An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

Matroid matching : the power of local search

We consider the classical matroid matching problem. Unweighted matroid matching for linearly represented matroids was solved by Lovász, and the problem is known to be intractable for general matroids. We present a polynomial-time approximation scheme for unweighted matroid matching for general matroids. In contrast, we show that natural linear-programming relaxations that have been studied have an $\Omega(n)$ integrality gap, and, moreover, $\Omega(n)$ rounds of the Sherali--Adams hierarchy are necessary to bring the gap down to a constant. More generally, for any fixed $k \geq 2$ and $\epsilon>0$, we obtain a $(k/2+\epsilon)$-approximation for matroid matching in $k$-uniform hypergraphs, also known as the matroid $k$-parity problem. As a consequence, we obtain a $(k/2+\epsilon)$-approximation for the problem of finding the maximum-cardinality set in the intersection of $k$ matroids. We also give a $3/2$-approximation for the weighted version of a special case of matroid matching, the matchoid problem