Abstract
Given a graph G = (V,E) with an edge cost and families \(\mathcal{V}_i\subseteq 2^V\), i = 1,2,...,m of disjoint subsets, an edge subset F ⊆ E is called a set connector if, for each \(\mathcal{V}_i\), the graph \((V,F)/\mathcal{V}_i\) obtained from (V,F) by contracting each \(X\in \mathcal{V}_i\) into a single vertex x has a property that every two contracted vertices x and x′ are connected in \((V,F)/\mathcal{V}_i\). In this paper, we introduce a problem of finding a minimum cost set connector, which contains several important network design problems such as the Steiner forest problem, the group Steiner tree problem, and the NA-connectivity augmentation problem as its special cases. We derive an approximate integer decomposition property from a fractional packing theorem of set connectors, and present a strongly polynomial 2α-approximation algorithm for the set connector problem, where \(\alpha=\max_{1 \leq i \leq m}(\sum_{X \in \mathcal{V}_i}|X|)-1\).
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Fukunaga, T., Nagamochi, H. (2007). The Set Connector Problem in Graphs. In: Fischetti, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2007. Lecture Notes in Computer Science, vol 4513. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72792-7_36
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DOI: https://doi.org/10.1007/978-3-540-72792-7_36
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