951 research outputs found
A Relaxation of the Directed Disjoint Paths Problem: A Global Congestion Metric Helps
In the Directed Disjoint Paths problem, we are given a digraph and a set
of requests , and the task is to find a
collection of pairwise vertex-disjoint paths such that
each is a path from to in . This problem is NP-complete
for fixed and W[1]-hard with parameter in DAGs. A few positive
results are known under restrictions on the input digraph, such as being planar
or having bounded directed tree-width, or under relaxations of the problem,
such as allowing for vertex congestion. Positive results are scarce, however,
for general digraphs. In this article we propose a novel global congestion
metric for the problem: we only require the paths to be "disjoint enough", in
the sense that they must behave properly not in the whole graph, but in an
unspecified part of size prescribed by a parameter. Namely, in the Disjoint
Enough Directed Paths problem, given an -vertex digraph , a set of
requests, and non-negative integers and , the task is to find a
collection of paths connecting the requests such that at least vertices of
occur in at most paths of the collection. We study the parameterized
complexity of this problem for a number of choices of the parameter, including
the directed tree-width of . Among other results, we show that the problem
is W[1]-hard in DAGs with parameter and, on the positive side, we give an
algorithm in time and a kernel of
size in general digraphs. This latter
result has consequences for the Steiner Network problem: we show that it is FPT
parameterized by the number of terminals and , where and
is the size of the solution.Comment: 25 pages, 9 figure
A Mobile Ambients-based Approach for Network Attack Modelling and Simulation
Attack Graphs are an important support for assessment and subsequent improvement of network security. They reveal possible paths an attacker can take to break through security perimeters and traverse a network to reach valuable assets deep inside the network. Although scalability is no longer the main issue, Attack Graphs still have some problems that make them less useful in practice. First, Attack Graphs remain difficult to relate to the network topology. Second, Attack Graphs traditionally only consider the exploitation of vulnerable hosts. Third, Attack Graphs do not rely on automatic identification of potential attack targets. We address these gaps in our MsAMS (Multi-step Attack Modelling and Simulation) tool, based on Mobile Ambients. The tool not only allows the modelling of more static aspects of the network, such as the network topology, but also the dynamics of network attacks. In addition to Mobile Ambients, we use the PageRank algorithm to determine targets and hub scores produced by the HITS (Hypertext Induced Topic Search) algorithm to guide the simulation of an attacker searching for targets
Designing technology-mediated tasks for language teaching: A methodological framework
PETALL (Pan-European Task-based Activities for Language Learning) is a European-funded project aiming at the promotion of foreign languages learning through ICT-based tasks. For that purpose, the project consortium has offered teacher training courses and has produced samples of best practices in which technologies play a major role. These tasks have been trialled and evaluated in the neighbouring countries in a network of collaborative partnerships in teaching and research, which allowed the designers of the tasks to receive constructive feedback from peers and end-users (teachers and learners). This article first provides an overview of the project (namely its rationale, literature review, implementation and evaluation processes, and the dissemination and exploitation strategies), before explaining in greater detail the procedures employed by the consortium in the setting-up of a methodological framework to be used in the designing and trialling of ICT-based tasks. The different stages of the designing process are described, as well as the criteria for the validation of the proposed samples. The template used by the designers is explained and an analysis of the set of tasks is also provided. In the end, some closing remarks based on the outcomes of the project are given.info:eu-repo/semantics/publishedVersio
Twin-Width VIII: Delineation and Win-Wins
We introduce the notion of delineation. A graph class C is said delineated by twin-width (or simply, delineated) if for every hereditary closure D of a subclass of C, it holds that D has bounded twin-width if and only if D is monadically dependent. An effective strengthening of delineation for a class C implies that tractable FO model checking on C is perfectly understood: On hereditary closures of subclasses D of C, FO model checking on D is fixed-parameter tractable (FPT) exactly when D has bounded twin-width. Ordered graphs [BGOdMSTT, STOC \u2722] and permutation graphs [BKTW, JACM \u2722] are effectively delineated, while subcubic graphs are not. On the one hand, we prove that interval graphs, and even, rooted directed path graphs are delineated. On the other hand, we observe or show that segment graphs, directed path graphs (with arbitrarily many roots), and visibility graphs of simple polygons are not delineated.
In an effort to draw the delineation frontier between interval graphs (that are delineated) and axis-parallel two-lengthed segment graphs (that are not), we investigate the twin-width of restricted segment intersection classes. It was known that (triangle-free) pure axis-parallel unit segment graphs have unbounded twin-width [BGKTW, SODA \u2721]. We show that K_{t,t}-free segment graphs, and axis-parallel H_t-free unit segment graphs have bounded twin-width, where H_t is the half-graph or ladder of height t. In contrast, axis-parallel H?-free two-lengthed segment graphs have unbounded twin-width. We leave as an open question whether unit segment graphs are delineated.
More broadly, we explore which structures (large bicliques, half-graphs, or independent sets) are responsible for making the twin-width large on the main classes of intersection and visibility graphs. Our new results, combined with the FPT algorithm for first-order model checking on graphs given with O(1)-sequences [BKTW, JACM \u2722], give rise to a variety of algorithmic win-win arguments. They all fall in the same framework: If p is an FO definable graph parameter that effectively functionally upperbounds twin-width on a class C, then p(G) ? k can be decided in FPT time f(k) ? |V(G)|^O(1). For instance, we readily derive FPT algorithms for k-Ladder on visibility graphs of 1.5D terrains, and k-Independent Set on visibility graphs of simple polygons. This showcases that the theory of twin-width can serve outside of classes of bounded twin-width
Clique number of tournaments
We introduce the notion of clique number of a tournament and investigate its
relation with the dichromatic number. In particular, it permits defining
\dic-bounded classes of tournaments, which is the paper's main topic
New Menger-like dualities in digraphs and applications to half-integral linkages
We present new min-max relations in digraphs between the number of paths
satisfying certain conditions and the order of the corresponding cuts. We
define these objects in order to capture, in the context of solving the
half-integral linkage problem, the essential properties needed for reaching a
large bramble of congestion two (or any other constant) from the terminal set.
This strategy has been used ad-hoc in several articles, usually with lengthy
technical proofs, and our objective is to abstract it to make it applicable in
a simpler and unified way. We provide two proofs of the min-max relations, one
consisting in applying Menger's Theorem on appropriately defined auxiliary
digraphs, and an alternative simpler one using matroids, however with worse
polynomial running time.
As an application, we manage to simplify and improve several results of
Edwards et al. [ESA 2017] and of Giannopoulou et al. [SODA 2022] about finding
half-integral linkages in digraphs. Concerning the former, besides being
simpler, our proof provides an almost optimal bound on the strong connectivity
of a digraph for it to be half-integrally feasible under the presence of a
large bramble of congestion two (or equivalently, if the directed tree-width is
large, which is the hard case). Concerning the latter, our proof uses brambles
as rerouting objects instead of cylindrical grids, hence yielding much better
bounds and being somehow independent of a particular topology.
We hope that our min-max relations will find further applications as, in our
opinion, they are simple, robust, and versatile to be easily applicable to
different types of routing problems in digraphs
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