Topological invariants of classification problems

AbstractThere is a general agreement that problems which are highly complex in any naive sense are also difficult from the computational point of view. It is therefore of great interest to find invariants and invariant structures which measure in some respect the complexity of the given problem. The question which we are going to consider in the following paper are classification problems, the “computations” are described by questionnaires [3, 10] or, as they are called nowadays, by “branching programs” [11]. The “complexity” of the problem is measured by classical topological invariants (Betti numbers, Euler-Poincaré characteristic) of topological structures (simplicial complexes, topological spaces)

More efficient periodic traversal in anonymous undirected graphs

We consider the problem of periodic graph exploration in which a mobile entity with constant memory, an agent, has to visit all n nodes of an arbitrary undirected graph G in a periodic manner. Graphs are supposed to be anonymous, that is, nodes are unlabeled. However, while visiting a node, the robot has to distinguish between edges incident to it. For each node v the endpoints of the edges incident to v are uniquely identified by different integer labels called port numbers. We are interested in minimisation of the length of the exploration period. This problem is unsolvable if the local port numbers are set arbitrarily. However, surprisingly small periods can be achieved when assigning carefully the local port numbers. Dobrev et al. described an algorithm for assigning port numbers, and an oblivious agent (i.e. agent with no memory) using it, such that the agent explores all graphs of size n within period 10n. Providing the agent with a constant number of memory bits, the optimal length of the period was previously proved to be no more than 3.75n (using a different assignment of the port numbers). In this paper, we improve both these bounds. More precisely, we show a period of length at most 4 1/3 n for oblivious agents, and a period of length at most 3.5n for agents with constant memory. Moreover, we give the first non-trivial lower bound, 2.8n, on the period length for the oblivious case