750 research outputs found
On the hardness of inclusion-wise minimal separators enumeration
Enumeration problems are often encountered as key subroutines in the exact
computation of graph parameters such as chromatic number, treewidth, or
treedepth. In the case of treedepth computation, the enumeration of
inclusion-wise minimal separators plays a crucial role. However and quite
surprisingly, the complexity status of this problem has not been settled since
it has been posed as an open direction by Kloks and Kratsch in 1998. Recently
at the PACE 2020 competition dedicated to treedepth computation, solvers have
been circumventing that by listing all minimal - separators and filtering
out those that are not inclusion-wise minimal, at the cost of efficiency.
Naturally, having an efficient algorithm for listing inclusion-wise minimal
separators would drastically improve such practical algorithms. In this note,
however, we show that no efficient algorithm is to be expected from an
output-sensitive perspective, namely, we prove that there is no
output-polynomial time algorithm for inclusion-wise minimal separators
enumeration unless P = NP.Comment: 12 pages, 3 figure
On the Enumeration of Minimal Dominating Sets and Related Notions
A dominating set in a graph is a subset of its vertex set such that each
vertex is either in or has a neighbour in . In this paper, we are
interested in the enumeration of (inclusion-wise) minimal dominating sets in
graphs, called the Dom-Enum problem. It is well known that this problem can be
polynomially reduced to the Trans-Enum problem in hypergraphs, i.e., the
problem of enumerating all minimal transversals in a hypergraph. Firstly we
show that the Trans-Enum problem can be polynomially reduced to the Dom-Enum
problem. As a consequence there exists an output-polynomial time algorithm for
the Trans-Enum problem if and only if there exists one for the Dom-Enum
problem. Secondly, we study the Dom-Enum problem in some graph classes. We give
an output-polynomial time algorithm for the Dom-Enum problem in split graphs,
and introduce the completion of a graph to obtain an output-polynomial time
algorithm for the Dom-Enum problem in -free chordal graphs, a proper
superclass of split graphs. Finally, we investigate the complexity of the
enumeration of (inclusion-wise) minimal connected dominating sets and minimal
total dominating sets of graphs. We show that there exists an output-polynomial
time algorithm for the Dom-Enum problem (or equivalently Trans-Enum problem) if
and only if there exists one for the following enumeration problems: minimal
total dominating sets, minimal total dominating sets in split graphs, minimal
connected dominating sets in split graphs, minimal dominating sets in
co-bipartite graphs.Comment: 15 pages, 3 figures, In revisio
On the Enumeration of all Minimal Triangulations
We present an algorithm that enumerates all the minimal triangulations of a
graph in incremental polynomial time. Consequently, we get an algorithm for
enumerating all the proper tree decompositions, in incremental polynomial time,
where "proper" means that the tree decomposition cannot be improved by removing
or splitting a bag
Enumeration of s-d separators in DAGs with application to reliability analysis in temporal graphs
Temporal graphs are graphs in which arcs have temporal labels, specifying at which time they can be traversed. Motivated by recent results concerning the reliability analysis of a temporal graph through the enumeration of minimal cutsets in the corresponding line graph, in this paper we attack the problem of enumerating minimal s-d separators in s-d directed acyclic graphs (in short, s-d DAGs), also known as 2-terminal DAGs or s-t digraphs. Our main result is an algorithm for enumerating all the minimal s-d separators in a DAG with O(nm) delay, where n and m are respectively the number of nodes and arcs, and the delay is the time between the output of two consecutive solutions. To this aim, we give a characterization of the minimal s-d separators in a DAG through vertex cuts of an expanded version of the DAG itself. As a consequence of our main result, we provide an algorithm for enumerating all the minimal s-d cutsets in a temporal graph with delay O(m3), where m is the number of temporal arcs
Efficient Enumerations for Minimal Multicuts and Multiway Cuts
Let be an undirected graph and let be a
set of terminal pairs. A node/edge multicut is a subset of vertices/edges of
whose removal destroys all the paths between every terminal pair in .
The problem of computing a {\em minimum} node/edge multicut is NP-hard and
extensively studied from several viewpoints. In this paper, we study the
problem of enumerating all {\em minimal} node multicuts. We give an incremental
polynomial delay enumeration algorithm for minimal node multicuts, which
extends an enumeration algorithm due to Khachiyan et al. (Algorithmica, 2008)
for minimal edge multicuts. Important special cases of node/edge multicuts are
node/edge {\em multiway cuts}, where the set of terminal pairs contains every
pair of vertices in some subset , that is, . We
improve the running time bound for this special case: We devise a polynomial
delay and exponential space enumeration algorithm for minimal node multiway
cuts and a polynomial delay and space enumeration algorithm for minimal edge
multiway cuts
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