On 1-factors and matching extension

AbstractWe prove the following: (1) Let G be a graph with a 1-factor and let F be an arbitrary 1-factor of G. If G⧹{a,b} is k-extendable for each ab∈F, then G is k-extendable. (2) Let G be a graph and let M be an arbitrary maximal matching of G. If G⧹{a,b} is k-factor-critical for each ab∈M, then G is k-factor-critical

Endoscopic transfer of orbital integrals in large residual characteristic

This article constructs Shalika germs in the context of motivic integration, both for ordinary orbital integrals and kappa-orbital integrals. Based on transfer principles in motivic integration and on Waldspurger's endoscopic transfer of smooth functions in characteristic zero, we deduce the endoscopic transfer of smooth functions in sufficiently large residual characteristic.Comment: 33 page

Proof of the 1-factorization and Hamilton decomposition conjectures III: approximate decompositions

In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, $\chi'(G)=D$. (ii) [Hamilton decomposition conjecture] Suppose that $D \ge \lfloor n/2 \rfloor$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) We prove an optimal result on the number of edge-disjoint Hamilton cycles in a graph of given minimum degree. According to Dirac, (i) was first raised in the 1950s. (ii) and (iii) answer questions of Nash-Williams from 1970. The above bounds are best possible. In the current paper, we show the following: suppose that $G$ is close to a complete balanced bipartite graph or to the union of two cliques of equal size. If we are given a suitable set of path systems which cover a set of `exceptional' vertices and edges of $G$, then we can extend these path systems into an approximate decomposition of $G$ into Hamilton cycles (or perfect matchings if appropriate).Comment: We originally split the proof into four papers, of which this was the third paper. We have now combined this series into a single publication [arXiv:1401.4159v2], which will appear in the Memoirs of the AMS. 29 pages, 2 figure

Is Sacral Extension a Risk Factor for Early Proximal Junctional Kyphosis in Adult Spinal Deformity Surgery?

Study designRetrospective cohort study.PurposeTo investigate the role of sacral extension (SE) for the development of proximal junctional kyphosis (PJK) in adult spinal deformity (ASD) surgery.Overview of literatureThe development of PJK is multifactorial and different risk factors have been identified. Of these, there is some evidence that SE also affects the development of PJK, but data are insufficient.MethodsUsing a combined database comprising two propensity-matched groups of fusions following ASD surgery, one with fixation to S1 or S1 and the ilium (SE) and one without SE but with a lower instrumented vertebra of L5 or higher (lumbar fixation, LF), PJK and the role of further parameters were analyzed. The propensity-matched variables included age, the upper-most instrumented vertebra (UIV), preoperative sagittal alignment, and the baseline to one year change of the sagittal alignment.ResultsPropensity matching led to two groups of 89 patients each. The UIV, pelvic incidence minus lumbar lordosis, sagittal vertical axis, pelvic tilt, age, and body mass index were similar in both groups (p >0.05). The incidence of PJK at postoperative one year was similar for SE (30.3%) and LF (22.5%) groups (p =0.207). The PJK angle was comparable (p =0.963) with a change of -8.2° (SE) and -8.3° (LF) from the preoperative measures (p =0.954). A higher rate of PJK after SE (p =0.026) was found only in the subgroup of patients with UIV levels between T9 and T12.ConclusionsInstrumentation to the sacrum with or without iliac extension did not increase the overall risk of PJK. However, an increased risk for PJK was found after SE with UIV levels between T9 and T12

Lightweight Lempel-Ziv Parsing

We introduce a new approach to LZ77 factorization that uses O(n/d) words of working space and O(dn) time for any d >= 1 (for polylogarithmic alphabet sizes). We also describe carefully engineered implementations of alternative approaches to lightweight LZ77 factorization. Extensive experiments show that the new algorithm is superior in most cases, particularly at the lowest memory levels and for highly repetitive data. As a part of the algorithm, we describe new methods for computing matching statistics which may be of independent interest.Comment: 12 page

Uncapacitated Flow-based Extended Formulations

An extended formulation of a polytope is a linear description of this polytope using extra variables besides the variables in which the polytope is defined. The interest of extended formulations is due to the fact that many interesting polytopes have extended formulations with a lot fewer inequalities than any linear description in the original space. This motivates the development of methods for, on the one hand, constructing extended formulations and, on the other hand, proving lower bounds on the sizes of extended formulations. Network flows are a central paradigm in discrete optimization, and are widely used to design extended formulations. We prove exponential lower bounds on the sizes of uncapacitated flow-based extended formulations of several polytopes, such as the (bipartite and non-bipartite) perfect matching polytope and TSP polytope. We also give new examples of flow-based extended formulations, e.g., for 0/1-polytopes defined from regular languages. Finally, we state a few open problems