The Complexity of the Simplex Method

The simplex method is a well-studied and widely-used pivoting method for solving linear programs. When Dantzig originally formulated the simplex method, he gave a natural pivot rule that pivots into the basis a variable with the most violated reduced cost. In their seminal work, Klee and Minty showed that this pivot rule takes exponential time in the worst case. We prove two main results on the simplex method. Firstly, we show that it is PSPACE-complete to find the solution that is computed by the simplex method using Dantzig's pivot rule. Secondly, we prove that deciding whether Dantzig's rule ever chooses a specific variable to enter the basis is PSPACE-complete. We use the known connection between Markov decision processes (MDPs) and linear programming, and an equivalence between Dantzig's pivot rule and a natural variant of policy iteration for average-reward MDPs. We construct MDPs and show PSPACE-completeness results for single-switch policy iteration, which in turn imply our main results for the simplex method

Variation and Change in Peruvian Spanish Word Order: Language Contact and Dialect Contact in Lima

Previous studies have revealed that the direct object/verb (OV) word order typical of Quechua and Aymara is also prevalent in Andean Spanish. The current study examines the frequency of such structures in Lima, Peru, where massive migration over the past 60 years has brought speakers of Andean indigenous languages and rural Andean Spanish into close contact with speakers of limeño Spanish. Goldvarb analysis of data from 34 participants (seven first-generation migrants, six 1.5-generation migrants, 10 second-generation migrants, and 11 native limeños) indicates that the pragmatic functions that motivated OV order among the participants include those found in noncontact varieties of Spanish, as well as others reported for rural Andean Spanish. Furthermore, L1 speakers of an indigenous language, who were almost all first- and 1.5-generation immigrants, were significantly more likely to use OV word order than L1 Spanish speakers. In contrast, in the speech of second-generation migrants, nearly all of whom spoke Spanish as an L1, the frequency of OV word order was similar to that documented for other non-contact varieties of Spanish

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

Finding Short Paths on Polytopes by the Shadow Vertex Algorithm

We show that the shadow vertex algorithm can be used to compute a short path between a given pair of vertices of a polytope P = {x : Ax \leq b} along the edges of P, where A \in R^{m \times n} is a real-valued matrix. Both, the length of the path and the running time of the algorithm, are polynomial in m, n, and a parameter 1/delta that is a measure for the flatness of the vertices of P. For integer matrices A \in Z^{m \times n} we show a connection between delta and the largest absolute value Delta of any sub-determinant of A, yielding a bound of O(Delta^4 m n^4) for the length of the computed path. This bound is expressed in the same parameter Delta as the recent non-constructive bound of O(Delta^2 n^4 \log (n Delta)) by Bonifas et al. For the special case of totally unimodular matrices, the length of the computed path simplifies to O(m n^4), which significantly improves the previously best known constructive bound of O(m^{16} n^3 \log^3(mn)) by Dyer and Frieze

Formation of 4-hydroxynonenal and further aldehydic mediators of inflammation during bromotrichlorornethane treatment of rat liver cells

Bromotrichloromethane (CBrCl3) treatment is a model for studies on molecular mechanisms of haloalkane toxicity with some advantages compared with CCl4 treatment. The formation of 4-hydroxynonenal and similar aldehydic products of lipid peroxidation, which play a role as mediators of inflammatory processes, was clearly demonstrated in rat hepatocytes treated with CBrCl3. It may be assumed that haloalkane toxicity is connected with the biological effects of those inflammation mediatory aldehydic compounds

Polytopality and Cartesian products of graphs

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we provide several families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.Comment: 21 pages, 10 figure