Abstract
We consider a framework for bi-objective network construction problems where one objective is to be maximized while the other is to be minimized. Given a host graph G = (V,E) with edge weights w e ∈ ℤ and edge lengths ℓ e ∈ ℕ for e ∈ E we define the density of a pattern subgraph H = (V′,E′) ⊆ G as the ratio \(\ensuremath{\varrho}(H)=\sum_{e\in E'}w_e/\sum_{e\in E'} \ell_e\). We consider the problem of computing a maximum density pattern H with weight at least W and and length at most L in a host G.
We consider this problem for different classes of hosts and patterns. We show that it is NP-hard even if the host has treewidth 2 and the pattern is a path. However, it can be solved in pseudo-polynomial linear time if the host has bounded treewidth and the pattern is a graph from a given minor-closed family of graphs. Finally, we present an FPTAS for a relaxation of the density maximization problem, in which we are allowed to violate the upper bound on the length at the cost of some penalty.
Supported by NSC-DFG Projects NSC98-2221-E-001-007-MY3 and WA 654/18.
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Kao, MJ., Katz, B., Krug, M., Lee, D.T., Rutter, I., Wagner, D. (2011). The Density Maximization Problem in Graphs. In: Fu, B., Du, DZ. (eds) Computing and Combinatorics. COCOON 2011. Lecture Notes in Computer Science, vol 6842. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22685-4_3
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DOI: https://doi.org/10.1007/978-3-642-22685-4_3
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