Polyhedral Characterization of Discrete Dynamic Programming

Abstract

Many interesting combinatorial problems can be optimized efficiently using recursive computations often termed discrete dynamic programming. In this paper, we develop a paradigm for a general class of such optimizations that yields a polyhedral description for each model in the class. The elementary concept of dynamic programs as shortest path problems in acyclic graphs is generalized to one seeking a least cost solution in a directed hypergraph. Sufficient conditions are then provided for binary integrality of the associated hyperflow problem. Given a polynomially solvable dynamic program, the result is a linear program, in polynomially many variables and constraints, having the solution vectors of the dynamic program as its extreme-point optima. That is, the linear program provides a succinct characterization of the solutions to the underlying optimization problem expressed through an appropriate change of variables. We also discuss projecting this formulation to recover constraints on the original variables and illustrate how some important dynamic programming solvable models fit easily into our paradigm. A classic multiechelon lot sizing problem in production and a host of optimization problems on recursively defined classes of graphs are included.

Column generation for extended formulations

“…Another important special case is when the extended formulation is stemming from a dynamic programming solver for the subproblem [16]. Most discrete dynamic program entails finding a shortest (or longest) path in a directed acyclic decision graph, where nodes correspond to states (representing partial solutions) and arcs correspond to transitions (associated with partial decisions to extend solutions).…”

“…These can be modeled by hyper-arcs with a single head but multiple tails. Then, the extended paradigm developed by [16] consists in seeing a dynamic programming solution as a hyper-path (associated to a unit flow incoming to the final state) in a hyper-graph that satisfy two properties:…”

“…The dynamic programs that can be modeled as a shortest path problem are a special case where the hyper-graph only has simple arcs with a single tail and hence the disjointness property does not have to be checked. Following [16], consider a directed hyper-graph G = (V, A), with hyper-arc set A = {(J, l) : J ⊂ V \ {l}, l ∈ V} and associated arc costs c(J, l), a node indexing σ : V → {1, . .…”

“…In this generalized context, [16] gives an explicit procedure to obtain a solution z, defining the hyper-arcs that are in the subproblem solution, from the solution of the dynamic programming recursion: the hyper-arc selection is a dual solution to the linear program (70); given the specific assumption on the hyper-graph (acyclic consistency and disjointness), this dual solution z can be obtained through a greedy procedure. So if one uses the dynamic program as oracle, one can recover a solution x and associated complementary solution z. Alternatively, if subproblem solution x is obtained by another algorithm, one can easily compute distance labels, u l , associated to nodes of the hyper-graph (u l = the cost of partial solution associated to node l if this partial solution is part of x and u l = ∞ otherwise) and apply the procedure of [16] to recover the complementary solution z.…”

“…So if one uses the dynamic program as oracle, one can recover a solution x and associated complementary solution z. Alternatively, if subproblem solution x is obtained by another algorithm, one can easily compute distance labels, u l , associated to nodes of the hyper-graph (u l = the cost of partial solution associated to node l if this partial solution is part of x and u l = ∞ otherwise) and apply the procedure of [16] to recover the complementary solution z. So, Property 4 is satisfied.…”