Matúš Mihalák

Robust optimization in the presence of uncertainty

We study optimization in the presence of uncertainty such as noise in measurements, and advocate a novel approach of tackling it. The main difference to any existing approach is that we do not assume any knowledge about the nature of the uncertainty (such as for instance a probability distribution). Instead, we are given several instances of the same optimization problem as input, and, assuming they are typical w.r.t. the uncertainty, we make use of it in order to compute a solution that is good for the sample instances as well as for future (unknown) typical instances.n We demonstrate our approach for the case of two typical input instances. We first propose a measure of similarity of instances with respect to an objective. This concept allows us to assess whether instances are indeed typical. Based on this concept, we then choose a solution randomly among all solutions that are near-optimum for both instances. We show that the exact notion of near-optimum is intertwined with the proposed measure of similarity. Furthermore, we will show that our measure of similarity also allows us to derive formal statements about the expected quality of the computed solution: If the given instances are not similar, or are too noisy, our approach will detect this. We demonstrate for a few optimization problems and real world data that our approach works well not only in theory, but also in practice.

Counting approximately-shortest paths in directed acyclic graphs

Given a directed acyclic graph with positive edge-weights, two vertices s and t, and a threshold-weight L, we present a fully-polynomial time approximation-scheme for the problem of counting the s-t paths of length at most L. We extend the algorithm for the case of two (or more) instances of the same problem. That is, given two graphs that have the same vertices and edges and differ only in edge-weights, and given two threshold-weights L 1 and L 2, we show how to approximately count the s-t paths that have length at most L 1 in the first graph and length not much larger than L 2 in the second graph. We believe that our algorithms should find application in counting approximate solutions of related optimization problems, where finding an (optimum) solution can be reduced to the computation of a shortest path in a purpose-built auxiliary graph.

Computing and Listing st-Paths in Public Transportation Networks

Given a set of directed paths called linesL, a public transportation network is a directed graph

Approximately Counting Approximately-Shortest Paths in Directed Acyclic Graphs

G_L=V_L,A_L

Robust Optimization in the Presence of Uncertainty: A Generic Approach

which contains exactly the vertices and arcs of every line

Computing and Listing st -Paths in Public Transportation Networks

lin L