Graph-Based Local Elimination Algorithms in Discrete Optimization

The use of discrete optimization (DO) models and algorithms makes it possible to solve many practical problems in scheduling theory, network optimization, routing in communication networks, facility location, optimization in enterprise resource planning, and logistics (in particular, in supply chain management [36]). The field of artificial intelligence includes aspects like theorem proving, SAT in propositional logic (see [23], [50]), robotics problems, inference calculation in Bayesian networks [66], scheduling, and others. Many real-life DO problems contain a huge number of variables and/or constraints that make the models intractable for currently available DO solvers. NP -hardness refers to the worst-case complexity of problems. Recognizing problem instances that are better (and easier for solving) than these ”worst cases” is a rewarding task given that better algorithms can be used for these easy cases. Complexity theory has proved that universality and effectiveness are contradictory requirements to algorithm complexity. But the complexity of some class of problems decreases if the class may be divided into subsets and the special structure of these subsets can be used in the algorithm design. To meet the challenge of solving large scale DO problems (DOPs) in reasonable time, there is an urgent need to develop new decomposition approaches [22], [82], [75]. Large-scale DOPs are characterized not only by huge size but also by special or sparse structure. The block form of many DO problems is usually caused by the weak connectedness of subsystems of real systems. One of the first examples of large sparse linear programming (LP) problems which Dantzig started to study was a class of staircase LP prob-