Online hypergraph coloring with rejection

In this paper we investigate the online hypergraph coloring problem with rejection, where the algorithm is allowed to reject a vertex instead of coloring it but each vertex has a penalty which has to be paid if it is not colored. The goal is to minimize the sum of the number of the used colors for the accepted vertices and the total penalty paid for the rejected ones. We study the online problem which means that the algorithm receives the vertices of the hypergraph in some order v₁, . . . , v_n and it must decide about v_i by only looking at the subhypergraph H_i = (V_i, E_i) where V_i = {v₁, . . . , v_i} and E_i contains the edges of the hypergraph which are subsets of V_i. We consider two models: in the full edge model only the edges where each vertex is accepted must be well-colored, in the trace model the subsets of the edges formed by the accepted vertices must be well colored as well. We consider proper and conflict free colorings. We present in each cases optimal online algorithms in the sense that they achieve asymptotically the smallest possible competitive ratio.