Generalized paths and cycles in semicomplete multipartite digraphs

It is well-known and easy to show that even the following version of the directed travelling salesman problem is NP-complete: Given a strongly connected complete digraph $D=(V,A)$, a cost function $w: A\rightarrow \{0,1\}$ and a natural number $K$; decide whether $D$ has a directed Hamiltonian cycle of cost at most $K$. We study the following variant of this problem for $\{0,1\}$-weighted semicomplete digraphs where the set of arcs which have cost 1 form a collection of vertex-disjoint complete digraphs. A digraph is \textbf{semicomplete multipartite} if it can be obtained from a semicomplete digraph $D$ by choosing a collection of vertex-disjoint subsets $X_1,\ldots{},X_c$ of $V(D)$ and then deleting all arcs both of whose end-vertices lie inside some $X_i$. Let $D$ be a semicomplete digraph with a cost function $w$ as above, where $w(a)=1$ precisely when $a$ is an arc inside one of the subsets $X_1,\ldots{},X_c$ and let $D^*$ be the corresponding \smd{} that we obtain by deleting all arcs inside the $X_i$'s. Then every cycle $C$ of $D$ corresponds to a {\bf generalized cycle} $C^g$ of $D^*$ which is either the cycle $C$ itself if $w(C)=0$ or a collection of two or more paths that we obtain by deleting all arcs of cost 1 on $C$. Similarly we can define a {\bf generalized path} $P^g$ in a semicomplete multipartite digraph. The purpose of this paper is to study structural and algorithmic properties of generalized paths and cycles in semicomplete multipartite digraphs. This allows us to identify classes of directed $\{0,1\}$-weighted TSP instances that can be solved in polynomial time as well as others for which we can get very close to the optimum in polynomial time. Along with these results we also show that two natural questions about properties of cycles meeting all partite sets in semicomplete multipartite digraphs are NP-complete.