551 research outputs found
On the complexity of strongly connected components in directed hypergraphs
We study the complexity of some algorithmic problems on directed hypergraphs
and their strongly connected components (SCCs). The main contribution is an
almost linear time algorithm computing the terminal strongly connected
components (i.e. SCCs which do not reach any components but themselves).
"Almost linear" here means that the complexity of the algorithm is linear in
the size of the hypergraph up to a factor alpha(n), where alpha is the inverse
of Ackermann function, and n is the number of vertices. Our motivation to study
this problem arises from a recent application of directed hypergraphs to
computational tropical geometry.
We also discuss the problem of computing all SCCs. We establish a superlinear
lower bound on the size of the transitive reduction of the reachability
relation in directed hypergraphs, showing that it is combinatorially more
complex than in directed graphs. Besides, we prove a linear time reduction from
the well-studied problem of finding all minimal sets among a given family to
the problem of computing the SCCs. Only subquadratic time algorithms are known
for the former problem. These results strongly suggest that the problem of
computing the SCCs is harder in directed hypergraphs than in directed graphs.Comment: v1: 32 pages, 7 figures; v2: revised version, 34 pages, 7 figure
Global optimal control of perturbed systems
We propose a new numerical method for the computation of the optimal value
function of perturbed control systems and associated globally stabilizing
optimal feedback controllers. The method is based on a set oriented
discretization of state space in combination with a new algorithm for the
computation of shortest paths in weighted directed hypergraphs. Using the
concept of a multivalued game, we prove convergence of the scheme as the
discretization parameter goes to zero
Finding hypernetworks in directed hypergraphs
The term ‘‘hypernetwork’’ (more precisely, s-hypernetwork and (s, d)-hypernetwork) has been recently adopted to denote some logical structures contained in a directed hypergraph. A hypernetwork identifies the core of a hypergraph model, obtained by filtering off redundant components. Therefore, finding hypernetworks has a notable relevance both from a theoretical and from a computational point of view.
In this paper we provide a simple and fast algorithm for finding s-hypernetworks, which substantially improves on a method previously proposed in the literature. We also point out two linearly solvable particular cases.
Finding an (s, d)-hypernetwork is known to be a hard problem, and only one polynomially solvable class has been found so far. Here we point out that this particular case is solvable in linear time
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