Shortest paths with ordinal weights

Abstract

We investigate the single-source-single-destination "shortest" paths problem in acyclic graphs with ordinal weighted arc costs. We define the concepts of ordinal dominance and efficiency for paths and their associated ordinal levels, respectively. Further, we show that the number of ordinally non-dominated paths vectors from the source node to every other node in the graph is polynomially bounded and we propose a polynomial time labeling algorithm for solving the problem of finding the set of ordinally non-dominated paths vectors from source to sink.

An experimental analysis of least-cost path models on ordinal-scaled raster surfaces

“…It is important to note that in almost all cases, there was more than one minimax-cost path between the selected terminal cells. Many of those most likely had 'ordinally non-dominant' (Schäfer et al 2020) sets of cost valuesroughly, a set (or vector) of values are said to 'ordinally dominate' (Schäfer et al 2020) another if each element of the former is at least as good as the corresponding element of the latterand an enumeration of them would be tractable only in some special cases, e.g., when the underlying graph is acyclic, as assumed by Schäfer et al (2020). Therefore, for the present experiment, we decided to employ a tie-breaking rule, which was chosen from possible tie-breaking rules (e.g., select one arbitrarily, select one that was found first, and select one with the shortest length) that remain valid in the case of ordinal cost values.…”

Multi-objective Matroid Optimization with Ordinal Weights

“…In order to define meaningful optimality concepts for problem (OWOP), we need to compare ordinal vectors in C r . The following definition is based on the concept first introduced in Schäfer et al (2020) and Schäfer et al (2021). Let y 1 , y 2 ∈ C r be two ordinal vectors.…”

“…We use the same notation in this case, and we say that y 1 ordinally dominates y 2 whenever y 1 y 2 . Note that what we consider here is a special case of the concept of ordinal dominance introduced in Schäfer et al (2020) who considered the more general case when feasible solutions -and hence their outcome vectors -may differ w.r.t. their number of elements.…”

“…Ordinal weights and ordinal objective functions have been first introduced in Schäfer et al (2020) for shortest path problems. Motivated by applications in civil security, edges are categorized, for example, as "secure", "neutral", or "insecure".…”

“…Motivated by applications in civil security, edges are categorized, for example, as "secure", "neutral", or "insecure". Schäfer et al (2020) introduce an ordinal preorder based on ordinal weights, analyze the complexity of the problem, and suggest a polynomial time labeling algorithm for its solution. Knapsack problems with ordinal weights are analyzed in Schäfer et al (2021).…”