Stable Fractional Matchings

We study a generalization of the classical stable matching problem that allows for cardinal preferences (as opposed to ordinal) and fractional matchings (as opposed to integral). After observing that, in this cardinal setting, stable fractional matchings can have much higher social welfare than stable integral ones, our goal is to understand the computational complexity of finding an optimal (i.e., welfare-maximizing) or nearly-optimal stable fractional matching. We present simple approximation algorithms for this problem with weak welfare guarantees and, rather unexpectedly, we furthermore show that achieving better approximations is hard. This computational hardness persists even for approximate stability. To the best of our knowledge, these are the first computational complexity results for stable fractional matchings. En route to these results, we provide a number of structural observations

A new iterative criterion for H-matrices: The reducible case

H-matrices appear in various areas of science and engineering and it is of vital importance to have an Algorithm to identify the H-matrix character of a certain matrix A epsilon C-n,C-n. Recently, the present authors have proposed a new iterative criterion (Algorithm AH) to completely identify the H-matrix property of an irreducible matrix. The present work extends the previous Algorithm to cover the reducible case as well. (C) 2008 Elsevier Inc. All rights reserved