No abstract

Let H = (T, U ) be a connected graph, V ⊇ T a set, and c a non-negative function on the unordered pairs of elements of V . In the minimum 0-extension problem ( * ), one is asked to minimize the inner product c · m over all metrics m on V such that (i) m coincides with the distance function of H within T ; and (ii) each v ∈ V is at zero distance from some s ∈ T , i.e. m (v, s) This problem is known to be NP-hard if H = K 3 (as being equivalent to the minimum 3-terminal cut problem), while it is polynomially solvable if H = K 2 (the minimum cut problem) or H = K 2,r (the minimum (2, r )-metric problem). We study problem ( * ) for all fixed H . More precisely, we consider the linear programming relaxation ( * * ) of ( * ) that is obtained by dropping condition (ii) above, and call H minimizable if the minima in ( * ) and ( * * ) coincide for all V and c. Note that for such an H problem ( * ) is solvable in strongly polynomial time.Our main theorem asserts that H is minimizable if and only if H is bipartite, has no isometric circuit with six or more nodes, and is orientable in the sense that H can be oriented so that nonadjacent edges of any 4-circuit are oppositely directed along this circuit. The proof is based on a combinatorial and topological study of tight and extreme extensions of graph metrics.Based on the idea of the proof of the NP-hardness for the minimum 3-terminal cut problem in [4], we then show that the minimum 0-extension problem is strongly NP-hard for many non-minimizable graphs H . Other results are also presented.

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We propose a method to construct "large" Condorcet domains by use of socalled rhombus tilings. Then we explain that this method fits to unify several previously known constructions of Condorcet domains. Finally, we discuss some conjectures on the size of such domains.

We study path problems in skew-symmetric graphs. These problems generalize the standard graph reachability and shortest path problems. We establish combinatorial solvability criteria and duality relations for the skew-symmetric path problems and use them to design efficient algorithms for these problems. The algorithms presented are competitive with the fastest algorithms for the standard problems.