On an Online Spanning Tree Problem in Randomly Weighted Graphs

This paper is devoted to an online variant of the minimum spanning tree problem in randomly weighted graphs. We assume that the input graph is complete and the edge weights are uniform distributed over [0, 1]. An algorithm receives the edges one by one and has to decide immediately whether to include the current edge into the spanning tree or to reject it. The corresponding edge sequence is determined by some adversary. We propose an algorithm which achieves E [ALG] /E [OPT] = O (1) and E [ALG/OPT] = O (1) against a fair adaptive adversary, i.e., an adversary which determines the edge order online and is fair in a sense that he does not know more about the edge weights than the algorithm. Furthermore, we prove that no online algorithm performs better than E [ALG] /E [OPT] =# (log n) if the adversary knows the edge weights in advance. This lower bound is tight, since there is an algorithm which yields E [ALG] /E [OPT] = O (log n) against the strongest imaginable adversary. 1

FAST LEARNING ON GRAPHS

We carry out a systematic study of classification problems on networked data, presenting novel techniques with good performance both in theory and in practice. We assess the power of node classification based on class-linkage information only. In particular, we propose four new algorithms that exploit the homiphilic bias (linked entities tend to belong to the same class) in different ways. The set of the algorithms we present covers diverse practical needs: some of them operate in an active transductive setting and others in an on-line transductive setting. A third group works within an explorative protocol, in which the vertices of an unknown graph are progressively revealed to the learner in an on-line fashion. Within the mistake bound learning model, for each of our algorithms we provide a rigorous theoretical analysis, together with an interpretation of the obtained performance bounds. We also design adversarial strategies achieving matching lower bounds. In particular, we prove optimality for all input graphs and for all fixed regularity values of suitable labeling complexity measures. We also analyze the computational requirements of our methods, showing that our algorithms can to handle very large data sets. In the case of the on-line protocol, for which we exhibit an optimal algorithm with constant amortized time per prediction, we validate our theoretical results carrying out experiments on real-world datasets

Online Optimization with Lookahead

The main contributions of this thesis consist of the development of a systematic groundwork for comprehensive performance evaluation of algorithms in online optimization with lookahead and the subsequent validation of the presented approaches in theoretical analysis and computational experiments