Large induced subgraphs via triangulations and CMSO

We obtain an algorithmic meta-theorem for the following optimization problem. Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an integer. For a given graph G, the task is to maximize |X| subject to the following: there is a set of vertices F of G, containing X, such that the subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X) models \phi. Some special cases of this optimization problem are the following generic examples. Each of these cases contains various problems as a special subcase: 1) "Maximum induced subgraph with at most l copies of cycles of length 0 modulo m", where for fixed nonnegative integers m and l, the task is to find a maximum induced subgraph of a given graph with at most l vertex-disjoint cycles of length 0 modulo m. 2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\ containing a planar graph, the task is to find a maximum induced subgraph of a given graph containing no graph from \Gamma\ as a minor. 3) "Independent \Pi-packing", where for a fixed finite set of connected graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G with the maximum number of connected components, such that each connected component of G[F] is isomorphic to some graph from \Pi. We give an algorithm solving the optimization problem on an n-vertex graph G in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential maximal cliques in G and f is a function depending of t and \phi\ only. We also show how a similar running time can be obtained for the weighted version of the problem. Pipelined with known bounds on the number of potential maximal cliques, we deduce that our optimization problem can be solved in time O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with polynomial number of minimal separators

Convex Polytopes: Extremal Constructions and f-Vector Shapes

These lecture notes treat some current aspects of two closely interrelated topics from the theory of convex polytopes: the shapes of f-vectors, and extremal constructions. The first lecture treats 3-dimensional polytopes; it includes a complete proof of the Koebe--Andreev--Thurston theorem, using the variational principle by Bobenko & Springborn (2004). In Lecture 2 we look at f-vector shapes of very high-dimensional polytopes. The third lecture explains a surprisingly simple construction for 2-simple 2-simplicial 4-polytopes, which have symmetric f-vectors. Lecture 4 sketches the geometry of the cone of f-vectors for 4-polytopes, and thus identifies the existence/construction of 4-polytopes of high ``fatness'' as a key problem. In this direction, the last lecture presents a very recent construction of ``projected products of polygons,'' whose fatness reaches 9-\eps.Comment: 73 pages, large file. Lecture Notes for PCMI Summer Course, Park City, Utah, 2004; revised and slightly updated final version, December 200

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Autonomous mobile robot navigation using fuzzy logic control

Traditionally the type of robot used in the workplace consisted mainly o f the fixed arm variety. Any mobile robots that were commercially available required that the environment be altered to accommodate them. This involved the installation of guide lanes or some form of sensor units placed at various locations around the workplace to facilitate the robot in determining its position within the environment. Such approaches are costly and limit the use of robots to environments where these methods are feasible. The inadequacies in this technology has led to research into autonomous mobile robots that offer greater flexibility and do not require changes in the enviromnent. There are many technical issues to be addressed in designing such a robot. These stem from the necessity that the robot must be able to navigate through an environment unaided. Other problems such as the cost of the vehicle must be considered so that prospective customers will not be put off. This thesis discusses the strategies taken in addressing the problems associated with navigation in an obstacle strewn environment. Such issues include position estimation, path planning, obstacle avoidance and the acquisition and interpretation of sensor information. It also discusses the suitability of fuzzy logic for controlling a robot. A graphical user interface runs on the PC which communicates with the robot over a radio link. The robot uses a fuzzy logic controller to follow a planned path and avoid unknown obstacles by controlling the velocity and steering angle o f the drive unit. It is a tracked vehicle which is suitable for indoor use only. The results of path planning and the robots attempts at following the paths and avoiding obstacles are illustrated and discussed

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum