We consider the genus-one curves which arise in cuts of sunrise and elliptic double-box Feynman integrals. compute compare invariants these a number ways, including parametrization, lightcone Baikov (in full loop-by-loop variants). find that same geometry for arises all cases, lends support to idea there exists an invariant notion geometry, independent on way it is computed. further indicate ho...

We deal with directed hypergraphs as a tool to model and solve some classes of problems arising in Operations Research and in Computer Science. Concepts such as connectivity, paths and cuts are defined. An extension of the main duality results to a special class of hypergraphs is presented. Algorithms to perform visits of hypergraphs and to find optimal paths are studied in detail. Some applica...

A way of implementing domain-specific cutting planes in branch-andcut based Mixed-Integer Programming (MIP) solvers is through solving so-called sub-IPs, solutions of which correspond to the actual cuts. We consider the suitability of using Maximum satisfiability solvers instead of MIP for solving subIPs. As a case study, we focus on the problem of learning optimal graphical models, namely, Bay...

This lecture covers the construction of optimal flows and cuts in networks, their relationship, and some applications. It paves the way to our subsequent treatment of linear programming. The network flow and cut algorithms do not follow any of the paradigms we discussed so far, not do they represent a paradigm of their own. The applications we will present illustrate the paradigm of efficient r...

We will analyze mixed-0/1 second-order cone programs where the continuous and binary variables are solely coupled via the conic constraints. We devise a cutting-plane framework based on an implicit Sherali-Adams reformulation. The resulting cuts are very effective as symmetric solutions are automatically cut off and each equivalence class of 0/1 solutions is visited at most once. Further, we pr...

We consider the reconstruction of shared secrets in communication networks, which are modelled by graphs whose components are subject to possible failure. The reconstruction probability can be approximated using minimal cuts, if the failure probabilities of vertices and edges are close to zero. As the main contribution of this paper, node separators are used to design a heuristic for the near-o...