This paper introduces an improved recursive algorithm to generate the set of all nondominated objective vectors for the Multi-Objective Integer Programming (MOIP) problem. We significantly improve the earlier recursive algorithm of Özlen and Azizoğlu by using the set of already solved subproblems and their solutions to avoid solving a large number of IPs. A numerical example is presented to explain the workings of the algorithm, and we conduct a series of computational experiments to show the savings that can be obtained. As our experiments show, the improvement becomes more significant as the problems grow larger in terms of the number of objectives.

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Abstract. Drawing together techniques from combinatorics and computer science, we improve the census algorithm for enumerating closed minimal P 2 -irreducible 3-manifold triangulations. In particular, new constraints are proven for face-pairing graphs, and pruning techniques are improved using a modification of the union-find algorithm. Using these results we catalogue all 136 closed non-orientable P 2 -irreducible 3-manifolds that can be formed from at most 10 tetrahedra.

Abstract. Regina is a software package for studying 3-manifold triangulations and normal surfaces. It includes a graphical user interface and Python bindings, and also supports angle structures, census enumeration, combinatorial recognition of triangulations, and high-level functions such as 3-sphere recognition, unknot recognition and connected sum decomposition. This paper brings 3-manifold topologists up-to-date with Regina as it appears today, and documents for the first time in the literature some of the key algorithms, heuristics and implementations that are central to Regina's performance. These include the all-important simplification heuristics, key choices of data structures and algorithms to alleviate bottlenecks in normal surface enumeration, modern implementations of 3-sphere recognition and connected sum decomposition, and more. We also give some historical background for the project, including the key role played by Rubinstein in its genesis 15 years ago, and discuss current directions for future development.

A census is presented of all closed non-orientable 3-manifold triangulations formed from at most seven tetrahedra satisfying the additional constraints of minimality and P 2 -irreducibility. The eight different 3-manifolds represented by these 41 different triangulations are identified and described in detail, with particular attention paid to the recurring combinatorial structures that are shared amongst the different triangulations. Using these recurring structures, the resulting triangulations are generalised to infinite families that allow similar triangulations of additional 3manifolds to be formed. Algorithms and techniques used in constructing the census are included. * The initial version of this paper was released in November 2003. The update from September 2005 corrects an offby-one error in the formulae of Theorems 4.6 and 4.9. It also includes general proofreading, particularly in the discussion of layered solid tori. on the resulting 3-manifolds and not their different triangulations, but again the non-computational nature of their work is remarkable.As with the previous closed censuses described above, we consider only triangulations satisfying the following constraints.• Closed: The triangulation is of a closed 3-manifold. In particular it has no boundary faces, and each vertex link is a 2-sphere.• P 2 -irreducible: The underlying 3-manifold has no embedded two-sided projective planes, and furthermore every embedded 2-sphere bounds a ball.• Minimal: The underlying 3-manifold cannot be triangulated using strictly fewer tetrahedra.