Nonserial Dynamic Programming and Tree Decomposition in Discrete Optimization

Abstract

Solving discrete optimization problems (DOP) can be a rather hard task. Many real DOPs contain a huge number of variables and/or constraints that make the models intractable for currently available solvers. There are few approaches for solving DOPs: tree search approaches (e.g., branch and bound), relaxation and decomposition methods. Large DOPs can be solved due to their special structure. Among decomposition approaches we can mention poorly known local decomposition algorithms using the special block matrix structure of constraints and half-forgotten nonserial dynamic programming algorithms which can exploit sparsity in the dependency graph of a DOP. One of the promising approaches to cope with NP-hardness in solving DOPs is the construction of decomposition methods [7]. Decomposition techniques usually determine subproblems, whose solutions can be combined to create a solution of the initial DOP problem. Usually, DOPs from applications have a special structure, and the matrices of constraints for large-scale problems have a lot of zero elements (sparse matrices). This paper reviews main results for local decomposition algorithms in discrete programming and establishes some links between them, tree decompositions and nonserial dynamic programming.