Shortest Paths in Nearly Conservative Digraphs

TL;DR

This paper studies shortest path problems in nearly conservative directed graphs, showing NP-hardness and coNP-completeness results, and presents fixed parameter tractability algorithms for the all pairs shortest path problem.

Contribution

It introduces the concept of nearly conservative digraphs and demonstrates fixed parameter tractability for shortest path computations in these graphs.

Findings

Deciding nearly conservativeness is coNP-complete.

Shortest path computation is NP-hard in nearly conservative digraphs.

All pairs shortest path problem is fixed parameter tractable under certain parameters.

Abstract

We introduce the following notion: a digraph $D=(V,A)$ with arc weights $c: A\rightarrow \R$ is called nearly conservative if every negative cycle consists of two arcs. Computing shortest paths in nearly conservative digraphs is NP-hard, and even deciding whether a digraph is nearly conservative is coNP-complete. We show that the "All Pairs Shortest Path" problem is fixed parameter tractable with various parameters for nearly conservative digraphs. The results also apply for the special case of conservative mixed graphs.