Exact Localisations of Feedback Sets

The feedback arc (vertex) set problem, shortened FASP (FVSP), is to transform a given multi digraph $G=(V,E)$ into an acyclic graph by deleting as few arcs (vertices) as possible. Due to the results of Richard M. Karp in 1972 it is one of the classic NP-complete problems. An important contribution of this paper is that the subgraphs $G_{\mathrm{el}}(e)$, $G_{\mathrm{si}}(e)$ of all elementary cycles or simple cycles running through some arc $e \in E$, can be computed in $\mathcal{O}\big(|E|^2\big)$ and $\mathcal{O}(|E|^4)$, respectively. We use this fact and introduce the notion of the essential minor and isolated cycles, which yield a priori problem size reductions and in the special case of so called resolvable graphs an exact solution in $\mathcal{O}(|V||E|^3)$. We show that weighted versions of the FASP and FVSP possess a Bellman decomposition, which yields exact solutions using a dynamic programming technique in times $\mathcal{O}\big(2^{m}|E|^4\log(|V|)\big)$ and $\mathcal{O}\big(2^{n}\Delta(G)^4|V|^4\log(|E|)\big)$, where $m \leq |E|-|V| +1$, $n \leq (\Delta(G)-1)|V|-|E| +1$, respectively. The parameters $m,n$ can be computed in $\mathcal{O}(|E|^3)$, $\mathcal{O}(\Delta(G)^3|V|^3)$, respectively and denote the maximal dimension of the cycle space of all appearing meta graphs, decoding the intersection behavior of the cycles. Consequently, $m,n$ equal zero if all meta graphs are trees. Moreover, we deliver several heuristics and discuss how to control their variation from the optimum. Summarizing, the presented results allow us to suggest a strategy for an implementation of a fast and accurate FASP/FVSP-SOLVER

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

Clash Resolution Optimization based on Component and Clash Dependent Networks

Effective coordination across multi-disciplines is crucial to make sure that the locations of building components meet physical and functional constraints. Building information modeling (BIM) has been increasingly applied for coordination and one of its most widely used applications is automatic clash detection. The realistic visualization function of BIM helps reduce ambiguity and expedites clash detection. However, many project participants criticize automatic clash detection, as many detected clashes are irrelevant with no significant impact on design or construction work, thereby decreasing the precision of clash results and the benefits of BIM. In addition, clash detection consists of discovering problems, but it does not entail solving these clashes. Even though some studies discussed automatic clash detection, they rarely discussed the dependence relationships between building components. However, a building is an inseparable whole, and the dependent relationships among building components propagate the impact of clashes. Relocating one object to correct one clash may result in other objects violating spatial constraints, which may directly cause new clashes or indirectly cause them through relocating other components. Therefore, figuring out the dependency among clash objects with peripheral building components is useful to optimizing clash solutions by avoiding change propagation. Algorithms are designed to automatically capture dependency relations from models to construct a component dependency network. The network is used as an input to distinguish irrelevant clashes for improving clash detection quality by analyzing the relations between clash components and the relations between clash components with their nearby components. The feasibility to harness the clash component network and graph theory are also explored to generate the clash component change list for minimizing clash change impact from a holistic perspective. In addition, this study demonstrates how to use BIM information to refine clash management, and specifically focus on designing a hybrid clash correction sequence to minimize potential iterative adjustments. The contributions of this study exist at three levels. The most straightforward contribution is that this research proposed a method to improve clash detection quality as well as to provide decision support for clash resolution, which can help project teams to focus on important clashes and improve design coordination efficiency. In addition, this research proposes a new perspective to view clashes, switching the clash management focus and inspiring researchers to focus on finding global optimal solutions for all clashes other than a single clash. The third level is that even though this research focuses on clash management, the optimization algorithms based on graph theory can be used in other interdependent systems to improve design and construction performance.Ph.D

Linear Orderings of Sparse Graphs

The Linear Ordering problem consists in finding a total ordering of the vertices of a directed graph such that the number of backward arcs, i.e., arcs whose heads precede their tails in the ordering, is minimized. A minimum set of backward arcs corresponds to an optimal solution to the equivalent Feedback Arc Set problem and forms a minimum Cycle Cover. Linear Ordering and Feedback Arc Set are classic NP-hard optimization problems and have a wide range of applications. Whereas both problems have been studied intensively on dense graphs and tournaments, not much is known about their structure and properties on sparser graphs. There are also only few approximative algorithms that give performance guarantees especially for graphs with bounded vertex degree. This thesis fills this gap in multiple respects: We establish necessary conditions for a linear ordering (and thereby also for a feedback arc set) to be optimal, which provide new and fine-grained insights into the combinatorial structure of the problem. From these, we derive a framework for polynomial-time algorithms that construct linear orderings which adhere to one or more of these conditions. The analysis of the linear orderings produced by these algorithms is especially tailored to graphs with bounded vertex degrees of three and four and improves on previously known upper bounds. Furthermore, the set of necessary conditions is used to implement exact and fast algorithms for the Linear Ordering problem on sparse graphs. In an experimental evaluation, we finally show that the property-enforcing algorithms produce linear orderings that are very close to the optimum and that the exact representative delivers solutions in a timely manner also in practice. As an additional benefit, our results can be applied to the Acyclic Subgraph problem, which is the complementary problem to Feedback Arc Set, and provide insights into the dual problem of Feedback Arc Set, the Arc-Disjoint Cycles problem