Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Log-Space

We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where `canonical' means that models of isomorphic graphs are equal. This implies that the recognition and the isomorphism problems for this class of graphs are solvable in logspace. For a broader class of concave-round graphs, that still possess (not necessarily proper) circular-arc models, we show that those can also be constructed canonically in logspace. As a building block for these results, we show how to compute canonical models of circular-arc hypergraphs in logspace, which are also known as matrices with the circular-ones property. Finally, we consider the search version of the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We solve it in logspace for the classes of proper circular-arc, concave-round, and co-convex graphs.Comment: 19 pages, 3 figures, major revisio

On the Disambiguation of Weighted Automata

We present a disambiguation algorithm for weighted automata. The algorithm admits two main stages: a pre-disambiguation stage followed by a transition removal stage. We give a detailed description of the algorithm and the proof of its correctness. The algorithm is not applicable to all weighted automata but we prove sufficient conditions for its applicability in the case of the tropical semiring by introducing the *weak twins property*. In particular, the algorithm can be used with all acyclic weighted automata, relevant to applications. While disambiguation can sometimes be achieved using determinization, our disambiguation algorithm in some cases can return a result that is exponentially smaller than any equivalent deterministic automaton. We also present some empirical evidence of the space benefits of disambiguation over determinization in speech recognition and machine translation applications

Algorithmic Aspects of a General Modular Decomposition Theory

A new general decomposition theory inspired from modular graph decomposition is presented. This helps unifying modular decomposition on different structures, including (but not restricted to) graphs. Moreover, even in the case of graphs, the terminology ``module'' not only captures the classical graph modules but also allows to handle 2-connected components, star-cutsets, and other vertex subsets. The main result is that most of the nice algorithmic tools developed for modular decomposition of graphs still apply efficiently on our generalisation of modules. Besides, when an essential axiom is satisfied, almost all the important properties can be retrieved. For this case, an algorithm given by Ehrenfeucht, Gabow, McConnell and Sullivan 1994 is generalised and yields a very efficient solution to the associated decomposition problem

The role of twins in computing planar supports of hypergraphs

A support or realization of a hypergraph $H$ is a graph $G$ on the same vertex as $H$ such that for each hyperedge of $H$ it holds that its vertices induce a connected subgraph of $G$. The NP-hard problem of finding a planar} support has applications in hypergraph drawing and network design. Previous algorithms for the problem assume that twins}---pairs of vertices that are in precisely the same hyperedges---can safely be removed from the input hypergraph. We prove that this assumption is generally wrong, yet that the number of twins necessary for a hypergraph to have a planar support only depends on its number of hyperedges. We give an explicit upper bound on the number of twins necessary for a hypergraph with $m$ hyperedges to have an $r$-outerplanar support, which depends only on $r$ and $m$. Since all additional twins can be safely removed, we obtain a linear-time algorithm for computing $r$-outerplanar supports for hypergraphs with $m$ hyperedges if $m$ and $r$ are constant; in other words, the problem is fixed-parameter linear-time solvable with respect to the parameters $m$ and $r$