8,140 research outputs found
Global offensive -alliances in digraphs
In this paper, we initiate the study of global offensive -alliances in
digraphs. Given a digraph , a global offensive -alliance in a
digraph is a subset such that every vertex outside of
has at least one in-neighbor from and also at least more in-neighbors
from than from outside of , by assuming is an integer lying between
two minus the maximum in-degree of and the maximum in-degree of . The
global offensive -alliance number is the minimum
cardinality among all global offensive -alliances in . In this article we
begin the study of the global offensive -alliance number of digraphs. For
instance, we prove that finding the global offensive -alliance number of
digraphs is an NP-hard problem for any value and that it remains NP-complete even when
restricted to bipartite digraphs when we consider the non-negative values of
given in the interval above. Based on these facts, lower bounds on
with characterizations of all digraphs attaining the bounds
are given in this work. We also bound this parameter for bipartite digraphs
from above. For the particular case , an immediate result from the
definition shows that for all digraphs ,
in which stands for the domination number of . We show that
these two digraph parameters are the same for some infinite families of
digraphs like rooted trees and contrafunctional digraphs. Moreover, we show
that the difference between and can be
arbitrary large for directed trees and connected functional digraphs
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
Parameterized complexity and approximability of directed odd cycle transversal
A directed odd cycle transversal of a directed graph (digraph) D is a vertex set S that intersects every odd directed cycle of D. In the Directed Odd Cycle Transversal (DOCT) problem, the input consists of a digraph D and an integer k. The objective is to determine whether there exists a directed odd cycle transversal of D of size at most k. In this paper, we settle the parameterized complexity of DOCT when parameterized by the solution size k by showing that DOCT does not admit an algorithm with running time f(k)nO(1) unless FPT = W[1]. On the positive side, we give a factor 2 fixed-parameter approximation (FPT approximation) algorithm for the problem. More precisely our algorithm takes as input D and k, runs in time 2O(k)nO(1), and either concludes that D does not have a directed odd cycle transversal of size at most k, or produces a solution of size at most 2k. Finally assuming gap-ETH, we show that there exists an Ï” > 0 such that DOCT does not admit a factor (1 + Ï”) FPT-approximation algorithm
A simple algorithm and min-max formula for the inverse arborescence problem
In 1998, Hu and Liu developed a strongly polynomial algorithm for solving the inverse arborescence problem that aims at minimally modifying a given cost-function on the edge-set of a digraph D so that an input spanning arborescence of D becomes a cheapest one. In this note, we develop a conceptually simpler algorithm along with a new min-max formula for the minimum modification of the cost-function. The approach is based on a link to a min-max theorem and a simple (two-phase greedy) algorithm by the first author from 1979 concerning the primal optimization problem of finding a cheapest subgraph of a digraph that covers an intersecting family along with the corresponding dual optimization problem, as well. (C) 2021 The Author(s). Published by Elsevier B.V
On packing spanning arborescences with matroid constraint
Let D = (V + s, A) be a digraph with a designated root vertex S. Edmondsâ seminal result (see J. Edmonds [4]) implies that D has a packing of k spanning s-arborescences if and only if D has a packing of k (s, t)-paths for all t â V, where a packing means arc-disjoint subgraphs. Let M be a matroid on the set of arcs leaving S. A packing of (s,t) -paths is called M-based if their arcs leaving S form a base of M while a packing of s-arborescences is called M -based if, for all t â V, the packing of (s, t) -paths provided by the arborescences is M -based. Durand de Gevigney, Nguyen, and Szigeti proved in [3] that D has an M-based packing of s -arborescences if and only if D has an M-based packing of (s,t) -paths for all t â V. BĂ©rczi and Frank conjectured that this statement can be strengthened in the sense of Edmondsâ theorem such that each S -arborescence is required to be spanning. Specifically, they conjectured that D has an M -based packing of spanning S -arborescences if and only if D has an M -based packing of (s,t) -paths for all t â V. In this paper we disprove this conjecture in its general form and we prove that the corresponding decision problem is NP-complete. We also prove that the conjecture holds for several fundamental classes of matroids, such as graphic matroids and transversal matroids. For all the results presented in this paper, the undirected counterpart also holds
The complexity of finding arc-disjoint branching flows
The concept of arc-disjoint flows in networks was recently introduced in \cite{bangTCSflow}. This is a very general framework within which many well-known and important problems can be formulated. In particular, the existence of arc-disjoint branching flows, that is, flows which send one unit of flow from a given source to all other vertices, generalizes the concept of arc-disjoint out-branchings (spanning out-trees) in a digraph. A pair of out-branchings from a root in a digraph on vertices corresponds to arc-disjoint branching flows (the arcs carrying flow in are those used in , ) in the network that we obtain from by giving all arcs capacity .It is then a natural question to ask how much we can lower the capacities on the arcs and still have, say, two arc-disjoint branching flows from the given root .We prove that for every fixed integer it is\begin{itemize}\item an NP-complete problem to decide whether a network where for every arc has two arc-disjoint branching flows rooted at .\item a polynomial problem to decide whether a network on vertices and for every arc has two arc-disjoint branching flows rooted at .\end{itemize}The algorithm for the later result generalizes the polynomial algorithm, due to Lov\'asz, for deciding whether a given input digraph has two arc-disjoint out-branchings rooted at a given vertex.Finally we prove that under the so-called Exponential Time Hypothesis (ETH), for every and for every with (and for every large we have for some ) there is no polynomial algorithm for deciding whether a given digraph contains twoarc-disjoint branching flows from the same root so that no arc carries flow larger than
The complexity of finding arc-disjoint branching flows
International audienceThe concept of arc-disjoint flows in networks was recently introduced in Bang-Jensen and Bessy (2014). This is a very general framework within which many well-known and important problems can be formulated. In particular, the existence of arc-disjoint branching flows, that is, flows which send one unit of flow from a given source s to all other vertices, generalizes the concept of arc-disjoint out-branchings (spanning out-trees) in a digraph. A pair of out-branchings B + s,1 , B + s,2 from a root s in a digraph D = (V , A) on n vertices corresponds to arc-disjoint branching flows x 1 , x 2 (the arcs carrying flow in x i are those used in B + s,i , i = 1, 2) in the network that we obtain from D by giving all arcs capacity n â 1. It is then a natural question to ask how much we can lower the capacities on the arcs and still have, say, two arc-disjoint branching flows from the given root s. We prove that for every fixed integer k â„ 2 it is âą an NP-complete problem to decide whether a network N = (V , A, u) where u ij = k for every arc ij has two arc-disjoint branching flows rooted at s. âą a polynomial problem to decide whether a network N = (V , A, u) on n vertices and u ij = n â k for every arc ij has two arc-disjoint branching flows rooted at s. The algorithm for the later result generalizes the polynomial algorithm, due to LovĂĄsz, for deciding whether a given input digraph has two arc-disjoint out-branchings rooted at a given vertex. Finally we prove that under the so-called Exponential Time Hypothesis (ETH), for every Ï” > 0 and for every k(n) with (log(n)) 1+Ï” †k(n) †n 2 (and for every large i we have k(n) = i for some n) there is no polynomial algorithm for deciding whether a given digraph contains two arc-disjoint branching flows from the same root so that no arc carries flow larger than n â k(n)
Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments
A digraph such that every proper induced subdigraph has a kernel is said to
be \emph{kernel perfect} (KP for short) (\emph{critical kernel imperfect} (CKI
for short) resp.) if the digraph has a kernel (does not have a kernel resp.).
The unique CKI-tournament is and the unique
KP-tournaments are the transitive tournaments, however bipartite tournaments
are KP. In this paper we characterize the CKI- and KP-digraphs for the
following families of digraphs: locally in-/out-semicomplete, asymmetric
arc-locally in-/out-semicomplete, asymmetric -quasi-transitive and
asymmetric -anti-quasi-transitive -free and we state that the problem
of determining whether a digraph of one of these families is CKI is polynomial,
giving a solution to a problem closely related to the following conjecture
posted by Bang-Jensen in 1998: the kernel problem is polynomially solvable for
locally in-semicomplete digraphs.Comment: 13 pages and 5 figure
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