Polynomial algorithms for sparse spanners on subcubic graphs

Let G be a connected graph and \(t \ge 1\) a (rational) constant. A t-spanner of G is a spanning subgraph of G in which the distance between any pair of vertices is at most t times its distance in G. We address two problems on spanners. The first one, known as the minimum t-spanner problem (MinS \(_t\)), seeks in a connected graph a t-spanner with the smallest possible number of edges. In the second one, called minimum cost tree t-spanner problem (MCTS \(_t\)), the input graph has costs assigned to its edges and seeks a t-spanner that is a tree with minimum cost. It is an optimization version of the tree t-spanner problem (TreeS \(_t\)), a decision problem concerning the existence of a t-spanner that is a tree. MinS \(_t\) is known to be \({\textsc {NP}}\)-hard for every \(t \ge 2\). On the other hand, TreeS \(_t\) admits a polynomial-time algorithm for \(t \le 2\) and is \({\textsc {NP}}\)-complete for \(t \ge 4\); but its complexity for \(t=3\) remains open. We focus on the class of subcubic graphs. First, we show that for such graphs MinS \(_3\) can be solved in polynomial time. These results yield a practical polynomial algorithm for TreeS \(_3\) that is of a combinatorial nature. We also show that MCTS \(_2\) can be solved in polynomial time. To obtain this last result, we prove a complete linear characterization of the polytope defined by the incidence vectors of the tree 2-spanners of a subcubic graph. A recent result showing that MinS \(_3\) on graphs with maximum degree at most 5 is NP-hard, together with the current result on subcubic graphs, leaves open only the complexity of MinS \(_3\) on graphs with maximum degree 4.