Finding Short Paths on Polytopes by the Shadow Vertex Algorithm

We show that the shadow vertex algorithm can be used to compute a short path between a given pair of vertices of a polytope P = {x : Ax \leq b} along the edges of P, where A \in R^{m \times n} is a real-valued matrix. Both, the length of the path and the running time of the algorithm, are polynomial in m, n, and a parameter 1/delta that is a measure for the flatness of the vertices of P. For integer matrices A \in Z^{m \times n} we show a connection between delta and the largest absolute value Delta of any sub-determinant of A, yielding a bound of O(Delta^4 m n^4) for the length of the computed path. This bound is expressed in the same parameter Delta as the recent non-constructive bound of O(Delta^2 n^4 \log (n Delta)) by Bonifas et al. For the special case of totally unimodular matrices, the length of the computed path simplifies to O(m n^4), which significantly improves the previously best known constructive bound of O(m^{16} n^3 \log^3(mn)) by Dyer and Frieze

Output-sensitive complexity of multiobjective combinatorial optimization

We study output-sensitive algorithms and complexity for multiobjective combinatorial optimization problems. In this computational complexity framework, an algorithm for a general enumeration problem is regarded efficient if it is output-sensitive, i.e., its running time is bounded by a polynomial in the input and the output size. We provide both practical examples of MOCO problems for which such an efficient algorithm exists as well as problems for which no efficient algorithm exists under mild complexity theoretic assumptions

Exact analysis for requirements selection and optimisation

Requirements engineering is the prerequisite of software engineering, and plays a crit- ically strategic role in the success of software development. Insufficient management of uncertainty in the requirements engineering process has been recognised as a key reason for software project failure. The essence of uncertainty may arise from partially observable, stochastic environments, or ignorance. To ease the impact of uncertainty in the software development process, it is important to provide techniques that explicitly manage uncertainty in requirements selection and optimisation. This thesis presents a decision support framework to exactly address the uncertainty in requirements selection and optimisation. Three types of uncertainty are managed. They are requirements uncertainty, algorithmic uncertainty, and uncertainty of resource constraints. Firstly, a probabilistic robust optimisation model is introduced to enable the manageability of requirements uncertainty. Requirements uncertainty is probabilis- tically simulated by Monte-Carlo Simulation and then formulated as one of the opti- misation objectives. Secondly, a probabilistic uncertainty analysis and a quantitative analysis sub-framework METRO is designed to cater for requirements selection deci- sion support under uncertainty. An exact Non-dominated Sorting Conflict Graph based Dynamic Programming algorithm lies at the heart of METRO to guarantee the elim- ination of algorithmic uncertainty and the discovery of guaranteed optimal solutions. Consequently, any information loss due to algorithmic uncertainty can be completely avoided. Moreover, a data analytic approach is integrated in METRO to help the deci- sion maker to understand the remaining requirements uncertainty propagation through- out the requirements selection process, and to interpret the analysis results. Finally, a more generic exact multi-objective integrated release and schedule planning approach iRASPA is introduced to holistically manage the uncertainty of resource constraints for requirements selection and optimisation. Software release and schedule plans are inte- grated into a single activity and solved simultaneously. Accordingly, a more advanced globally optimal result can be produced by accommodating and managing the inherent additional uncertainty due to resource constraints as well as that due to requirements. To settle the algorithmic uncertainty problem and guarantee the exactness of results, an ε-constraint Quadratic Programming approach is used in iRASPA