In the multiterminal cut problem one is given an edge-weighted graph and a subset of the vertices called terminals, and is asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the mincut, max-flow problem, and can be solved in polynomial time. It is shown that the problem becomes NP-hard as soon as $k = 3$, but can be solved in polynomial time for planar graphs for any fixed k. The planar problem is NP-hard, however, if k is not fixed. A simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of ${{2 - 2} / k}$ of the optimal cut weight is also described.

A fast parallel algorithm for the recognition of ultrametrics is presented. Its time-processor product is of the same order as the time bound of the known sequential algorithm of Culberson and Rudnicki [Inform. Process. Lett., 30 (1990), pp. 215–220] (compare also [SIAM J. Disc. Math., 3 (1990), pp. 1–6] and [Quart. Appl. Math., 26 (1968), pp. 607–609]. By the same way, tree metrics also can be recognized.

In the Multiway Cut problem we are given an edge-weighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the min-cut, max-flow problem, and can be solved in polynomial time. We show that the problem becomes NP-hard as soon as k = 3, but can be solved in polynomial time for planar graphs for any fixed k. The planar problem is NP-hard, however, if k is not fixed. We also describe a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2–2/k of the optimal cut weight.

The first efficient parallel algorithm for computing minimal elimination ordering (MEO) of an arbitrary graph is designed. The algorithm works in O(log/sup 3/n) parallel time and O(nm) processors on a concurrent-read-concurrent-write parallel random-access machine (CRCW PRAM) for an n-vertex, m-edge graph and is optimal up to polylogarithmic factor with respect to the best sequential algorithm of D. Rose et. al. (SIAM J. Comput., vol.5, p.266-83, 1976). As an application, the first efficient parallel solution to the problem of minimal fill-in for arbitrary graphs is given. The method of solution involves the development of new techniques for solving the connected minimal set system problem and combining them with some new divide-and-conquer methods.>

G.A. Dirac's classical theorem (1952) asserts that if every vertex of a graph G on n vertices has degree at least n/2, the G has a Hamiltonian cycle. A fast parallel algorithm on a concurrent-read-exclusive-write parallel random-access machine (CREW PRAM) is given to find a Hamiltonian cycle in such graphs. The algorithm uses a linear number of processors and is optimal up to a polylogarithmic factor. It works in O(log/sup 4/n) parallel time and uses linear number of processors on a CREW PRAM. It is also proved that a perfect matching in dense graphs can be found in NC/sup 2/. The cost of improved time is a quadratic number of processors. It is also proved that finding an NC algorithm for perfect matching in slightly less dense graphs is as hard as the same problem for all graphs, and the problem of finding a Hamiltonian cycle becomes NP-complete.>