Computations by fly-automata beyond monadic second-order logic

We present logically based methods for constructing XP and FPT graph algorithms, parametrized by tree-width or clique-width. We will use fly-automata introduced in a previous article. They make possible to check properties that are not monadic second-order expressible because their states may include counters, so that their sets of states may be infinite. We equip these automata with output functions, so that they can compute values associated with terms or graphs. Rather than new algorithmic results we present tools for constructing easily certain dynamic programming algorithms by combining predefined automata for basic functions and properties.Comment: Accepted for publication in Theoretical Computer Scienc

Tracking Paths in Planar Graphs

We consider the NP-complete problem of tracking paths in a graph, first introduced by Banik et al. [Banik et al., 2017]. Given an undirected graph with a source s and a destination t, find the smallest subset of vertices whose intersection with any s-t path results in a unique sequence. In this paper, we show that this problem remains NP-complete when the graph is planar and we give a 4-approximation algorithm in this setting. We also show, via Courcelle\u27s theorem, that it can be solved in linear time for graphs of bounded-clique width, when its clique decomposition is given in advance

Fly-automata for checking MSO 2 graph properties

A more descriptive but too long title would be : Constructing fly-automata to check properties of graphs of bounded tree-width expressed by monadic second-order formulas written with edge quantifications. Such properties are called MSO2 in short. Fly-automata (FA) run bottom-up on terms denoting graphs and compute "on the fly" the necessary states and transitions instead of looking into huge, actually unimplementable tables. In previous works, we have constructed FA that process terms denoting graphs of bounded clique-width, in order to check their monadic second-order (MSO) properties (expressed by formulas without edge quan-tifications). Here, we adapt these FA to incidence graphs, so that they can check MSO2 properties of graphs of bounded tree-width. This is possible because: (1) an MSO2 property of a graph is nothing but an MSO property of its incidence graph and (2) the clique-width of the incidence graph of a graph is linearly bounded in terms of its tree-width. Our constructions are actually implementable and usable. We detail concrete constructions of automata in this perspective.Comment: Submitted for publication in December 201

On defining linear orders by automata

We define linear orders ≤Z on product sets Z := X1 × X2 × ... × Xn and on subsets Z of X1 × X2 where each composing set Xi is [0, p] or N, and ordered in the natural way. We require that (Z, ≤Z) be isomor-phic to (N, ≤) if it is infinite. We want linear orderings of Z such that, in two consecutive tuples z = (z1, ..., zn) and z = (z 1 , ..., z n), we have |zi − z i | ≤ 1 for each i. Furthermore, we define their distance d(z, z) as the number of indices i such that zi = z i. We will consider orderings where the distance of two consecutive tuples is at most 2. We are interested in algorithms that determine the tuple in Z following z by using local information, where "local" is meant with respect to graphs associated with Z, and that work as well for finite and infinite components Xi, without knowing whether the components Xi are finite or not. We will formalize these algorithms by deterministic graph-walking automata

Computations by fly-automata beyond monadic second-order logic

We present logically based methods for constructing XP and FPT graph algorithms, parametrized by tree-width or clique-width. We will use fly-automata introduced in a previous article. They make possible to check properties that are not monadic second-order expressible because their states may include counters, so that their sets of states may be infinite. We equip these automata with output functions, so that they can compute values associated with terms or graphs. Rather than new algorithmic results we present tools for constructing easily certain dynamic programming algorithms by combining predefined automata for basic functions and properties