On the Parameterized Complexity of the Connected Flow and Many Visits TSP Problem
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We study a variant of Min Cost Flow in which the flow needs to be connected. Specifically, in the Connected Flow problem one is given a directed graph G, along with a set of demand vertices D ⊆ V(G) with demands 𝖽𝖾𝗆: D →ℕ, and costs and capacities for each edge. The goal is to find a minimum cost flow that satisfies the demands, respects the capacities and induces a (strongly) connected subgraph. This generalizes previously studied problems like the (Many Visits) TSP. We study the parameterized complexity of Connected Flow parameterized by |D|, the treewidth tw and by vertex cover size k of G and provide: (i) 𝖭𝖯-completeness already for the case |D|=2 with only unit demands and capacities and no edge costs, and fixed-parameter tractability if there are no capacities, (ii) a fixed-parameter tractable 𝒪^⋆(k^𝒪(k)) time algorithm for the general case, and a kernel of size polynomial in k for the special case of Many Visits TSP, (iii) a |V(G)|^𝒪(tw) time algorithm and a matching |V(G)|^o(tw) time conditional lower bound conditioned on the Exponential Time Hypothesis. To achieve some of our results, we significantly extend an approach by Kowalik et al. [ESA'20].
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