Linked Tree-Decompositions of Infinite Represented Matroids

<p>It is natural to try to extend the results of Robertson and Seymour's Graph Minors Project to other objects. As linked tree-decompositions (LTDs) of graphs played a key role in the Graph Minors Project, establishing the existence of ltds of other objects is a useful step towards such extensions. There has been progress in this direction for both infinite graphs and matroids. Kris and Thomas proved that infinite graphs of finite tree-width have LTDs. More recently, Geelen, Gerards and Whittle proved that matroids have linked branch-decompositions, which are similar to LTDs. These results suggest that infinite matroids of finite treewidth should have LTDs. We answer this conjecture affirmatively for the representable case. Specifically, an independence space is an infinite matroid, and a point configuration (hereafter configuration) is a represented independence space. It is shown that every configuration having tree-width has an LTD k E w (kappa element of omega) of width at most 2k. Configuration analogues for bridges of X (also called connected components modulo X) and chordality in graphs are introduced to prove this result. A correspondence is established between chordal configurations only containing subspaces of dimension at most k E w (kappa element of omega) and configuration tree-decompositions having width at most k. This correspondence is used to characterise finite-width LTDs of configurations by their local structure, enabling the proof of the existence result. The theory developed is also used to show compactness of configuration tree-width: a configuration has tree-width at most k E w (kappa element of omega) if and only if each of its finite subconfigurations has tree-width at most k E w (kappa element of omega). The existence of LTDs for configurations having finite tree-width opens the possibility of well-quasi-ordering (or even better-quasi-ordering) by minors those independence spaces representable over a fixed finite field and having bounded tree-width.</p>

Linked Tree-Decompositions of Infinite Represented Matroids

<p>It is natural to try to extend the results of Robertson and Seymour's Graph Minors Project to other objects. As linked tree-decompositions (LTDs) of graphs played a key role in the Graph Minors Project, establishing the existence of ltds of other objects is a useful step towards such extensions. There has been progress in this direction for both infinite graphs and matroids. Kris and Thomas proved that infinite graphs of finite tree-width have LTDs. More recently, Geelen, Gerards and Whittle proved that matroids have linked branch-decompositions, which are similar to LTDs. These results suggest that infinite matroids of finite treewidth should have LTDs. We answer this conjecture affirmatively for the representable case. Specifically, an independence space is an infinite matroid, and a point configuration (hereafter configuration) is a represented independence space. It is shown that every configuration having tree-width has an LTD k E w (kappa element of omega) of width at most 2k. Configuration analogues for bridges of X (also called connected components modulo X) and chordality in graphs are introduced to prove this result. A correspondence is established between chordal configurations only containing subspaces of dimension at most k E w (kappa element of omega) and configuration tree-decompositions having width at most k. This correspondence is used to characterise finite-width LTDs of configurations by their local structure, enabling the proof of the existence result. The theory developed is also used to show compactness of configuration tree-width: a configuration has tree-width at most k E w (kappa element of omega) if and only if each of its finite subconfigurations has tree-width at most k E w (kappa element of omega). The existence of LTDs for configurations having finite tree-width opens the possibility of well-quasi-ordering (or even better-quasi-ordering) by minors those independence spaces representable over a fixed finite field and having bounded tree-width.</p>

A SAT Approach to Branchwidth

Branch decomposition is a prominent method for structurally decomposing a graph, hypergraph or CNF formula. The width of a branch decomposition provides a measure of how well the object is decomposed. For many applications it is crucial to compute a branch decomposition whose width is as small as possible. We propose a SAT approach to finding branch decompositions of small width. The core of our approach is an efficient SAT encoding which determines with a single SAT-call whether a given hypergraph admits a branch decomposition of certain width. For our encoding we develop a novel partition-based characterization of branch decompositions. The encoding size imposes a limit on the size of the given hypergraph. In order to break through this barrier and to scale the SAT approach to larger instances, we develop a new heuristic approach where the SAT encoding is used to locally improve a given candidate decomposition until a fixed-point is reached. This new method scales now to instances with several thousands of vertices and edges.

Hierarchical decomposition of metabolic networks using k-modules

The optimal solutions obtained by flux balance analysis (FBA) are typically not unique. Flux modules have recently been shown to be a very useful tool to simplify and decompose the space of FBA-optimal solutions. Since yield-maximization is sometimes not the primary objective encountered in vivo, we are also interested in understanding the space of sub-optimal solutions. Unfortunately, the flux modules are too restrictive and not suited for this task. We present a generalization, called k-module, which compensates the limited applicability of flux modules to the space of sub-optimal solutions. Intuitively, a k-module is a sub-network with low connectivity to the rest of the network. Recursive application of k-modules yields a hierarchical decomposition of the metabolic network, which is also known as branch decomposition in matroid theory. In particular, decompositions computed by existing methods, like the null-space-based approach, introduced by Poolman et al. [(2007) J. Theor. Biol. 249, 691–705] can be interpreted as branch decompositions. With k-modules we can now compare alternative decompositions of metabolic networks to the classical sub-systems of glycolysis, tricarboxylic acid (TCA) cycle, etc. They can be used to speed up algorithmic problems [theoretically shown for elementary flux modes (EFM) enumeration] and have the potential to present computational solutions in a more intuitive way independently from the classical sub-systems.