We analyse approximation algorithms (greedy heuristics) for the classical domination number and two multiple domination numbers in simple graphs. First, we present a short self-contained proof of the known result that the minimum domination problem in any graph G with maximum degree ∆ can be solved within the approximation ratio of ln(∆ + 1) + 1. The proof is based on an analysis of a simple greedy heuristic. Then, by analysing more advanced greedy heuristic techniques and using ideas from our self-contained proof for the classical domination number, we fix a gap in the existing proof of a similar result for the k-tuple domination number. That is, we prove that the minimum k-tuple domination problem indeed can be approximated within the ratio of ln(∆+1)+1. The proof of this result is self-contained, direct, and much shorter than the existing proof, which contains the gap. Finally, we show that the known approximation ratio of ln(2∆)+1 for the minimum k-domination problem can be improved to a better ratio.

The Kuratowski graphs K3,3 and K5 characterize planarity. Counting distinct 2-cell embeddings of these two graphs on orientable surfaces was previously done by Mull (1999) and Mull et al. (2008), using Burnside’s Lemma and automorphism groups of K3,3 and K5, without actually constructing the embeddings. We obtain all 2-cell embeddings of these graphs on the double torus, using a constructive approach. This shows that there is a unique non-orientable 2-cell embedding of K3,3, and 14 orientable and 17 non-orientable 2-cell embeddings of K5 on the double torus, which are explicitly obtained using an algorithmic procedure of expanding from minors. Therefore we confirm the numbers of embeddings obtained by Mull (1999) and Mull et al. (2008). As a consequence, several new polygonal representations of the double torus are presented. Rotation systems for the one-face embeddings of K5 on the triple torus are also found, using exhaustive search.

The fixed parameter tractable (FPT) approach is a powerful tool in tackling computationally hard problems. In this paper, we link FPT results to classic artificial intelligence (AI) techniques to show how they complement each other. Specifically, we consider the workflow satisfiability problem (WSP) which asks whether there exists an assignment of authorised users to the steps in a workflow specification, subject to certain constraints on the assignment. It was shown by Cohen et al. (JAIR 2014) that WSP restricted to the class of user-independent constraints (UI), covering many practical cases, admits FPT algorithms, i.e. can be solved in time exponential only in the number of steps k and polynomial in the number of users n. Since usually k << n in WSP, such FPT algorithms are of great practical interest. We present a new interpretation of the FPT nature of the WSP with UI constraints giving a decomposition of the problem into two levels. Exploiting this two-level split, we develop a new FPT algorithm that is by many orders of magnitude faster than the previous state-of-the-art WSP algorithm and also has only polynomial-space complexity. We also introduce new pseudo-Boolean (PB) and Constraint Satisfaction (CSP) formulations of the WSP with UI constraints which efficiently exploit this new decomposition of the problem and raise the novel issue of how to use general-purpose solvers to tackle FPT problems in a fashion that meets FPT efficiency expectations. In our computational study, we investigate, for the first time, the phase transition (PT) properties of the WSP, under a model for generation of random instances. We show how PT studies can be extended, in a novel fashion, to support empirical evaluation of scaling of FPT algorithms.

A workflow specification defines a set of steps, a set of users, and an access control policy. The policy determines which steps a user is authorized to perform and imposes constraints on which sets of users can perform which sets of steps. The workflow satisfiability problem (WSP) is the problem of determining whether there exists an assignment of users to workflow steps that satisfies the policy. Given the computational hardness of WSP and its importance in the context of workflow management systems, it is important to develop algorithms that are as efficient as possible to solve WSP. In this article, we study the fixed-parameter tractability of WSP in the presence of class-independent constraints, which enable us to (1) model security requirements based on the groups to which users belong and (2) generalize the notion of a user-independent constraint. Class-independent constraints are defined in terms of equivalence relations over the set of users. We consider sets of nested equivalence relations because this enables us to model security requirements in hierarchical organizations. We prove that WSP is fixed-parameter tractable (FPT) for class-independent constraints defined over nested equivalence relations and develop an FPT algorithm to solve WSP instances incorporating such constraints. We perform experiments to evaluate the performance of our algorithm and compare it with that of SAT4J, an off-the-shelf pseudo-Boolean SAT solver. The results of these experiments demonstrate that our algorithm significantly outperforms SAT4J for many instances of WSP.

A workflow specification defines sets of steps and users. An authorization policy determines for each user a subset of steps the user is allowed to perform. Other security requirements, such as separation-of-duty, impose constraints on which subsets of users may perform certain subsets of steps. The workflow satisfiability problem (WSP) is the problem of determining whether there exists an assignment of users to workflow steps that satisfies all such authorizations and constraints. An algorithm for solving WSP is important, both as a static analysis tool for workflow specifications, and for the construction of run-time reference monitors for workflow management systems. Given the computational difficulty of WSP, it is important, particularly for the second application, that such algorithms are as efficient as possible. We introduce class-independent constraints, enabling us to model scenarios where the set of users is partitioned into groups, and the identities of the user groups are irrelevant to the satisfaction of the constraint. We prove that solving WSP is fixed-parameter tractable (FPT) for this class of constraints and develop an FPT algorithm that is useful in practice. We compare the performance of the FPT algorithm with that of SAT4J (a pseudo-Boolean SAT solver) in computational experiments, which show that our algorithm significantly outperforms SAT4J for many instances of WSP. User-independent constraints, a large class of constraints including many practical ones, are a special case of class-independent constraints for which WSP was proved to be FPT (Cohen et al., J. Artif. Intel. Res. 2014). Thus our results considerably extend our knowledge of the fixed-parameter tractability of WSP.

The Workflow Satisfiability Problem (WSP) is a problem of practical interest that arises whenever tasks need to be performed by authorized users, subject to constraints defined by business rules. We are required to decide whether there exists a plan - an assignment of tasks to authorized users - such that all constraints are satisfied. It is natural to see the WSP as a subclass of the Constraint Satisfaction Problem (CSP) in which the variables are tasks and the domain is the set of users. What makes the WSP distinctive is that the number of tasks is usually very small compared to the number of users, so it is appropriate to ask for which constraint languages the WSP is fixed-parameter tractable (FPT), parameterized by the number of tasks. This novel approach to the WSP, using techniques from CSP, has enabled us to design a generic algorithm which is FPT for several families of workflow constraints considered in the literature. Furthermore, we prove that the union of FPT languages remains FPT if they satisfy a simple compatibility condition. Lastly, we identify a new FPT constraint language, user-independent constraints, that includes many of the constraints of interest in business processing systems. We demonstrate that our generic algorithm has provably optimal running time O*(2^(klog k)), for this language, where k is the number of tasks.

MOTIVATION: High-throughput screening (HTS) is an early-stage process in drug discovery which allows thousands of chemical compounds to be tested in a single study. We report a method for correcting HTS data prior to the hit selection process (i.e. selection of active compounds). The proposed correction minimizes the impact of systematic errors which may affect the hit selection in HTS. The introduced method, called a well correction, proceeds by correcting the distribution of measurements within wells of a given HTS assay. We use simulated and experimental data to illustrate the advantages of the new method compared to other widely-used methods of data correction and hit selection in HTS. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: High-throughput screening (HTS) plays a central role in modern drug discovery, allowing for testing of >100,000 compounds per screen. The aim of our work was to develop and implement methods for minimizing the impact of systematic error in the analysis of HTS data. To the best of our knowledge, two new data correction methods included in HTS-Corrector are not available in any existing commercial software or freeware. RESULTS: This paper describes HTS-Corrector, a software application for the analysis of HTS data, detection and visualization of systematic error, and corresponding correction of HTS signals. Three new methods for the statistical analysis and correction of raw HTS data are included in HTS-Corrector: background evaluation, well correction and hit-sigma distribution procedures intended to minimize the impact of systematic errors. We discuss the main features of HTS-Corrector and demonstrate the benefits of the algorithms.