arXiv subjects

Ignasi Sau

Publications and source records attributed to Ignasi Sau.

79 records · Page 5Linked to original sources

arXiv2013

Hitting and Harvesting Pumpkins

The "c-pumpkin" is the graph with two vertices linked by c>0 parallel edges. A c-pumpkin-model in a graph G is a pair A,B of disjoint subsets of vertices of G, each inducing a connected subgraph of G, such that there are at least c edges in G between A and B. We focus on covering and packing c-pumpkin-models in a given graph: On the one hand, we provide an FPT algorithm running in time 2^O(k) n^O(1) deciding, for any fixed c>0, whether all c-pumpkin-models can be covered by at most k vertices. This generalizes known single-exponential FPT algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the cases c=1,2 respectively. On the other hand, we present a O(log n)-approximation algorithm for both the problems of covering all c-pumpkin-models with a smallest number of vertices, and packing a maximum number of vertex-disjoint c-pumpkin-models.

cs.DS↗

arXiv2013

The role of planarity in connectivity problems parameterized by treewidth

For some years it was believed that for "connectivity" problems such as Hamiltonian Cycle, algorithms running in time 2^{O(tw)}n^{O(1)} -called single-exponential- existed only on planar and other sparse graph classes, where tw stands for the treewidth of the n-vertex input graph. This was recently disproved by Cygan et al. [FOCS 2011], Bodlaender et al. [ICALP 2013], and Fomin et al. [SODA 2014], who provided single-exponential algorithms on general graphs for essentially all connectivity problems that were known to be solvable in single-exponential time on sparse graphs. In this article we further investigate the role of planarity in connectivity problems parameterized by treewidth, and convey that several problems can indeed be distinguished according to their behavior on planar graphs. Known results from the literature imply that there exist problems, like Cycle Packing, that cannot be solved in time 2^{o(tw logtw)}n^{O(1)} on general graphs but that can be solved in time 2^{O(tw)}n^{O(1)} when restricted to planar graphs. Our main contribution is to show that there exist problems that can be solved in time 2^{O(tw logtw)}n^{O(1)} on general graphs but that cannot be solved in time 2^{o(tw logtw)}n^{O(1)} even when restricted to planar graphs. Furthermore, we prove that Planar Cycle Packing and Planar Disjoint Paths cannot be solved in time 2^{o(tw)}n^{O(1)}. The mentioned negative results hold unless the ETH fails. We feel that our results constitute a first step in a subject that can be further exploited.

cs.DS↗

arXiv2012

Linear kernels and single-exponential algorithms via protrusion decompositions

A \emph{$t$-treewidth-modulator} of a graph $G$ is a set $X \subseteq V(G)$ such that the treewidth of $G-X$ is at most some constant $t-1$. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs $G$ that come equipped with a $t$-treewidth-modulator. This decomposition, called a \emph{protrusion decomposition}, is the cornerstone in obtaining the following two main results. We first show that any parameterized graph problem (with parameter $k$) that has \emph{finite integer index} and is \emph{treewidth-bounding} admits a linear kernel on $H$-topological-minor-free graphs, where $H$ is some arbitrary but fixed graph. A parameterized graph problem is called treewidth-bounding if all positive instances have a $t$-treewidth-modulator of size $O(k)$, for some constant $t$. This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus [Bodlaender et al., FOCS 2009] and $H$-minor-free graphs [Fomin et al., SODA 2010]. Our second application concerns the Planar-$\mathcal{F}$-Deletion problem. Let $\mathcal{F}$ be a fixed finite family of graphs containing at least one planar graph. Given an $n$-vertex graph $G$ and a non-negative integer $k$, Planar-$\mathcal{F}$-Deletion asks whether $G$ has a set $X\subseteq V(G)$ such that $|X|\leq k$ and $G-X$ is $H$-minor-free for every $H\in \mathcal{F}$. Very recently, an algorithm for Planar-$\mathcal{F}$-Deletion with running time $2^{O(k)} n \log^2 n$ (such an algorithm is called \emph{single-exponential}) has been presented in [Fomin et al., FOCS 2012] under the condition that every graph in $\mathcal{F}$ is connected. Using our algorithm to construct protrusion decompositions as a building block, we get rid of this connectivity constraint and present an algorithm for the general Planar-$\mathcal{F}$-Deletion problem running in time $2^{O(k)} n^2$.

cs.DS↗

arXiv2012

Parameterized Domination in Circle Graphs

A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Applied Mathematics, 42(1):51-63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction: - Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs, parameterized by the size of the solution. - Whereas both Connected Dominating Set and Acyclic Dominating Set are W[1]-hard in circle graphs, it turns out that Connected Acyclic Dominating Set is polynomial-time solvable in circle graphs. - If T is a given tree, deciding whether a circle graph has a dominating set isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by |V(T)|. We prove that the FPT algorithm is subexponential.

cs.DS↗

arXiv2011

Dynamic Programming for Graphs on Surfaces

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2^{O(k log k)} n steps. Our approach combines tools from topological graph theory and analytic combinatorics. In particular, we introduce a new type of branch decomposition called "surface cut decomposition", generalizing sphere cut decompositions of planar graphs introduced by Seymour and Thomas, which has nice combinatorial properties. Namely, the number of partial solutions that can be arranged on a surface cut decomposition can be upper-bounded by the number of non-crossing partitions on surfaces with boundary. It follows that partial solutions can be represented by a single-exponential (in the branchwidth k) number of configurations. This proves that, when applied on surface cut decompositions, dynamic programming runs in 2^{O(k)} n steps. That way, we considerably extend the class of problems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve most previous results in this direction.

cs.DS↗

arXiv2011

Asymptotic Enumeration of Non-crossing Partitions on Surfaces

We generalize the notion of non-crossing partition on a disk to general surfaces with boundary. For this, we consider a surface $Σ$ and introduce the number $C_Σ(n)$ of non-crossing partitions of a set of $n$ points laying on the boundary of $Σ$. Our proofs use bijective techniques arising from map enumeration, joint with the symbolic method and singularity analysis on generating functions. An outcome of our results is that the exponential growth of $C_Σ(n)$ is the same as the one of the $n$-th Catalan number, i.e., does not change when we move from the case where $Σ$ is a disk to general surfaces with boundary.

math.CO↗

arXiv2010

The Recognition of Tolerance and Bounded Tolerance Graphs

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This subclass of perfect graphs has been extensively studied, due to both its interesting structure and its numerous applications. Several efficient algorithms for optimization problems that are NP-hard on general graphs have been designed for tolerance graphs. In spite of this, the recognition of tolerance graphs - namely, the problem of deciding whether a given graph is a tolerance graph - as well as the recognition of their main subclass of bounded tolerance graphs, have been the most fundamental open problems on this class of graphs (cf. the book on tolerance graphs \cite{GolTol04}) since their introduction in 1982 \cite{GoMo82}. In this article we prove that both recognition problems are NP-complete, even in the case where the input graph is a trapezoid graph. The presented results are surprising because, on the one hand, most subclasses of perfect graphs admit polynomial recognition algorithms and, on the other hand, bounded tolerance graphs were believed to be efficiently recognizable as they are a natural special case of trapezoid graphs (which can be recognized in polynomial time) and share a very similar structure with them. For our reduction we extend the notion of an \emph{acyclic orientation} of permutation and trapezoid graphs. Our main tool is a new algorithm that uses \emph{vertex splitting} to transform a given trapezoid graph into a permutation graph, while preserving this new acyclic orientation property. This method of vertex splitting is of independent interest; very recently, it has been proved a powerful tool also in the design of efficient recognition algorithms for other classes of graphs \cite{MC-Trapezoid}.

cs.CC↗