New Results in the Theory of Linear and Integer Programming

The polynomial Hirsch conjecture states that the vertex-edge diameter of a d-dimensional polyhedron with n facets is bounded by a polynomial in d and n. For the special case where the polyhedron is defined as the set of points satisfying a system Ax ≤ b of linear inequalities where A is an integer matrix with subdeterminants bounded by ∆, we prove an upper bound of O(∆2d4 logd∆) on the diameter. This special case includes the case of totally unimodular matrices, which are common in fundamental problems of combinatorial optimization such as network flows. Our bound can be improved slightly when the polyhedron is known to be bounded. We obtain this result by assigning a volume to every vertex of the polyhedron via its normal fan and proving a volume expansion property using isoperimetric inequalities. We show a (1 + ε)-approximation algorithm for the closest lattice vector problem in the l∞- norm with a running time bounded by (2log(1/ε))O(d) ·bO(1), where d is the dimension and b thebinaryencodinglengthoftheprobleminstance. Thisimprovesuponthe(2+1/ε)O(d)·bO(1) running time of the previously fastest known algorithm. Our approach is based on a geometric covering problem: find a family of parallelepipeds that cover the cube [−1 + ε, 1 − ε]d , while at the same time every parallelepiped remains within the cube [−1, 1]d even when scaled around its centroid by a factor of 2. We then reduce the original problem to the problem of solving an approximate integer program for each of the parallelepipeds, which can be done using existing approximation algorithms for the closest vector problem. We also discuss some variants of the geometric covering problem. We investigate two questions related to parameterized families of integer programs. First, we relatetheadditiveintegralitygapofafamilyofintegerprogramsmin{cTx : Ax=b,x≥0,x∈ Zn }, where b ∈ Zd is a parameter, to a generalization of the Hilbert basis of a convex cone. We show that under certain conditions, such as when d is fixed and c = 1 is the all ones vector, we can decide in polynomial time whether the additive integrality gap is bounded by a constant γ for all choices of parameter. This is achieved by partitioning the space of parameters using a hyperplane arrangement, and reducing the problem to one integer program for every cell in the arrangement. This decision procedure also yields an alternative method for deciding whether a set of generating vectors of a cone is a Hilbert basis. On the other hand, the general problem of deciding whether the additive integrality gap of such a family of integer programs is bounded is N P -hard even when d is fixed. We show this by a reduction from the coin problem, also known as the Frobenius problem. Second, consider a family of integer programs Ax ≤ b,x ∈ Zd, where b ∈ B is a parameter. Parametric integer programming is the problem of deciding whether all integer programs in vii the family are feasible. We investigate the related problem of finding a parameter b ∈ B that minimizes the number of integer solutions x ∈ Zd of Ax ≤ b. We show that, unlike parametric integer programming, this problem is already N P -hard when d = 2 and B ∼= R. In fact, the following special case is N P -hard: given a convex polygon P and a direction v ∈ Z2 , find t ∈ R that minimizes the number of integer points in P + t v . We show this by a reduction from simultaneous Diophantine approximation. On the positive side, we give a polynomial time approximation scheme for this particular problem.