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Abstract. We introduce a general method to count unlabeled combinatorial structures and to efficiently generate them at random. The approach is based on pointing unlabeled structures in an "unbiased" way that a structure of size n gives rise to n pointed structures. We extend Pólya theory to the corresponding pointing operator, and present a random sampling framework based on both the principles of Boltzmann sampling and on Pólya operators. All previously known unlabeled construction principles for Boltzmann samplers are special cases of our new results. Our method is illustrated on several examples: in each case, we provide enumerative results and efficient random samplers. The approach applies to unlabeled families of plane and nonplane unrooted trees, and tree-like structures in general, but also to families of graphs (such as cacti graphs and outerplanar graphs) and families of planar maps. This is the extended and revised journal version of a conference paper with the title "An unbiased pointing operator for unlabeled structures, with applications to counting and sam-

The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming framework SCIP . The focus of this paper is on the role of the SCIP Optimization Suite in supporting research. SCIP ’s main design principles are discussed, followed by a presentation of the latest performance improvements and developments in version 8.0, which serve both as examples of SCIP ’s application as a research tool and as a platform for further developments. Further, the paper gives an overview of interfaces to other programming and modeling languages, new features that expand the possibilities for user interaction with the framework, and the latest developments in several extensions built upon SCIP .

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This paper is concerned with optimal operation of pressurized water supply networks at a fixed point in time. We use a mixed-integer nonlinear programming (MINLP) model incorporating both the nonlinear physical laws and the discrete decisions such as switching pumps on and off. We demonstrate that for instances from our industry partner, these stationary models can be solved to ε-global optimality within small running times using problem-specific presolving and state-of-the-art MINLP algorithms.In our modeling, we emphasize the importance of distinguishing between what we call real and imaginary flow, i.e., taking into account that the law of Darcy-Weisbach correlates pressure difference and flow along a pipe if and only if water is available at the high pressure end of a pipe. Our modeling solution extends to the dynamic operative planning problem.

In this article we give a brief overview of the start-of-the-art in software for the solution of mixed integer nonlinear programs (MINLP). We establish several groupings with respect to various features and give concise individual descriptions for each solver. The provided information may guide the selection of a best solver for a particular MINLP problem.

Abstract. This paper discusses how to build a solver for mixed integer quadratically constrained programs (MIQCPs) by extending a framework for constraint integer programming (CIP). The advantage of this approach is that we can utilize the full power of advanced MILP and CP technologies, in particular for the linear relaxation and the discrete components of the problem. We use an outer approximation generated by linearization of convex constraints and linear underestimation of nonconvex constraints to relax the problem. Further, we give an overview of the reformulation, separation, and propagation techniques that are used to handle the quadratic constraints efficiently.We implemented these methods in the branch-cut-and-price framework SCIP. Computational experiments indicating the potential of the approach and evaluating the impact of the algorithmic components are provided.Key words. mixed integer quadratically constrained programming, constraint integer programming, branch-and-cut, convex relaxation, domain propagation, primal heuristic, nonconvex AMS(MOS) subject classifications. 90C11, 90C20, 90C26, 90C27, 90C57.1. Introduction. In recent years, substantial progress has been made in the solvability of generic mixed integer linear programs (MILPs) [2,12]. Furthermore, it has been shown that successful MILP solving techniques can often be extended to the more general case of mixed integer nonlinear programs (MINLPs) [1,6,13]. Analogously, several authors have shown that an integrated approach of constraint programming (CP) and MILP can help to solve optimization problems that were intractable with either of the two methods alone, for an overview see [17].The paradigm of constraint integer programming (CIP) [2, 4] combines modeling and solving techniques from the fields of constraint programming (CP), mixed integer programming, and satisfiability testing (SAT). The concept of CIP aims at restricting the generality of CP modeling as little as needed while still retaining the full performance of MILP solving techniques. Such a paradigm allows us to address a wide range of optimization problems. For example, in [2], it is shown that CIP includes MILP and constraint programming over finite domains as special cases.The goal of this paper is to show, how a framework for CIPs can be extended towards a competitive solver for mixed integer quadratically constrained programs (MIQCPs), which are an important subclass of MINLPs. This framework allows us to utilize the power of already existing MILP and

We determine the exact and asymptotic number of unlabeled outerplanar graphs. The exact number gn of unlabeled outerplanar graphs on n vertices can be computed in polynomial time, and gn is asymptotically g n −5/2 ρ −n , where g ≈ 0.00909941 and ρ −1 ≈ 7.50360 can be approximated. Using our enumerative results we investigate several statistical properties of random unlabeled outerplanar graphs on n vertices, for instance concerning connectedness, chromatic number, and the number of edges. To obtain the results we combine classical cycle index enumeration with recent results from analytic combinatorics.3 Research supported by the Deutsche Forschungsgemeinschaft (DFG Pr 296/7-3) graphs on n vertices (i.e. taken uniformly at random among all unlabeled outerplanar graphs on n vertices), for example connectedness, chromatic number, the number of components, and the number of edges.Before we provide a more detailed exposition of the results obtained in this paper, we would like to give a brief survey on the vast literature on enumerative results for planar structures. The exact and asymptotic number of embedded planar graphs (i.e., planar maps) has been studied intensively, starting with Tutte's seminal work on the number of rooted oriented planar maps [26]. The number of three-connected planar maps is related to the number of three-connected planar graphs [19,26], since a three-connected planar graph has a unique embedding on the sphere [30]. Bender, Gao, and Wormald used this property to count labeled two-connected planar graphs [1], and Giménez and Noy recently extended this work to the enumeration of labeled planar graphs [14]. The growth constant for labeled planar graphs can be computed with arbitrary precision and its first digits are 27.2269. Many interesting properties of a random labeled planar graph were studied by McDiarmid, Steger, and Welsh [18]. It is also known how to generate labeled three-connected planar graphs, labeled planar maps, and labeled planar graphs uniformly at random [4,7,13,24].The asymptotic number of general unlabeled planar graphs has not yet been determined, but has been studied for quite some time [29]. Moreover, no polynomial time algorithm for the computation of the exact number of unlabeled planar graphs on n vertices is known. Such an algorithm is only known for unlabeled rooted two-connected planar graphs [5], and for unlabeled rooted cubic planar graphs [6].An outerplanar graph is a graph that can be embedded in the plane such that every vertex lies on the outer face. Such graphs can also be characterized in terms of forbidden minors [9], namely K 2,3 and K 4 . The class of outerplanar graphs is often used as a first non-trivial test-case for results about the class of all planar graphs; apart from that, this class appears frequently in various applications of graph theory. The asymptotic number of labeled outerplanar graphs was recently determined in [3]. In this paper, we determine the number of unlabeled outerplanar graphs, i.e., we enumerate outerplanar graphs up to isomorp...