@misc{Bley, author = {Bley, Andreas}, title = {On the Complexity of Vertex-Disjoint Length-Restricted Path Problems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3639}, number = {SC-98-20}, abstract = {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.}, language = {en} }