On the reduction of the CSP dichotomy conjecture to digraphs

It is well known that the constraint satisfaction problem over general relational structures can be reduced in polynomial time to digraphs. We present a simple variant of such a reduction and use it to show that the algebraic dichotomy conjecture is equivalent to its restriction to digraphs and that the polynomial reduction can be made in logspace. We also show that our reduction preserves the bounded width property, i.e., solvability by local consistency methods. We discuss further algorithmic properties that are preserved and related open problems.Comment: 34 pages. Article is to appear in CP2013. This version includes two appendices with proofs of claims omitted from the main articl

Join-Reachability Problems in Directed Graphs

For a given collection G of directed graphs we define the join-reachability graph of G, denoted by J(G), as the directed graph that, for any pair of vertices a and b, contains a path from a to b if and only if such a path exists in all graphs of G. Our goal is to compute an efficient representation of J(G). In particular, we consider two versions of this problem. In the explicit version we wish to construct the smallest join-reachability graph for G. In the implicit version we wish to build an efficient data structure (in terms of space and query time) such that we can report fast the set of vertices that reach a query vertex in all graphs of G. This problem is related to the well-studied reachability problem and is motivated by emerging applications of graph-structured databases and graph algorithms. We consider the construction of join-reachability structures for two graphs and develop techniques that can be applied to both the explicit and the implicit problem. First we present optimal and near-optimal structures for paths and trees. Then, based on these results, we provide efficient structures for planar graphs and general directed graphs

Tight Localizations of Feedback Sets

The classical NP-hard feedback arc set problem (FASP) and feedback vertex set problem (FVSP) ask for a minimum set of arcs $\varepsilon \subseteq E$ or vertices $\nu \subseteq V$ whose removal $G\setminus \varepsilon$, $G\setminus \nu$ makes a given multi-digraph $G=(V,E)$ acyclic, respectively. Though both problems are known to be APX-hard, approximation algorithms or proofs of inapproximability are unknown. We propose a new $\mathcal{O}(|V||E|^4)$-heuristic for the directed FASP. While a ratio of $r \approx 1.3606$ is known to be a lower bound for the APX-hardness, at least by empirical validation we achieve an approximation of $r \leq 2$. The most relevant applications, such as circuit testing, ask for solving the FASP on large sparse graphs, which can be done efficiently within tight error bounds due to our approach.Comment: manuscript submitted to AC

On-line and Dynamic Shortest Paths through Graph Decompositions (Preliminary Version)

We describe algorithms for finding shortest paths and distances in a planar digraph which exploit the particular topology of the input graph. We give both sequential and parallel algorithms that work on a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. For outerplanar digraphs, for instance, the data structures can be updated after any such change in only $O(\log n)$ time, where $n$ is the number of vertices of the digraph. The parallel algorithms presented here are the first known ones for solving this problem. Our results can be extended to hold for digraphs of genus $o(n)$

Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts

A skew-symmetric graph $(D=(V,A),\sigma)$ is a directed graph $D$ with an involution $\sigma$ on the set of vertices and arcs. In this paper, we introduce a separation problem, $d$-Skew-Symmetric Multicut, where we are given a skew-symmetric graph $D$, a family of $\cal T$ of $d$-sized subsets of vertices and an integer $k$. The objective is to decide if there is a set $X\subseteq A$ of $k$ arcs such that every set $J$ in the family has a vertex $v$ such that $v$ and $\sigma(v)$ are in different connected components of $D'=(V,A\setminus (X\cup \sigma(X))$. In this paper, we give an algorithm for this problem which runs in time $O((4d)^{k}(m+n+\ell))$, where $m$ is the number of arcs in the graph, $n$ the number of vertices and $\ell$ the length of the family given in the input. Using our algorithm, we show that Almost 2-SAT has an algorithm with running time $O(4^kk^4\ell)$ and we obtain algorithms for {\sc Odd Cycle Transversal} and {\sc Edge Bipartization} which run in time $O(4^kk^4(m+n))$ and $O(4^kk^5(m+n))$ respectively. This resolves an open problem posed by Reed, Smith and Vetta [Operations Research Letters, 2003] and improves upon the earlier almost linear time algorithm of Kawarabayashi and Reed [SODA, 2010]. We also show that Deletion q-Horn Backdoor Set Detection is a special case of 3-Skew-Symmetric Multicut, giving us an algorithm for Deletion q-Horn Backdoor Set Detection which runs in time $O(12^kk^5\ell)$. This gives the first fixed-parameter tractable algorithm for this problem answering a question posed in a paper by a superset of the authors [STACS, 2013]. Using this result, we get an algorithm for Satisfiability which runs in time $O(12^kk^5\ell)$ where $k$ is the size of the smallest q-Horn deletion backdoor set, with $\ell$ being the length of the input formula

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Web browsing automation for applications quality control

Context: Quality control comprises the set of activities aimed to evaluate that software meets its specification and delivers the functionality expected by the consumers. These activities are often removed in the development process and, as a result, the final software product usually lacks quality. Objective: We propose a set of techniques to automate the quality control for web applications from the client-side, guiding the process by functional and nonfunctional requirements (performance, security, compatibility, usability and accessibility). Method: The first step to achieve automation is to define the structure of the web navigation. Existing software artifacts in the phase of analysis and design are reused. Then, the independent paths of navigation are found, and each path is traversed automatically using real browsers while different kinds of assessments are carried out. Results: The processes and methods proposed in this paper have been implemented by means of a reference architecture and open source tools. A laboratory experiment and an industrial case study have been performed in order to validate the proposal. Conclusion: The definition of navigation paths is a rich approach to model web applications. Grey-box (black-box and white-box) methods have been proved to be very valuable for web assessment. The Chinese Postman Problem (CPP) is an optimal way to find the independent paths in a web navigation modeled as a directed graph