Finding branch-decompositions of matroids, hypergraphs, and more

Given $n$ subspaces of a finite-dimensional vector space over a fixed finite field $\mathcal F$, we wish to find a "branch-decomposition" of these subspaces of width at most $k$, that is a subcubic tree $T$ with $n$ leaves mapped bijectively to the subspaces such that for every edge $e$ of $T$, the sum of subspaces associated with leaves in one component of $T-e$ and the sum of subspaces associated with leaves in the other component have the intersection of dimension at most $k$. This problem includes the problems of computing branch-width of $\mathcal F$-represented matroids, rank-width of graphs, branch-width of hypergraphs, and carving-width of graphs. We present a fixed-parameter algorithm to construct such a branch-decomposition of width at most $k$, if it exists, for input subspaces of a finite-dimensional vector space over $\mathcal F$. Our algorithm is analogous to the algorithm of Bodlaender and Kloks (1996) on tree-width of graphs. To extend their framework to branch-decompositions of vector spaces, we developed highly generic tools for branch-decompositions on vector spaces. The only known previous fixed-parameter algorithm for branch-width of $\mathcal F$-represented matroids was due to Hlin\v{e}n\'y and Oum (2008) that runs in time $O(n^3)$ where $n$ is the number of elements of the input $\mathcal F$-represented matroid. But their method is highly indirect. Their algorithm uses the non-trivial fact by Geelen et al. (2003) that the number of forbidden minors is finite and uses the algorithm of Hlin\v{e}n\'y (2005) on checking monadic second-order formulas on $\mathcal F$-represented matroids of small branch-width. Our result does not depend on such a fact and is completely self-contained, and yet matches their asymptotic running time for each fixed $k$.Comment: 73 pages, 10 figure

Classification of the BPS states in Bagger-Lambert Theory

We classify, in a group theoretical manner, the BPS configurations in the multiple M2-brane theory recently proposed by Bagger and Lambert. We present three types of BPS equations preserving various fractions of supersymmetries: in the first type we have constant fields and the interactions are purely algebraic in nature; in the second type the equations are invariant under spatial rotation SO(2), and the fields can be time-dependent; in the third class the equations are invariant under boost SO(1,1) and provide the eleven-dimensional generalizations of the Nahm equations. The BPS equations for different number of supersymmetries exhibit the division algebra structures: octonion, quarternion or complex.Comment: 28+1 pages, No figure; v2 Sec.3.3 slightly expanded, typos fixed; v3 some comments added, to appear in JHE

Development of Navigation Control Algorithm for AGV Using D* Search Algorithm

In this paper, we present a navigation control algorithm for Automatic Guided Vehicles (AGV) that move in industrial environments including static and moving obstacles using D* algorithm. This algorithm has ability to get paths planning in unknown, partially known and changing environments efficiently. To apply the D* search algorithm, the grid map represent the known environment is generated. By using the laser scanner LMS-151 and laser navigation sensor NAV-200, the grid map is updated according to the changing of environment and obstacles. When the AGV finds some new map information such as new unknown obstacles, it adds the information to its map and re-plans a new shortest path from its current coordinates to the given goal coordinates. It repeats the process until it reaches the goal coordinates. This algorithm is verified through simulation and experiment. The simulation and experimental results show that the algorithm can be used to move the AGV successfully to reach the goal position while it avoids unknown moving and static obstacles. [Keywords— navigation control algorithm; Automatic Guided Vehicles (AGV); D* search algorithm