Worst case and probabilistic analysis of the 2-Opt algorithm for the TSP

2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on “real world” Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2-dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every p∈N , a family of L p instances on which 2-Opt can take an exponential number of steps. Previous probabilistic analyses were restricted to instances in which n points are placed uniformly at random in the unit square [0,1]2, where it was shown that the expected number of steps is bounded by O~(n10) for Euclidean instances. We consider a more advanced model of probabilistic instances in which the points can be placed independently according to general distributions on [0,1] d , for an arbitrary d≥2. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number n of points and the maximal density ϕ of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of O~(n4+1/3⋅ϕ8/3) . When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to O~(n4+1/3−1/d⋅ϕ8/3) . If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by O~(n4−1/d⋅ϕ) . In addition, we prove an upper bound of O(ϕ√d) on the expected approximation factor with respect to all L p metrics. Let us remark that our probabilistic analysis covers as special cases the uniform input model with ϕ=1 and a smoothed analysis with Gaussian perturbations of standard deviation σ with ϕ∼1/σ d

Smoothed Complexity Theory

Smoothed analysis is a new way of analyzing algorithms introduced by Spielman and Teng (J. ACM, 2004). Classical methods like worst-case or average-case analysis have accompanying complexity classes, like P and AvgP, respectively. While worst-case or average-case analysis give us a means to talk about the running time of a particular algorithm, complexity classes allows us to talk about the inherent difficulty of problems. Smoothed analysis is a hybrid of worst-case and average-case analysis and compensates some of their drawbacks. Despite its success for the analysis of single algorithms and problems, there is no embedding of smoothed analysis into computational complexity theory, which is necessary to classify problems according to their intrinsic difficulty. We propose a framework for smoothed complexity theory, define the relevant classes, and prove some first hardness results (of bounded halting and tiling) and tractability results (binary optimization problems, graph coloring, satisfiability). Furthermore, we discuss extensions and shortcomings of our model and relate it to semi-random models.Comment: to be presented at MFCS 201

Autoantibodies against NMDA receptor 1 modify rather than cause encephalitis

The etiology and pathogenesis of “anti-N-methyl-D-aspartate-receptor (NMDAR) encephalitis” and the role of autoantibodies (AB) in this condition are still obscure. While NMDAR1-AB exert NMDAR-antagonistic properties by receptor internalization, no firm evidence exists to date that NMDAR1-AB by themselves induce brain inflammation/encephalitis. NMDAR1-AB of all immunoglobulin classes are highly frequent across mammals with multiple possible inducers and boosters. We hypothesized that “NMDAR encephalitis” results from any primary brain inflammation coinciding with the presence of NMDAR1-AB, which may shape the encephalitis phenotype. Thus, we tested whether following immunization with a “cocktail” of 4 NMDAR1 peptides, induction of a spatially and temporally defined sterile encephalitis by diphtheria toxin-mediated ablation of pyramidal neurons (“DTA” mice) would modify/aggravate the ensuing phenotype. In addition, we tried to replicate a recent report claiming that immunizing just against the NMDAR1-N368/G369 region induced brain inflammation. Mice after DTA induction revealed a syndrome comprising hyperactivity, hippocampal learning/memory deficits, prefrontal cortical network dysfunction, lasting blood brain-barrier impairment, brain inflammation, mainly in hippocampal and cortical regions with pyramidal neuronal death, microgliosis, astrogliosis, modest immune cell infiltration, regional atrophy, and relative increases in parvalbumin-positive interneurons. The presence of NMDAR1-AB enhanced the hyperactivity (psychosis-like) phenotype, whereas all other readouts were identical to control-immunized DTA mice. Non-DTA mice with or without NMDAR1-AB were free of any encephalitic signs. Replication of the reported NMDAR1-N368/G369-immunizing protocol in two large independent cohorts of wild-type mice completely failed. To conclude, while NMDAR1-AB can contribute to the behavioral phenotype of an underlying encephalitis, induction of an encephalitis by NMDAR1-AB themselves remains to be proven

Comparing and taming the reactivity of HWE and Wittig reagents with cyclic hemiacetals

A practical solution to the formation of mixtures of E/Z and open/cyclic isomers in the reaction of (2R,4S)-4-hydroxy-2-methylpentanal (as its hemiacetal, a lactol) with conjugated phosphoranes (stabilised Wittig reagents) and Horner-Wadsworth-Emmons reagents is disclosed. The HWE reaction has a strong bias to give oxolanes. On the other hand, stabilised Wittig reagents give unsaturated carboxyl derivatives of configuration E (major) and oxolanes (minor); the latter can be avoided by addition of CF3CH2OH or using morpholine amide phosphorane

Demonstration of CO2-Enhanced Oil Recovery Potential in the Illinois Basin: Technical Report

U.S. DOE Cooperative Agreement: DE-FC26-05NT42588Ope

The Civil Rights Movement in Dallas

The Civil Rights Movement was a defining movement in the United States of America and shaped the country as we know it today. Before the Montgomery Bus Boycotts, Rosa Parks, or Emmett Till, segregation in the southern states was a constant reminder of the inequalities that African-Americans had to endure. Traditionally Dallas, Texas has been a conservative city and was against the Civil Rights movement, but in the 1960s, we start to see the city shift towards being a progressive and desegregated area. The transition of Dallas, Texas from a conservative and segregated city to providing equality and guaranteeing African-Americans their Civil Rights is an area of history that has not been extensively studied. This transition was spearheaded by the actions of Mayor Earle Cabell (1961-1965), Juanita J. Craft (Civil Rights leader in Texas), Bob Cullum (grocery store owner), and Stanley Marcus (high end department store owner). These men and women greatly impacted the desegregation of Dallas and its communities. There was both opposition and support for Mayor Cabell's reforms and Craft, Cullum, and Marcus's push for desegregation, by comparing these reactions from the people of Dallas on both sides of the Civil Rights Movement one can gauge how the city was able to handle the Civil Rights Movement. One of the more surprising supporters of integration in Dallas were Cullum and Marcus, because they were wealthy white businessmen, but they played a key role in making Dallas a unique area of study in the Civil Rights Movement. Through studying and analyzing letters sent directly to Earle Cabell about desegregation, along with anti-segregation pamphlets, which compared desegregation to communist and anti-Christ ideals, one can see how the change in Dallas, Texas transitioned from a traditionally conservative city to a more progressive one. While there was certainly opposition to Mayor Cabell, Craft, Cullum, and Marcus, there were those who called for an end to segregation in the city both white and black

Black Bears

The black bear (Ursus americanus) is the smallest and most widely distributed of the North American bears. Black bears east of the Mississippi are predominantly black, but in the Rocky Mountains and westward various shades of brown, cinnamon, and even blond are common. The head is moderately sized with a straight profile and tapering nose. The ears are relatively small, rounded, and erect. It is important to be able to distinguish between black bears and grizzly/brown bears (Ursus arctos). The grizzly/brown bear is typically much larger than the black bear

Uncoordinated Two-Sided Markets

Various economic interactions can be modeled as two-sided markets. A central solution concept to these markets are stable matchings, introduced by Gale and Shapley. It is well known that stable matchings can be computed in polynomial time, but many real-life markets lack a central authority to match agents. In those markets, matchings are formed by actions of self-interested agents. Knuth introduced uncoordinated two-sided markets and showed that the uncoordinated better response dynamics may cycle. However, Roth and Vande Vate showed that the random better response dynamics converges to a stable matching with probability one, but did not address the question of convergence time. In this paper, we give an exponential lower bound for the convergence time of the random better response dynamics in two-sided markets. We also extend these results to the best response dynamics, i. e., we present a cycle of best responses, and prove that the random best response dynamics converges to a stable matching with probability one, but its convergence time is exponential. Additionally, we identify the special class of correlated two-sided markets with real-life applications for which we prove that the random best response dynamics converges in expected polynomial time