The Steiner Shortest Path Tree Problem

We introduce and study a novel problem of computing a shortest path tree with a minimum number of non-terminals. It can be viewed as an (unweighted) Steiner Shortest Path Tree (SSPT) that spans a given set of terminal vertices by shortest paths from a given source while minimizing the number of nonterminal vertices included in the tree. This problem is motivated by applications where shortest-path connections from a source are essential, and where reducing the number of intermediate vertices helps limit cost, complexity, or overhead. We show that the SSPT problem is NP-hard. To approximate it, we introduce and study the shortest path subgraph of a graph. Using it, we show an approximation-preserving reduction of SSPT to the uniform vertex-weighted variant of the Directed Steiner Tree (DST) problem, termed UVDST. Consequently, the algorithm of Grandoni et al., 2023 approximating DST implies a quasi-polynomial polylog-approximation algorithm for SSPT. We present a polynomial polylog-approximation algorithm for UVDST, and thus for SSPT, for a restricted class of graphs.