Convex Combinatorial Optimization

We introduce the convex combinatorial optimization problem, a far reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and discuss several applications

Generalized scalings satisfying linear equations

AbstractWe unify and generalize a broad class of problems referred in the literature as “scaling problems,” by extending the applicability of a formulation suggested recently by Bapat and Raghavan. Specifically, let a ∈ Rn,b ∈Rm, and C ∈ Rm × n be given, where a is strictly positive. A C-scaling of the vector a is defined to be a vector a'∈Rn with a'i= aiΠmk=1uCkjk for some strictly positive vector u Rm. The problem of finding a C-scaling of the vector a which satisfies the linear system Cx = b will be called a generalized scaling problem. In this paper it is shown that previously studied matrix-scaling problems, (e.g., finding scalings with prespecified row sums and column sums, or finding scalings with row sums equaling the corresponding column sums, or finding scalings of multidimensional matrices with prespecified margins) are special instances of generalized scaling problems. Generalized scaling problems are reduced to convex optimization problems, and the reduction is used to characterize solutions, to develop necessary and sufficient conditions for their existence, to establish uniqueness results and to characterize approximate solutions

Scalings of matrices which have prespecified row sums and column sums via optimization

AbstractThe problem of scaling a matrix so that it has given row and column sums is transformed into a convex minimization problem. In particular, we use this transformation to characterize the existence of such scaling or corresponding approximations. We obtain new results as well as new, streamlined proofs of known results

A proof of the convexity of the range of a nonatomic vector measure using linear inequalities

AbstractThis note shows how a standard result about linear inequality systems can be used to give a simple proof of the fact that the range of a nonatomic vector measure is convex, a result that is due to Liapounoff

Linear Problems and Linear Algorithms

AbstractUsing predicate logic, the concept of a linear problem is formalized. The class of linear problems is huge, diverse, complex, and important. Linear and randomized linear algorithms are formalized. For each linear problem, a linear algorithm is constructed that solves the problem and a randomized linear algorithm is constructed that completely solves it, that is, for any data of the problem, the output set of the randomized linear algorithm is identical to the solution set of the problem. We obtain a single machine, referred to as the Universal (Randomized) Linear Machine, which (completely) solves every instance of every linear problem. Conversely, for each randomized linear algorithm, a linear problem is constructed that the algorithm completely solves. These constructions establish a one-to-one and onto correspondence from equivalence classes of linear problems to equivalence classes of randomized linear algorithms.Our construction of (randomized) linear algorithms to (completely) solve linear problems as well as the algorithms themselves are based on Fourier Elimination and have superexponential complexity. However, there is no evidence that the inefficiency of our methods is unavoidable relative to the difficulty of the problem

Bounds on distances between eigenvalues

AbstractExplicit (computable) lower and upper bounds on the distances between a given real eigenvalue of a real square matrix and the remaining (not necessarily real) eigenvalues of the matrix are developed

Paths to marriage stability

AbstractWe obtain a family of algorithms that determine stable matchings for the stable marriage problem by starting with an arbitrary matching and iteratively satisfying blocking pairs, that is, matching couples who both prefer to be together over the outcome of the current matching. The existence of such an algorithm is related to a question raised by Knuth (1976) and was recently resolved positively by Roth and Vande Vate (1992). The basic version of our method depends on a fixed ordering of all mutually acceptable man-woman pairs which is consistent with the preferences of either all men or of all women. Given such an ordering, we show that starting with an arbitrary matching and iteratively satisfying the highest blocking pair at each iteration will eventually yield a stable matching. We show that the single-proposal variant of the Gale-Shapley algorithm as well as the Roth-Vande Vate algorithm are instances of our approach. We also demonstrate that an arbitrary decentralized system does not guarantee convergence to a stable matching

An upper bound for the minimum rank of a graph

For a graph G of order n, the minimum rank of G is defined to be the smallest possible rank over all real symmetric n×n matrices A whose (i,j)th entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. We prove an upper bound for minimum rank in terms of minimum degree of a vertex is valid for many graphs, including all bipartite graphs, and conjecture this bound is true over for all graphs, and prove a related bound for all zero-nonzero patterns of (not necessarily symmetric) matrices. Most of the results are valid for matrices over any infinite field, but need not be true for matrices over finite fields