Polynomial algorithms for sparse spanners on subcubic graphs

Abstract

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.