TY - GEN A1 - Bley, Andreas T1 - On the Complexity of Vertex-Disjoint Length-Restricted Path Problems N2 - Let $G=(V,E)$ be a simple graph and $s$ and $t$ be two distinct vertices of $G$. A path in $G$ is called $\ell$-bounded for some $\ell\in\mathbb{N}$, if it does not contain more than $\ell$ edges. We study the computational complexity of approximating the optimum value for two optimization problems of finding sets of vertex-disjoint $\ell$-bounded $s,t$-paths in $G$. First, we show that computing the maximum number of vertex-disjoint $\ell$-bounded $s,t$-paths is $\mathcal{AP\kern-1pt X}$--complete for any fixed length bound $\ell\geq 5$. Second, for a given number $k\in\mathbb{N}$, $1\leq k \leq |V|-1$, and non-negative weights on the edges of $G$, the problem of finding $k$ vertex-disjoint $\ell$-bounded $s,t$-paths with minimal total weight is proven to be $\mathcal{NPO}$--complete for any length bound $\ell\geq 5$. Furthermore, we show that, even if $G$ is complete, it is $\mathcal{NP}$--complete to approximate the optimal solution value of this problem within a factor of $2^{\langle\phi\rangle^\epsilon}$ for any constant $0<\epsilon<1$, where $\langle\phi\rangle$ denotes the encoding size of the given problem instance $\phi$. We prove that these results are tight in the sense that for lengths $\ell\leq 4$ both problems are polynomially solvable, assuming that the weights satisfy a generalized triangle inequality in the weighted problem. All results presented also hold for directed and non-simple graphs. For the analogous problems where the path length restriction is replaced by the condition that all paths must have length equal to $\ell$ or where vertex-disjointness is replaced by edge-disjointness we obtain similar results. T3 - ZIB-Report - SC-98-20 KW - disjoint paths KW - length bounded paths KW - approximation KW - reducibility KW - completeness Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3639 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Karbstein, Marika T1 - A Primal-Dual Approximation Algorithm for the Steiner Connectivity Problem N2 - We extend the primal-dual approximation technique of Goemans and Williamson to the Steiner connectivity problem, a kind of Steiner tree problem in hypergraphs. This yields a (k+1)-approximation algorithm for the case that k is the minimum of the maximal number of nodes in a hyperedge minus 1 and the maximal number of terminal nodes in a hyperedge. These results require the proof of a degree property for terminal nodes in hypergraphs which generalizes the well-known graph property that the average degree of terminal nodes in Steiner trees is at most 2. T3 - ZIB-Report - 13-54 KW - Primal-Dual Approximation KW - Steiner Connectivity Problem KW - Degree Property KW - Hypergraph Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-42430 SN - 1438-0064 ER -