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The traveling salesman (or salesperson) problem, short TSP, is a problem of strong interest to many researchers from mathematics, economics, and computer science. Manifold TSP variants occur in nearly every scientific field and application domain: engineering, physics, biology, life sciences, and manufacturing just to name a few. Several thousand papers are published on theoretical research or application-oriented results each year. This paper provides the first systematic survey on the best currently known approximability and inapproximability results for well-known TSP variants such as the "standard" TSP, Path TSP, Bottleneck TSP, Maximum Scatter TSP, Generalized TSP, Clustered TSP, Traveling Purchaser Problem, Profitable Tour Problem, Quota TSP, Prize-Collecting TSP, Orienteering Problem, Time-dependent TSP, TSP with Time Windows, and the Orienteering Problem with Time Windows. The foundation of our survey is the definition scheme T3CO, which we propose as a uniform, easy-to-use and extensible means for the formal and precise definition of TSP variants. Applying T3CO to formally define the variant studied by a paper reveals subtle differences within the same named variant and also brings out the differences between the variants more clearly. We achieve the first comprehensive, concise, and compact representation of approximability results by using T3CO definitions. This makes it easier to understand the approximability landscape and the assumptions under which certain results hold. Open gaps become more evident and results can be compared more easily.

The formalism of the generalized quantum master equation (GQME) is an effective tool to simultaneously increase the accuracy and the efficiency of quasiclassical trajectory methods in the simulation of nonadiabatic quantum dynamics. The GQME expresses correlation functions in terms of a non-Markovian equation of motion, involving memory kernels which are typically fast-decaying and can therefore be computed by short-time quasiclassical trajectories. In this paper we study the approximate solution of the GQME, obtained by calculating the kernels with two methods, namely Ehrenfest mean-field theory and spin mapping. We test the approaches on a range of spin--boson models with increasing energy bias between the two electronic levels and place a particular focus on the long-time limits of the populations. We find that the accuracy of the predictions of the GQME depends strongly on the specific technique used to calculate the kernels. In particular, spin mapping outperforms Ehrenfest for all systems studied. The problem of unphysical negative electronic populations affecting spin mapping is resolved by coupling the method with the master equation. Conversely, Ehrenfest in conjunction with the GQME can predict negative populations, despite the fact that the populations calculated from direct dynamics are positive definite.

Domain-specific constraint patterns are introduced, which form the counterpart to design patterns in software engineering for the constraint programming setting. These patterns describe the expert knowledge and best-practice solution to recurring problems and include example implementations. We aim to reach a stage where, for common problems, the modelling process consists of simply picking the applicable patterns from a library of patterns and combining them in a model. This vastly simplifies the modelling process and makes the models simple to adapt. By making the patterns domain-specific we can further include problem-specific modelling ideas, including specific global constraints and search strategies that are known for the problem, into the pattern description. This ensures that the model we obtain from patterns is not only correct but also of high quality. We introduce domain-specific constraint patterns on the example of job shop and flow shop, discuss their advantages and show how the occurrence of patterns can automatically be checked in an event log., Comment: 15 pages

Given a `genus' function $g=g(n)$, we let $\mathcal{E}^g$ be the class of all graphs $G$ such that if $G$ has order $n$ (that is, has $n$ vertices) then it is embeddable in a surface of Euler genus at most $g(n)$. Let the random graph $R_n$ be sampled uniformly from the graphs in $\mathcal{E}^g$ on vertex set $[n]=\{1,\ldots,n\}$. Observe that if $g(n)$ is 0 then $R_n$ is a random planar graph, and if $g(n)$ is sufficiently large then $R_n$ is a binomial random graph $G(n,\tfrac12)$. We investigate typical properties of $R_n$. We find that for \emph{every} genus function $g$, with high probability at most one component of $R_n$ is non-planar. In contrast, we find a transition for example for connectivity: if $g$ is non-decreasing and $g(n) = O(n/\log n)$ then $\liminf_{n \to \infty} \mathbb{P}(R_n \mbox{ is connected}) < 1$, and if $g(n) \gg n$ then with high probability $R_n$ is connected. These results also hold when we consider orientable and non-orientable surfaces separately. We also investigate random graphs sampled uniformly from the `hereditary part' or the `minor-closed' part of $\mathcal{E}^g$, and briefly consider corresponding results for unlabelled graphs., Comment: 33 pages