Total dual integrality

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In mathematical optimization, total dual integrality is a sufficient condition for the integrality of a polyhedron. Thus, the optimization of a linear objective over the integral points of such a polyhedron can be done using techniques from linear programming.

A linear system , where and are rational, is called totally dual integral (TDI) if for any such that there is a feasible, bounded solution to the linear program

there is an integer optimal dual solution.[1][2]

Edmonds and Giles[2] showed that if a polyhedron is the solution set of a TDI system , where has all integer entries, then every vertex of is integer-valued. Thus, if a linear program as above is solved by the simplex algorithm, the optimal solution returned will be integer. Further, Giles and Pulleyblank[1] showed that if is a polytope whose vertices are all integer valued, then is the solution set of some TDI system , where is integer valued.

Note that TDI is a weaker sufficient condition for integrality than total unimodularity.[3]

References[edit]

  1. ^ a b Giles, F.R.; W.R. Pulleyblank (1979). "Total Dual Integrality and Integer Polyhedra". Linear algebra and its applications. 25: 191–196. 
  2. ^ a b Edmonds, J.; R. Giles (1977). "A min-max relation for submodular functions on graphs". Annals of Discrete Mathematics. 1: 185–204. 
  3. ^ Chekuri, C. "Combinatorial Optimization Lecture Notes" (PDF).