Approximating Oracle

For every k, an oracle Turing machine M k , and rational polytopes P k (S) for all n and S f0; 1g n , are constructed; querying from the set S given as an oracle, M S k solves the separation problem over P k (S) in strongly polynomial time, performing O(n 3k) arithmetic operations. Each of the polytopes P k (S) approximates S in the sense P k (S) \ f0; 1g n = S and P k (S) 0; 1] n , and for all k, P k+1 (S) is contained in P k (S). As a result, other oracle machines are obtained that, querying from the oracle S, maximize in polynomial time linear functionals over approximations for S obtained from P k (S) by applying the cones-of-matrices cutting operator of Lovv asz and Schrijver a constant (possibly zero) number of times. Thus, our construction enables a systematic application of the cones-of-matrices scheme to any combina-torial optimization problem. Abbreviated title. Approximating machines for combinatorial optimization