Online Steiner Cover Problems in Hypergraphs

Abstract

The online Steiner cover problem in hypergraphs (\treePN) is a generalization of the online Steiner tree problem in graphs. An edge-weighted hypergraph $H = (V, E, w)$ is given offline and a set of terminal vertices $R\subseteq V$ is requested sequentially online. Upon receiving each request $r_i\in R$ an algorithm for the problem must buy some edges $P_i\subseteq E$ which connect $r_i$ to the previous solution. The solution after satisfying the $i^{\text{th}}$ request is then the union over all edges bought up to that point, $F_i = \bigcup_{j = 1}^i P_j$. The goal is to minimize the total cost of the solution to connect the requests, i.e., for $|R| = n$ let $F = F_n$, then we want to minimize $\sum_{e\in F}w(e)$. The generalized \treePN (\forestPN) is a generalization of both the \treePN and the Steiner forest problem in graphs. Again, we are given an edge-weighted hypergraph $H = (V, E, w)$ offline, but instead of a set of terminals as the online portion of the input we are given a set of terminal pairs $R\subseteq V\times V$. Upon receiving the $i^{\text{th}}$ request pair $p_i\in R$ an algorithm for this second problem must buy some set of edges $P_i\subset E$ which connect the terminals from $p_i$. We define the instantaneous solution after connecting request $p_i$ as before, so $F_i = \bigcup_{j = 1}^i P_i$ and the final solution is again denoted $F = F_n$ for a request sequence of size $|R|=n$. The worst-case performance of an online algorithm is measured by the competitive ratio, which is the ratio between the cost of a solution obtained by the online algorithm to that of an optimal offline solution. Besides some simpler preliminary results, we obtain a lower bound on the competitive ratio for \treePN (which also applies to \forestPN) of $\Omega(k\log{n})$, where $k$ is the rank of the hypergraph, and a matching upper bound for the simple $Greedy$ algorithm. For \forestPN we show that the simple $Greedy$ algorithm is $O(k\log^2{n})$-competitive and provide another algorithm, called $GGSC$, which achieves a competitive ratio of $O(k\log{n})$