Nonlinear Matroid Optimization and Experimental Design

We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multi-criteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection, and an algebraic algorithm for vectorial matroids. Our work is partly motivated by applications to minimum-aberration model-fitting in experimental design in statistics, which we discuss and demonstrate in detail

Approximation Schemes for Multi-Budgeted Independence Systems

A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Some classical optimization problems, such as spanning tree and forest, shortest path, (perfect) matching, independent set (basis) in a matroid or in the intersection of two matroids, become NP-hard even with one budget constraint. Still, for most of these problems efficient deterministic and randomized approximation schemes are known. For two or more bud-gets, typically only multi-criteria approximation schemes are available, which return slightly infeasible solutions. Not much is known however for strict budget constraints: filling this gap is the main goal of this paper. It is not hard to see that the above-mentioned problems whose solution sets do not correspond to independence systems are inapproximable al-ready for two budget constraints. For the remaining problems, we present approximation schemes for a constant number k of budget constraints using a variety of techniques: i) we present a simple and powerful mech-anism to transform multi-criteria approximation schemes into pure ap-proximation schemes. This leads to deterministic and randomized ap-proximation schemes for various of the above-mentioned problems; ii) we show that points in low-dimensional faces of any matroid polytope are almost integral, an interesting result on its own. This gives a de-terministic approximation scheme for k-budgeted matroid independent set; iii) we present a deterministic approximation scheme for 2-budgeted matching. The backbone of this result is a purely topological property of curves in R2

Matchings, matroids and submodular functions

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 111-118).This thesis focuses on three fundamental problems in combinatorial optimization: non-bipartite matching, matroid intersection, and submodular function minimization. We develop simple, efficient, randomized algorithms for the first two problems, and prove new lower bounds for the last two problems. For the matching problem, we give an algorithm for constructing perfect or maximum cardinality matchings in non-bipartite graphs. Our algorithm requires O(n") time in graphs with n vertices, where w < 2.38 is the matrix multiplication exponent. This algorithm achieves the best-known running time for dense graphs, and it resolves an open question of Mucha and Sankowski (2004). For the matroid intersection problem, we give an algorithm for constructing a common base or maximum cardinality independent set for two so-called "linear" matroids. Our algorithm has running time O(nrw-1) for matroids with n elements and rank r. This is the best-known running time of any linear matroid intersection algorithm. We also consider lower bounds on the efficiency of matroid intersection algorithms, a question raised by Welsh (1976). Given two matroids of rank r on n elements, it is known that O(nr1.5) oracle queries suffice to solve matroid intersection. However, no non-trivial lower bounds are known. We make the first progress on this question. We describe a family of instances for which (log2 3)n - o(n) queries are necessary to solve these instances. This gives a constant factor improvement over the trivial lower bound for a certain range of parameters. Finally, we consider submodular functions, a generalization of matroids. We give three different proofs that [omega](n) queries are needed to find a minimizer of a submodular function, and prove that [omega](n2/ log n) queries are needed to find all minimizers.by Nicholas James Alexander Harvey.Ph.D