1,160 research outputs found
Hitting Diamonds and Growing Cacti
We consider the following NP-hard problem: in a weighted graph, find a
minimum cost set of vertices whose removal leaves a graph in which no two
cycles share an edge. We obtain a constant-factor approximation algorithm,
based on the primal-dual method. Moreover, we show that the integrality gap of
the natural LP relaxation of the problem is \Theta(\log n), where n denotes the
number of vertices in the graph.Comment: v2: several minor changes
A tabu search heuristic based on k-diamonds for the weighted feedback vertex set problem
Given an undirected and vertex weighted graph G = (V,E,w), the Weighted Feedback Vertex Problem (WFVP) consists of finding a subset F ⊆ V of vertices of minimum weight such that each cycle in G contains at least one vertex in F. The WFVP on general graphs is known to be NP-hard and to be polynomially solvable on some special classes of graphs (e.g., interval graphs, co-comparability graphs, diamond graphs). In this paper we introduce an extension of diamond graphs, namely the k-diamond graphs, and give a dynamic programming algorithm to solve WFVP in linear time on this class of graphs. Other than solving an open question, this algorithm allows an efficient exploration of a neighborhood structure that can be defined by using such a class of graphs. We used this neighborhood structure inside our Iterated Tabu Search heuristic. Our extensive experimental show the effectiveness of this heuristic in improving the solution provided by a 2-approximate algorithm for the WFVPon general graphs
Exact Localisations of Feedback Sets
The feedback arc (vertex) set problem, shortened FASP (FVSP), is to transform
a given multi digraph into an acyclic graph by deleting as few arcs
(vertices) as possible. Due to the results of Richard M. Karp in 1972 it is one
of the classic NP-complete problems. An important contribution of this paper is
that the subgraphs , of all elementary
cycles or simple cycles running through some arc , can be computed in
and , respectively. We use
this fact and introduce the notion of the essential minor and isolated cycles,
which yield a priori problem size reductions and in the special case of so
called resolvable graphs an exact solution in . We show
that weighted versions of the FASP and FVSP possess a Bellman decomposition,
which yields exact solutions using a dynamic programming technique in times
and
, where , , respectively. The parameters can
be computed in , ,
respectively and denote the maximal dimension of the cycle space of all
appearing meta graphs, decoding the intersection behavior of the cycles.
Consequently, equal zero if all meta graphs are trees. Moreover, we
deliver several heuristics and discuss how to control their variation from the
optimum. Summarizing, the presented results allow us to suggest a strategy for
an implementation of a fast and accurate FASP/FVSP-SOLVER
Kernelization for Counting Problems on Graphs: Preserving the Number of Minimum Solutions
A kernelization for a parameterized decision problem is a
polynomial-time preprocessing algorithm that reduces any parameterized instance
into an instance whose size is bounded by a function of
alone and which has the same yes/no answer for . Such
preprocessing algorithms cannot exist in the context of counting problems, when
the answer to be preserved is the number of solutions, since this number can be
arbitrarily large compared to . However, we show that for counting minimum
feedback vertex sets of size at most , and for counting minimum dominating
sets of size at most in a planar graph, there is a polynomial-time
algorithm that either outputs the answer or reduces to an instance of
size polynomial in with the same number of minimum solutions. This shows
that a meaningful theory of kernelization for counting problems is possible and
opens the door for future developments. Our algorithms exploit that if the
number of solutions exceeds , the size of the input is
exponential in terms of so that the running time of a parameterized
counting algorithm can be bounded by . Otherwise, we can use
gadgets that slightly increase to represent choices among
options by only vertices.Comment: Extended abstract appears in the proceedings of IPEC 202
Consensus clustering in complex networks
The community structure of complex networks reveals both their organization
and hidden relationships among their constituents. Most community detection
methods currently available are not deterministic, and their results typically
depend on the specific random seeds, initial conditions and tie-break rules
adopted for their execution. Consensus clustering is used in data analysis to
generate stable results out of a set of partitions delivered by stochastic
methods. Here we show that consensus clustering can be combined with any
existing method in a self-consistent way, enhancing considerably both the
stability and the accuracy of the resulting partitions. This framework is also
particularly suitable to monitor the evolution of community structure in
temporal networks. An application of consensus clustering to a large citation
network of physics papers demonstrates its capability to keep track of the
birth, death and diversification of topics.Comment: 11 pages, 12 figures. Published in Scientific Report
Directed Symmetric Multicut is W[1]-hard
Given a directed graph and a set of vertex pairs , the Directed Symmetric Multicut problem asks to delete the
minimum number of edges from to separate every pair into
distinct strong components. Eiben, Rambaud and Wahlstr\"om [IPEC 2022]
initiated the study of this problem parameterized by the solution size. They
gave a fixed-parameter tractable 2-approximation algorithm, and left the exact
parameterized complexity status as an open question. We answer their question
in negative, showing that Directed Symmetric Multicut is W[1]-hard
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