Multipath Spanners

International audienceThis paper concerns graph spanners that approximate multipaths between pair of vertices of an undirected graphs with $n$ vertices. Classically, a spanner $H$ of stretch $s$ for a graph $G$ is a spanning subgraph such that the distance in $H$ between any two vertices is at most $s$ times the distance in $G$. We study in this paper spanners that approximate short cycles, and more generally $p$ edge-disjoint paths with $p>1$, between any pair of vertices. For every unweighted graph $G$, we construct a $2$-multipath $3$-spanner of $O(n^3/2)$ edges. In other words, for any two vertices $u,v$ of $G$, the length of the shortest cycle (with no edge replication) traversing $u,v$ in the spanner is at most thrice the length of the shortest one in $G$. This construction is shown to be optimal in term of stretch and of size. In a second construction, we produce a $2$-multipath $(2,8)$-spanner of $O(n^3/2)$ edges, i.e., the length of the shortest cycle traversing any two vertices have length at most twice the shortest length in $G$ plus eight. For arbitrary $p$, we observe that, for each integer $k\ge 1$, every weighted graph has a $p$-multipath $p(2k-1)$-spanner with $O(p n^1+1/k)$ edges, leaving open the question whether, with similar size, the stretch of the spanner can be reduced to $2k-1$ for all $p>1$