Boolean dimension and tree-width

The dimension of a partially ordered set P is an extensively studied parameter. Small dimension allows succinct encoding. Indeed if P has dimension d, then to know whether x < y in P it is enough to check whether x < y in each of the d linear extensions of a witnessing realizer. Focusing on the encoding aspect Nešetřil and Pudlák defined the boolean dimension so that P has boolean dimension at most d if it is possible to decide whether x < y in P by looking at the relative position of x and y in only d permutations of the elements of P . Our main result is that posets with cover graphs of bounded tree-width have bounded boolean dimension. This stays in contrast with the fact that there are posets with cover graphs of tree-width three and arbitrarily large dimension. As a corollary, we obtain a labeling scheme of size log(n) for reachability queries on n-vertex digraphs with bounded tree-width. Our techniques seem to be very different from the usual approach for labeling schemes which is based on divide-and-conquer decompositions of the input graphs. (S. Felsner) Institut für Mathematik, Technische Universität Berlin, (T. Mészáros) Institut für Mathematik, Freie Universität Berlin, (P. Micek) Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków and Freie Universität Berlin, E-mail addresses: [email protected], [email protected], [email protected].