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

Path trading : fast algorithms, smoothed analysis, and hardness results

The Border Gateway Protocol (BGP) serves as the main routing protocol of the Internet and ensures network reachability among autonomous systems (ASes). When traffic is forwarded between the many ASes on the Internet according to that protocol, each AS selfishly routes the traffic inside its own network according to some internal protocol that supports the local objectives of the AS. We consider possibilities of achieving higher global performance in such systems while maintaining the objectives and costs of the individual ASes. In particular, we consider how path trading, i.e. deviations from routing the traffic using individually optimal protocols, can lead to a better global performance. Shavitt and Singer ("Limitations and Possibilities of Path Trading between Autonomous Systems", INFOCOM 2010) were the first to consider the computational complexity of finding such path trading solutions. They show that the problem is weakly NP-hard and provide a dynamic program to find path trades between pairs of ASes. In this paper we improve upon their results, both theoretically and practically. First, we show that finding path trades between sets of ASes is also strongly NP-hard. Moreover, we provide an algorithm that finds all Pareto-optimal path trades for a pair of two ASes. While in principal the number of Pareto-optimal path trades can be exponential, in our experiments this number was typically small. We use the framework of smoothed analysis to give theoretical evidence that this is a general phenomenon, and not only limited to the instances on which we performed experiments. The computational results show that our algorithm yields far superior running times and can solve considerably larger instances than the previous dynamic program

Internet routing between autonomous systems: Fast algorithms for path trading

Routing traffic on the internet efficiently has become an important research topic over the past decade. In this article we consider a generalization of the shortest path problem, the path-trading problem, which has applications in inter-domain traffic routing. When traffic is forwarded between autonomous systems (ASes), such as competing internet providers, each AS selfishly routes the traffic inside its own network. Efficient solutions to the path trading problem can lead to higher global performance in such systems, while maintaining the objectives and costs of the individual ASes. First, we extend a previous hardness result for the path trading problem. Moreover, we provide an algorithm that finds all Pareto-optimal path trades for a pair of two ASes. While in principal the number of Pareto-optimal path trades can be exponential, in our experiments this number was typically small. We use the framework of smoothed analysis to give a theoretical explanation for that fact. The computational results show that our algorithm yields far superior running times and can solve considerably larger instances than a previously known algorithm

Path trading : fast algorithms, smoothed analysis, and hardness results

The Border Gateway Protocol (BGP) serves as the main routing protocol of the Internet and ensures network reachability among autonomous systems (ASes). When traffic is forwarded between the many ASes on the Internet according to that protocol, each AS selfishly routes the traffic inside its own network according to some internal protocol that supports the local objectives of the AS. We consider possibilities of achieving higher global performance in such systems while maintaining the objectives and costs of the individual ASes. In particular, we consider how path trading, i.e. deviations from routing the traffic using individually optimal protocols, can lead to a better global performance. Shavitt and Singer ("Limitations and Possibilities of Path Trading between Autonomous Systems", INFOCOM 2010) were the first to consider the computational complexity of finding such path trading solutions. They show that the problem is weakly NP-hard and provide a dynamic program to find path trades between pairs of ASes. In this paper we improve upon their results, both theoretically and practically. First, we show that finding path trades between sets of ASes is also strongly NP-hard. Moreover, we provide an algorithm that finds all Pareto-optimal path trades for a pair of two ASes. While in principal the number of Pareto-optimal path trades can be exponential, in our experiments this number was typically small. We use the framework of smoothed analysis to give theoretical evidence that this is a general phenomenon, and not only limited to the instances on which we performed experiments. The computational results show that our algorithm yields far superior running times and can solve considerably larger instances than the previous dynamic program

Searching for relationships between feathering and egg white genotypes in laying hens

In a load balancing network each processor has an initial collection of unit-size jobs, tokens, and in each round, pairs of processors connected by balancers split their load as evenly as possible. An excess token (if any) is placed according to some predefined rule. As it turns out, this rule crucially effects the performance of the network. In this work we propose a model that studies this effect. We suggest a model bridging the uniformly-random assignment rule, and the arbitrary one (in the spirit of smoothed-analysis) by starting from an arbitrary assignment of balancer directions, then flipping each assignment with probability α independently. For a large class of balancing networks our result implies that after O(log n) rounds the discrepancy is whp O((1/2−α) log n+log log n). This matches and generalizes the known bounds for α = 0 and α = 1/2