**Abstract** : Consider a graph G=(V,E) with n vertices, m
edges and a positive weight (or capacity) on each edge. Let X be a set of vertices in V called terminals. The multiterminal (or
multiway) cut problem is to find a minimum weight set of edges that disconnects each pair of terminals. The multiterminal cut
problem is a special case of the multicut problem where one wants to separate K given pairs of vertices. Let us recall the duality
relationship linking both multicut and multicommodity flow problems and then associate a commodity with each vertex pair in X.
Now we consider the multiterminal flow problem associated to the multiterminal cut problem: it consists in maximizing the total
amount of flow routed between any pair of nodes in X. The multicut and integer multiflow problems are polynomial in directed
trees but are NP-hard in undirected trees. On the other hand, the muliterminal cut problem, which is NP-hard for K2 in
unrestricted graphs becomes polynomial if the graph is a tree. P.L. Erdos and L.A. Szekely proposed an O(n2)algorithm to solve
the problem in undirected trees. In fact their algorithm solves a more general problem which is to separate r disjoint subsets of
vertices.
Here, we present a procedure in O(n) to solve the multiterminal cut problem in undirected trees and a procedure in O(n2)
to solve the multiterminal flow problem in undirected trees and we show that most often there exists a
duality gap between the optimal integer multiterminal cut and flow values. Both algorithms are independant but their general
schemes are similar: we first solve the problems on trees of height 1, i.e. stars. Then we consider a star connected to the tree by
an only edge; we give a sub-solution on this star and reduce the tree before applying the same method recursively until we obtain
a lonely star.
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