Satisfiability threshold for random regular NAE-SAT

We consider the random regular $k$-NAE-SAT problem with $n$ variables each appearing in exactly $d$ clauses. For all $k$ exceeding an absolute constant $k_0$, we establish explicitly the satisfiability threshold $d_*=d_*(k)$. We prove that for $d<d_*$ the problem is satisfiable with high probability while for $d>d_*$ the problem is unsatisfiable with high probability. If the threshold $d_*$ lands exactly on an integer, we show that the problem is satisfiable with probability bounded away from both zero and one. This is the first result to locate the exact satisfiability threshold in a random constraint satisfaction problem exhibiting the condensation phenomenon identified by Krzakala et al. (2007). Our proof verifies the one-step replica symmetry breaking formalism for this model. We expect our methods to be applicable to a broad range of random constraint satisfaction problems and combinatorial problems on random graphs

The Impact of Road Configuration on V2V-based Cooperative Localization

Cooperative localization with map matching has been shown to reduce Global Navigation Satellite System (GNSS) localization error from several meters to sub-meter level by fusing the GNSS measurements of four vehicles in our previous work. While further error reduction is expected to be achievable by increasing the number of vehicles, the quantitative relationship between the estimation error and the number of connected vehicles has neither been systematically investigated nor analytically proved. In this work, a theoretical study is presented that analytically proves the correlation between the localization error and the number of connected vehicles in two cases of practical interest. More specifically, it is shown that, under the assumption of small non-common error, the expected square error of the GNSS common error correction is inversely proportional to the number of vehicles, if the road directions obey a uniform distribution, or inversely proportional to logarithm of the number of vehicles, if the road directions obey a Bernoulli distribution. Numerical simulations are conducted to justify these analytic results. Moreover, the simulation results show that the aforementioned error decrement rates hold even when the assumption of small non-common error is violated