Semidefinite Ranking on Graphs

Semidefinite Ranking on Graphs Shankar Vembu, Thomas Gärtner, and Stefan Wrobel 1 Fraunhofer IAIS, Schloß Birlinghoven, 53754 Sankt Augustin, Germany 2 Department of Computer Science III, University of Bonn, Germany {shankar.vembu, thomas.gaertner, stefan.wrobel}@iais.fraunhofer.de We consider the problem of ranking the vertices of an undirected graph given some preference relation. Without inconsistent preferences in the data, the preferences would form a partial order and we could aim at finding the linear extension that conforms best with the undirected graph. However, in real data there are also inconsistent preferences and hence we have to allow for a few backward edges. This ‘ranking on graphs’ problem has been tackled before using spectral relaxations. Recently, it has been shown that semidefinite relaxations offer in many cases better solutions than spectral ones for clustering [1] and transductive classification [2]. In this paper, we investigate semidefinite relaxations of ranking on graphs. We incorporate the preferences by fixing certain angles between the metric embedding of the vertices. The final linear extension is obtained by the random projection method. Experiments on real world data sets show the expected improvements over spectral relaxations. Several special cases of ‘ranking on graphs’ are well investigated problems like ‘topological sort’, ‘minimum feedback arc set’, and ‘minimum-length ordering’. The latter problems are known to be NP-hard. The best known approximation algorithms for ‘minimum-length ordering’ are based on semidefinite relaxations and random projection of the resulting point set. s s s s s s s s s s s s s s s s s s s s 1j 5