Algorithms and Models for the On-Line Vertex-Covering

Abstract

In on-line computation, the instance of a problem is revealed step-by-step and one has, at the end of each step, to irrevocably decide on the part of the final solution dealing with this step. We first study the minimum vertex-covering problem under two on-line models corresponding to two different ways vertices are revealed. The former one implies that the input-graph is revealed vertex-by-vertex. The second model implies that the input-graph is revealed per clusters, i.e., per induced subgraphs of the final graph. Under the cluster-model, we then relax the constraint that the choice of the part of the final solution dealing with each cluster has to be irrevocable, by allowing backtracking. We assume that one can change decisions upon a vertex membership of the final solution, this change implying, however, some cost depending on the number of the vertices changed. Finally we study simple model where instance is revealed edge-by-edge. Most of the results we present are tight and optimal, or asymptotically optimal.