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Online hypergraph coloring with rejection. (English) Zbl 1334.68305
Summary: 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 \(v_1,\dots,v_n\) and it must decide about \(v_i\) by only looking at the subhypergraph \(H_i = (V_i,E_i)\) where \(V_i=\{v_1,\dots,v_i\}\) 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.
MSC:
68W27 Online algorithms; streaming algorithms
05C15 Coloring of graphs and hypergraphs
05C65 Hypergraphs
05C85 Graph algorithms (graph-theoretic aspects)
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