@article{EiflerGleixner, author = {Eifler, Leon and Gleixner, Ambros}, title = {Safe and verified Gomory mixed integer cuts in a rational MIP framework}, series = {SIAM Journal on Optimization}, volume = {34}, journal = {SIAM Journal on Optimization}, number = {1}, doi = {10.1137/23M156046X}, abstract = {This paper is concerned with the exact solution of mixed-integer programs (MIPs) over the rational numbers, i.e., without any roundoff errors and error tolerances. Here, one computational bottleneck that should be avoided whenever possible is to employ large-scale symbolic computations. Instead it is often possible to use safe directed rounding methods, e.g., to generate provably correct dual bounds. In this work, we continue to leverage this paradigm and extend an exact branch-and-bound framework by separation routines for safe cutting planes, based on the approach first introduced by Cook, Dash, Fukasawa, and Goycoolea in 2009 [INFORMS J. Comput., 21 (2009), pp. 641-649]. Constraints are aggregated safely using approximate dual multipliers from an LP solve, followed by mixed-integer rounding to generate provably valid, although slightly weaker inequalities. We generalize this approach to problem data that is not representable in floating-point arithmetic, add routines for controlling the encoding length of the resulting cutting planes, and show how these cutting planes can be verified according to the VIPR certificate standard. Furthermore, we analyze the performance impact of these cutting planes in the context of an exact MIP framework, showing that we can solve 21.5\% more instances to exact optimality and reduce solving times by 26.8\% on the MIPLIB 2017 benchmark test set.}, language = {en} } @inproceedings{EiflerWitzigGleixner, author = {Eifler, Leon and Witzig, Jakob and Gleixner, Ambros}, title = {Branch and cut for partitioning a graph into a cycle of clusters}, series = {Combinatorial Optimization. ISCO 2024}, volume = {14594}, booktitle = {Combinatorial Optimization. ISCO 2024}, doi = {10.1007/978-3-031-60924-4_8}, abstract = {In this paper we study formulations and algorithms for the cycle clustering problem, a partitioning problem over the vertex set of a directed graph with nonnegative arc weights that is used to identify cyclic behavior in simulation data generated from nonreversible Markov state models. Here, in addition to partitioning the vertices into a set of coherent clusters, the resulting clusters must be ordered into a cycle such as to maximize the total net flow in the forward direction of the cycle. We provide a problem-specific binary programming formulation and compare it to a formulation based on the reformulation-linearization technique (RLT). We present theoretical results on the polytope associated with our custom formulation and develop primal heuristics and separation routines for both formulations. In computational experiments on simulation data from biology we find that branch and cut based on the problem-specific formulation outperforms the one based on RLT.}, language = {en} }