Abstract. Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations

We study the problem of optimizing nonlinear objective functions over bipartite matchings. While the problem is generally intractable, we provide several efficient algorithms for it, including a deterministic algorithm for maximizing convex objectives, approximative algorithms for norm minimization and maximization, and a randomized algorithm for optimizing arbitrary objectives.

The convex dimension of a graph G = (V, E) is the smallest dimension d for which G admits an injective map f: V − → R d of its vertices into d-space, such that the barycenters of the images of the edges of G are in convex position. The strong convex dimension of G is the smallest d for which G admits a map as above such that the images of the vertices of G are also in convex position. In this paper we study the convex and strong convex dimensions of graphs. 1

We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on R d and w1x,..., wdx are linear forms on R n, max {c(w1x,..., wdx) : Ax = b, x ∈ N n}. This method works for arbitrary input data A, b, d, w1,..., wd, c. Moreover, for fixed d and several important classes of programs in variable dimension, we prove that our algorithm runs in polynomial time. As a consequence, we obtain polynomial time algorithms for various types of multi-way transportation problems, packing problems, and partitioning problems in variable dimension.

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Abstract. Recent results show that edge-directions of polyhedra play an important role in (combinatorial) optimization; in particular, a d-dimensional polyhedron with |D | distinct edge-directions has at most O(|D | d−1) vertices. Here, we obtain a characterization of the directions of edges that are adjacent to a given vertex of a standard polyhedron of the form P ={x: Ax = b,l � x � u}, tightening a standard necessary condition which asserts that such directions must be minimal support solutions of the homogenous equation Ax =0 which are feasible at the given vertex. We specialize the characterization for polyhedra that correspond to network flows, obtaining a graph characterization of circuits which correspond to edgedirections. Applications to partitioning polyhedra are discussed. Key words: Edge-directions, network flows, polyhedra 1.

Convexly independent subsets of the

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Let P and Q be finite sets of points in the plane. In this note we consider the largest cardinality of a subset of the Minkowski sum S ⊆ P ⊕ Q which consist of convexly independent points. We show that, if |P | = m and |Q | = n then |S | = O(m 2/3 n 2/3 + m + n). 1