Alternating Reachability

We consider a graph with colored edges. A trail (vertices may repeat but not edges) is called \emph{alternating} when successive edges have different colors. Given a set of vertices called \emph{terminals}, the \emph{alternating reachability} problem is to find an alternating trail connecting distinct terminals, if one exists. A special case with two colors is searching for an augmenting path with respect to a given matching. In another special case with two colors red and blue, the \emph{alternating cone} is defined as the set of assignments of nonnegative weights to the edges such that at each vertex, the total red weight equals the total blue weight; in a companion paper we showed how the search for an integral weight vector within a given box in the alternating cone can be reduced to the alternating reachability problem in a 2-colored graph. We define an obstacle, called a \emph{Tutte set}, to the existence of an alternating trail connecting distinct terminals in a colored graph, and give a polynomial-time algorithm, generalizing the blossom algorithm of Edmonds, that finds either an alternating trail connecting distinct terminals or a Tutte set. We use Tutte sets to show that an an edge-colored bridgeless graph where each vertex has incident edges of at least two different colors has a closed alternating trail. A special case with two colors one of which forms a matching yields a combinatorial result of Giles and Seymour. We show that in a 2-colored graph, the cone generated by the characteristic vectors of closed alternating trails is the intersection of the alternating cone with the cone generated by the characteristic vectors of cycles in the underlying graph.