A-Tint: A polymake extension for algorithmic tropical intersection theory

In this paper we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus of this paper is the study of moduli spaces, where the underlying combinatorics of the varieties involved allow a much more efficient way of computing certain tropical cycles. The algorithms discussed here have been implemented in an extension for polymake, a software for polyhedral computations.Comment: 32 pages, 5 figures, 4 tables. Second version: Revised version, to be published in European Journal of Combinatoric

Comparison of matroid intersection algorithms for large circuit analysis

This paper presents two approaches to symbolic analysis of large analog integrated circuits via simplification during the generation of the symbolic expressions. Both techniques are examined from the point of view of matroid theory. Finally, a new approach which combines the positive features of both approaches is introduced

Randomized Strategies for Robust Combinatorial Optimization

In this paper, we study the following robust optimization problem. Given an independence system and candidate objective functions, we choose an independent set, and then an adversary chooses one objective function, knowing our choice. Our goal is to find a randomized strategy (i.e., a probability distribution over the independent sets) that maximizes the expected objective value. To solve the problem, we propose two types of schemes for designing approximation algorithms. One scheme is for the case when objective functions are linear. It first finds an approximately optimal aggregated strategy and then retrieves a desired solution with little loss of the objective value. The approximation ratio depends on a relaxation of an independence system polytope. As applications, we provide approximation algorithms for a knapsack constraint or a matroid intersection by developing appropriate relaxations and retrievals. The other scheme is based on the multiplicative weights update method. A key technique is to introduce a new concept called $(\eta,\gamma)$-reductions for objective functions with parameters $\eta, \gamma$. We show that our scheme outputs a nearly $\alpha$-approximate solution if there exists an $\alpha$-approximation algorithm for a subproblem defined by $(\eta,\gamma)$-reductions. This improves approximation ratio in previous results. Using our result, we provide approximation algorithms when the objective functions are submodular or correspond to the cardinality robustness for the knapsack problem