Cutting-Plane Proofs in Polynomial Space

The integer programming problem is to decide if a given system of linear inequalities has an integral solution. Recent progress on this algorithmic question has involved techniques from the geometry of numbers, in the celebrated paper of Lenstra [20] and in results of Babai [1], Gr6tschel, Lovfisz and Schrijver [14] and Kannan [16]. One of the things that is apparent in these results is the importance of the fact that if a polyhedron contains no integral vectors then there must be some direction in which it is not very 'wide'. This idea has been developed more fully by Kannan and Lovfisz [17], who obtained a theorem which provides much more information on the appearance of such polyhedra. These 'width' results have consequences for the construction and analysis of p roof systems for verifying that a polyhedron contains no integral vectors. Whereas the integer programming problem is directly related to the question of the equality of P and NP, the existence of a polynomiallength proof system for integer programming is equivalent to NP = co-NP. One of the fundamental concepts in the theory of integer programming is that of cutting planes, going back to the work of Dantzig, Fulkerson and Johnson [11] and Gomory [12]. On the practical side, cutting-plane techniques are the basis of very successful algorithms for the solution of large-scale combinatorial and 0-1 programming problems in Crowder, Johnson and Padberg [9], Crowder and Padberg [10], Gr/Stschel, Jiinger and Reinelt [13], Padberg, van Roy and Wolsey [21] and elsewhere. On the theoretical side, Chvfital [3, 4, 5, 6] has shown that the notion of cutting planes leads to many nice results and proofs in combinatorics. We will adopt Chvfital's point of view and consider cutting planes as a proof system, in our case for verifying that polyhedra contain no integral vectors. Perhaps the best known of all proof systems is the resolution method for proving the unsatisfiability of formulas in the proposit ional calculus. Haken [15] settled a