# On the Enumeration and Counting of Bicriteria Temporal Paths

## TL;DR

This paper explores the complexity of enumerating and counting efficient bicriteria paths in weighted temporal graphs, providing polynomial algorithms for enumeration and Pareto set computation, but proving counting is #P-complete.

## Contribution

It introduces polynomial-time algorithms for enumerating and identifying Pareto-optimal paths in weighted temporal graphs with positive or nonnegative costs, and proves counting is #P-complete.

## Key findings

- Polynomial delay enumeration of efficient paths with positive costs
- Polynomial-time computation of Pareto-optimal solutions with nonnegative costs
- Counting efficient paths is #P-complete

## Abstract

We discuss the complexity of path enumeration and counting in weighted temporal graphs. In a weighted temporal graph, each edge has an availability time, a traversal time and some real cost. We introduce two bicriteria temporal min-cost path problems in which we are interested in the set of all efficient paths with low cost and short duration or early arrival time, respectively. However, the number of efficient paths can be exponential in the size of the input. For the case of strictly positive edge costs we are able to provide algorithms that enumerate the set of efficient paths with polynomial time delay and polynomial space. If we are only interested in the set of Pareto-optimal solutions and not in the paths themselves, then these can be determined in polynomial time if all edge costs are nonnegative. In addition, for each Pareto-optimal solution, we are able to find an efficient path in polynomial time. On the negative side, we prove that counting the number of efficient paths is #P-complete, even in the non-weighted single criterion case.