-time solvable on $H$-free graphs if and only if $H$ is an induced subgraph of either $kP_3$ or $P_5+kP_1$, for some $k \geq 1$. Second, it makes considerable progress toward a complexity dichotomy for \textsc{Odd Cycle Transversal} on $H$-free graphs, allowing to answer a question of Agrawal, Lima, Lokshtanov, Rz{ą}{ż}ewski, Saurabh, and Sharma [TALG 2024]. Third, it gives a short and self-contained proof of the known result of Chudnovsky, Hajebi, and Spirkl [Combinatorica 2024] that \textsc{List $r$-Coloring} on $kP_3$-free graphs is polynomial-time solvable for every fixed $r$ and $k$. We also consider two natural distance-$d$ generalizations of \textsc{Max-Weight Independent Set} and \textsc{List $r$-Coloring} and provide polynomial-time algorithms on $kP_3$-free graphs for every fixed integers $r$, $k$, and $d \geq 6$.

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arXiv2024

Non-empty intersection of longest paths in $P_5$-free and claw-free graphs

A family $\mathcal{F}$ of graphs is a \textit{Gallai family} if for every connected graph $G\in \mathcal{F}$, all longest paths in $G$ have a common vertex. While it is not known whether $P_5$-free graphs are a Gallai family, Long Jr., Milans, and Munaro [The Electronic Journal of Combinatorics, 2023] showed that this is \emph{not} the case for the class of claw-free graphs. We give a complete characterization of the graphs $H$ of size at most five for which $(\text{claw}, H)$-free graphs form a Gallai family. We also show that $(P_5, H)$-free graphs form a Gallai family if $H$ is a triangle, a paw, or a diamond. Both of our results are constructive.

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arXiv2024

Tree decompositions meet induced matchings: beyond Max Weight Independent Set

For a tree decomposition $\mathcal{T}$ of a graph $G$, by $μ(\mathcal{T})$ we denote the size of a largest induced matching in $G$ all of whose edges intersect one bag of $\mathcal{T}$. Induced matching treewidth of a graph $G$ is the minimum value of $μ(\mathcal{T})$ over all tree decompositions $\mathcal{T}$ of $G$. Yolov [SODA 2018] proved that Max Weight Independent Set can be solved in polynomial time for graphs of bounded induced matching treewidth. In this paper we explore what other problems are tractable in such classes of graphs. As our main result, we give a polynomial-time algorithm for Min Weight Feedback Vertex Set. We also provide some positive results concerning packing induced subgraphs, which in particular imply a PTAS for the problem of finding a largest induced subgraph of bounded treewidth. These results suggest that in graphs of bounded induced matching treewidth, one could find in polynomial time a maximum-weight induced subgraph of bounded treewidth satisfying a given CMSO$_2$ formula. We conjecture that such a result indeed holds and prove it for graphs of bounded tree-independence number, which form a rich and important family of subclasses of graphs of bounded induced matching treewidth. We complement these algorithmic results with a number of complexity and structural results concerning induced matching treewidth.

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arXiv2024

Odd Cycle Transversal on $P_5$-free Graphs in Polynomial Time

An independent set in a graph G is a set of pairwise non-adjacent vertices. A graph $G$ is bipartite if its vertex set can be partitioned into two independent sets. In the Odd Cycle Transversal problem, the input is a graph $G$ along with a weight function $w$ associating a rational weight with each vertex, and the task is to find a smallest weight vertex subset $S$ in $G$ such that $G - S$ is bipartite; the weight of $S$, $w(S) = \sum_{v\in S} w(v)$. We show that Odd Cycle Transversal is polynomial-time solvable on graphs excluding $P_5$ (a path on five vertices) as an induced subgraph. The problem was previously known to be polynomial-time solvable on $P_4$-free graphs and NP-hard on $P_6$-free graphs [Dabrowski, Feghali, Johnson, Paesani, Paulusma and Rzążewski, Algorithmica 2020]. Bonamy, Dabrowski, Feghali, Johnson and Paulusma [Algorithmica 2019] posed the existence of a polynomial-time algorithm on $P_5$-free graphs as an open problem, this was later re-stated by Rzążewski [Dagstuhl Reports, 9(6): 2019] and by Chudnovsky, King, Pilipczuk, Rzążewski, and Spirkl [SIDMA 2021], who gave an algorithm with running time $n^{O(\sqrt{n})}$.

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arXiv2023

Treewidth is NP-Complete on Cubic Graphs (and related results)

In this paper, we give a very simple proof that Treewidth is NP-complete; this proof also shows NP-completeness on the class of co-bipartite graphs. We then improve the result by Bodlaender and Thilikos from 1997 that Treewidth is NP-complete on graphs with maximum degree at most 9, by showing that Treewidth is NP-complete on cubic graphs.

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arXiv2022

$b$-Coloring Parameterized by Pathwidth is XNLP-complete

We show that the $b$-Coloring problem is complete for the class XNLP when parameterized by the pathwidth of the input graph. Besides determining the precise parameterized complexity of this problem, this implies that b-Coloring parameterized by pathwidth is $W[t]$-hard for all $t$, and resolves the parameterized complexity of $b$-Coloring parameterized by treewidth.

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arXiv2022

b-Coloring Parameterized by Clique-Width

We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This unifies and extends nearly all previously known polynomial time results on graph classes, and answers open questions posed by Campos and Silva [Algorithmica, 2018] and Bonomo et al. [Graphs Combin., 2009]. This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is FPT when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound. The running times of the clique-width based algorithms for b-Coloring and Fall Coloring are tight under the Exponential Time Hypothesis.

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arXiv2022

Fixed-parameter tractability of Directed Multicut with three terminal pairs parameterized by the size of the cutset: twin-width meets flow-augmentation

We show fixed-parameter tractability of the Directed Multicut problem with three terminal pairs (with a randomized algorithm). This problem, given a directed graph $G$, pairs of vertices (called terminals) $(s_1,t_1)$, $(s_2,t_2)$, and $(s_3,t_3)$, and an integer $k$, asks to find a set of at most $k$ non-terminal vertices in $G$ that intersect all $s_1t_1$-paths, all $s_2t_2$-paths, and all $s_3t_3$-paths. The parameterized complexity of this case has been open since Chitnis, Cygan, Hajiaghayi, and Marx proved fixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, and Pilipczuk and Wahlström proved the W[1]-hardness of the 4-terminal-pairs case at SODA 2016. On the technical side, we use two recent developments in parameterized algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch, Pilipczuk, Wahlström, STOC 2022] we cast the problem as a CSP problem with few variables and constraints over a large ordered domain.We observe that this problem can be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. We look at this problem through the lenses of twin-width, a recently introduced structural parameter [Bonnet, Kim, Thomassé, Watrigant, FOCS 2020]: By a recent characterization [Bonnet, Giocanti, Ossona de Mendes, Simon, Thomassé, Toruńczyk, STOC 2022] the said FO model-checking task can be done in FPT time if the said matrices have bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If any of the matrices in the said encoding has a large grid minor, a vertex corresponding to the ``middle'' box in the grid minor can be proclaimed irrelevant -- not contained in the sought solution -- and thus reduced.

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arXiv2022

A tight quasi-polynomial bound for Global Label Min-Cut

We study a generalization of the classic Global Min-Cut problem, called Global Label Min-Cut (or sometimes Global Hedge Min-Cut): the edges of the input (multi)graph are labeled (or partitioned into color classes or hedges), and removing all edges of the same label (color or from the same hedge) costs one. The problem asks to disconnect the graph at minimum cost. While the $st$-cut version of the problem is known to be NP-hard, the above global cut version is known to admit a quasi-polynomial randomized $n^{O(\log \mathrm{OPT})}$-time algorithm due to Ghaffari, Karger, and Panigrahi [SODA 2017]. They consider this as ``strong evidence that this problem is in P''. We show that this is actually not the case. We complete the study of the complexity of the Global Label Min-Cut problem by showing that the quasi-polynomial running time is probably optimal: We show that the existence of an algorithm with running time $(np)^{o(\log n/ (\log \log n)^2)}$ would contradict the Exponential Time Hypothesis, where $n$ is the number of vertices, and $p$ is the number of labels in the input. The key step for the lower bound is a proof that Global Label Min-Cut is W[1]-hard when parameterized by the number of uncut labels. In other words, the problem is difficult in the regime where almost all labels need to be cut to disconnect the graph. To turn this lower bound into a quasi-polynomial-time lower bound, we also needed to revisit the framework due to Marx [Theory Comput. 2010] of proving lower bounds assuming Exponential Time Hypothesis through the Subgraph Isomorphism problem parameterized by the number of edges of the pattern. Here, we provide an alternative simplified proof of the hardness of this problem that is more versatile with respect to the choice of the regimes of the parameters.

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arXiv2022

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

In 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 [ToCT 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, and Bipartite Bandwidth.

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arXiv2022

Taming graphs with no large creatures and skinny ladders

We confirm a conjecture of Gartland and Lokshtanov [arXiv:2007.08761]: if for a hereditary graph class $\mathcal{G}$ there exists a constant $k$ such that no member of $\mathcal{G}$ contains a $k$-creature as an induced subgraph or a $k$-skinny-ladder as an induced minor, then there exists a polynomial $p$ such that every $G \in \mathcal{G}$ contains at most $p(|V(G)|)$ minimal separators. By a result of Fomin, Todinca, and Villanger [SIAM J. Comput. 2015] the latter entails the existence of polynomial-time algorithms for Maximum Weight Independent Set, Feedback Vertex Set and many other problems, when restricted to an input graph from $\mathcal{G}$. Furthermore, as shown by Gartland and Lokshtanov, our result implies a full dichotomy of hereditary graph classes defined by a finite set of forbidden induced subgraphs into tame (admitting a polynomial bound of the number of minimal separators) and feral (containing infinitely many graphs with exponential number of minimal separators).

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arXiv2022

Reducing the Vertex Cover Number via Edge Contractions

The CONTRACTION(vc) problem takes as input a graph $G$ on $n$ vertices and two integers $k$ and $d$, and asks whether one can contract at most $k$ edges to reduce the size of a minimum vertex cover of $G$ by at least $d$. Recently, Lima et al. [JCSS 2021] proved, among other results, that unlike most of the so-called blocker problems, CONTRACTION(vc) admits an XP algorithm running in time $f(d) \cdot n^{O(d)}$. They left open the question of whether this problem is FPT under this parameterization. In this article, we continue this line of research and prove the following results: 1. 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 $f(k + d) \cdot n^{o(k + d)}$ for any function $f$. In particular, this answers the open question stated in Lima et al. [JCSS 2021] in the negative. 2. It is NP-hard to decide whether an instance $(G, k, d)$ of CONTRACTION(vc) is a yes-instance even when $k = d$, hence enhancing our understanding of the classical complexity of the problem. 3. CONTRACTION(vc) can be solved in time $2^{O(d)} \cdot n^{k - d + O(1)}$. This XP algorithm improves the one of Lima et al. [JCSS 2021], which uses Courcelle's theorem as a subroutine and hence, the $f(d)$-factor in the running time is non-explicit and probably very large. On the other hard, it shows that when $k=d$, the problem is FPT parameterized by $d$ (or by $k$).

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arXiv2021

Using edge contractions to reduce the semitotal domination number

In this paper, we consider the problem of reducing the semitotal domination number of a given graph by contracting $k$ edges, for some fixed $k \geq 1$. We show that this can always be done with at most 3 edge contractions and further characterise those graphs requiring 1, 2 or 3 edge contractions, respectively, to decrease their semitotal domination number. We then study the complexity of the problem for $k=1$ and obtain in particular a complete complexity dichotomy for monogenic classes.

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arXiv2021

Reducing graph transversals via edge contractions

For a graph invariant $π$, the Contraction($π$) problem consists in, given a graph $G$ and two positive integers $k,d$, deciding whether one can contract at most $k$ edges of $G$ to obtain a graph in which $π$ has dropped by at least $d$. Galby et al. [ISAAC 2019, MFCS 2019] recently studied the case where $π$ is the size of a minimum dominating set. We focus on graph invariants defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection ${\cal H}$ according to a fixed containment relation. We prove co-NP-hardness results under some assumptions on the graphs in ${\cal H}$, which in particular imply that Contraction($π$) is co-NP-hard even for fixed $k=d=1$ when $π$ is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, we show that when $π$ is the size of a minimum vertex cover, the problem is in XP parameterized by $d$.

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arXiv2020

Graph Square Roots of Small Distance from Degree One Graphs

Given a graph class $\mathcal{H}$, the task of the $\mathcal{H}$-Square Root problem is to decide, whether an input graph $G$ has a square root $H$ from $\mathcal{H}$. We are interested in the parameterized complexity of the problem for classes $\mathcal{H}$ that are composed by the graphs at vertex deletion distance at most $k$ from graphs of maximum degree at most one, that is, we are looking for a square root $H$ such that there is a modulator $S$ of size $k$ such that $H-S$ is the disjoint union of isolated vertices and disjoint edges. We show that different variants of the problems with constraints on the number of isolated vertices and edges in $H-S$ are FPT when parameterized by $k$ by demonstrating algorithms with running time $2^{2^{O(k)}}\cdot n^{O(1)}$. We further show that the running time of our algorithms is asymptotically optimal and it is unlikely that the double-exponential dependence on $k$ could be avoided. In particular, we prove that the VC-$k$ Root problem, that asks whether an input graph has a square root with vertex cover of size at most $k$, cannot be solved in time $2^{2^{o(k)}}\cdot n^{O(1)}$ unless Exponential Time Hypothesis fails. Moreover, we point out that VC-$k$ Root parameterized by $k$ does not admit a subexponential kernel unless $P=NP$.

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arXiv2020

Algorithms for the rainbow vertex coloring problem on graph classes

Given a vertex-colored graph, we say a path is a rainbow vertex path if all its internal vertices have distinct colors. The graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. In the Rainbow Vertex Coloring (RVC) problem we want to decide whether the vertices of a given graph can be colored with at most $k$ colors so that the graph becomes rainbow vertex-connected. This problem is known to be NP-complete even in very restricted scenarios, and very few efficient algorithms are known for it. In this work, we give polynomial-time algorithms for RVC on permutation graphs, powers of trees and split strongly chordal graphs. The algorithm for the latter class also works for the strong variant of the problem, where the rainbow vertex paths between each vertex pair must be shortest paths. We complement the polynomial-time solvability results for split strongly chordal graphs by showing that, for any fixed $p\geq 3$ both variants of the problem become NP-complete when restricted to split $(S_3,\ldots,S_p)$-free graphs, where $S_q$ denotes the $q$-sun graph.

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arXiv2020

Structural Parameterizations of Clique Coloring

A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with $q$ colors. For fixed $q \ge 2$, we give an $\mathcal{O}^{\star}(q^{tw})$-time algorithm when the input graph is given together with one of its tree decompositions of width $tw$. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is XP parameterized by clique-width.

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